CRYPTOGRAPHY MATH 0209A
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GAME THEORY Thomas S Ferguson University of California at Los Angeles Contents Introduction References Part I Impartial Combinatorial Games 11 Take Away Games 12 The Game of Nim 13 Graph Games 14 Sums of Combinatorial Games 15 Coin Turning Games 16 Green Hackenbush References Part II TWO Person Zero Sum Games 21 The Strategic Form of a Game 22 Matrix Games Domination 23 The Principle of Indifference 24 Solving Finite Games 25 The Extensive Form of a Game 26 Recursive and Stochastic Games 27 Continuous Poker Models Part III Two Person General Sum Games 31 Bimatrix Games 7 Safety Levels 32 Noncooperative Games 7 Equilibria 33 Models of Duopoly 34 Cooperative Games Part IV Games in Coalitional Form 41 Many Person TU Games 42 lrnputations and the Core 43 The Shapley Value 44 The Nucleolus Appendixes Al Utility Theory A2 Contraction Maps and Fixed Points A3 Existence of Equilibria in Finite Games INTRODUCTION Game theory is a fascinating subject We all know many entertaining games such as chess poker tic tac toe bridge baseball computer games 7 the list is quite varied and almost endless In addition there is a vast area of economic games discussed in Myerson 1991 and Kreps 1990 and the related political games Ordeshook 1986 Shubik 1982 and Taylor 1995 The competition between firms the conflict between management and labor the fight to get bills through congress the power of the judiciary war and peace negotiations between countries and so on all provide examples of games in action There are also psychological games played on a personal level where the weapons are words and the payoffs are good or bad feelings Berne 1964 There are biological games the competition between species where natural selection can be modeled as a game played between genes Smith 1982 There is a connection between game theory and the mathematical areas of logic and computer science One may view theoretical statistics as a two person game in which nature takes the role of one of the players as in Blackwell and Girshick 1954 and Ferguson 1968 Games are characterized by a number of players or decision makers who interact possibly threaten each other and form coalitions take actions under uncertain conditions and finally receive some benefit or reward or possibly some punishment or monetary loss In this text we present various mathematical models of games and study the phenomena that arise In some cases we will be able to suggest what courses of action should be taken by the players In others we hope simply to be able to understand what is happening in order to make better predictions about the future As we outline the contents of this text we introduce some of the key words and terminology used in game theory First there is the number of players which will be denoted by 11 Let us label the players with the integers 1 to n and denote the set of players by N 12 n We study mostly two person games 11 2 where the concepts are clearer and the conclusions are more definite When specialized to one player the theory is simply called decision theory Games of solitaire and puzzles are examples of one person games as are various sequential optimization problems found in operations research and optimization see Papadimitriou and Steiglitz 1982 for example or linear programming see Chvatal 1983 or gambling see Dubins and Savage1965 There are even things called zero person games such as the game of life77 of Conway see Berlekamp et al 1982 Chap 25 once an automaton gets set in motion it keeps going without any person making decisions We assume throughout that there are at least two players that is n 2 2 ln macroeconomic models the number of players can be very large ranging into the millions In such models it is often preferable to assume that there are an infinite number of players In fact it has been found useful in many situations to assume there are a continuum of players with each player having an infinitesimal in uence on the outcome as in Aumann and Shapley 1974 In this course we take 11 to be finite There are three main mathematical models or forms used in the study of games the extensive form the strategic form and the coalitional form These differ in the amount of detail on the play of the game built into the model The most detail is given 3 in the extensive form where the structure closely follows the actual rules of the game In the extensive form of a game we are able to speak of a position in the game and of a move of the game as moving from one position to another The set of possible moves from a position may depend on the player whose turn it is to move from that position In the extensive form of a game some of the moves may be random moves such as the dealing of cards or the rolling of dice The rules of the game specify the probabilities of the outcomes of the random moves One may also speak of the information players have when they move Do they know all past moves in the game by the other players Do they know the outcomes of the random moves When the players know all past moves by all the players and the outcomes of all past random moves the game is said to be of perfect information Two person games of perfect information with win or lose outcome and no chance moves are known as combi natorial games There is a beautiful and deep mathematical theory of such games You may find an exposition of it in Conway 1976 and in Berlekamp et al 1982 Such a game is said to be impartial if the two players have the same set of legal moves from each position and it is said to be partizan otherwise Part 1 of this text contains an introduc tion to the theory of impartial combinatorial games For another elementary treatment of impartial games see the book by Guy 1989 We begin Part 11 by describing the strategic form or normal form of a game In the strategic form many of the details of the game such as position and move are lost the main concepts are those of a strategy and a payoff 1n the strategic form each player chooses a strategy from a set of possible strategies We denote the strategy set or action space of player i by A for i 1 2 n Each player considers all the other players and their possible strategies and then chooses a specific strategy from his strategy set All players make such a choice simultaneously the choices are revealed and the game ends with each player receiving some payoff Each player7s choice may in uence the final outcome for all the players We model the payoffs as taking on numerical values In general the payoffs may be quite complex entities such as you receive a ticket to a baseball game tomorrow when there is a good chance of rain and your raincoat is torn The mathematical and philosophical justification behind the assumption that each player can replace such payoffs with numerical values is discussed in the Appendix under the title Utility Theory This theory is treated in detail in the books of Savage 1954 and of Fishburn 1988 We therefore assume that each player receives a numerical payoff that depends on the actions chosen by all the players Suppose player 1 chooses a1 6 Ai player 2 chooses a2 6 A2 etc and player 11 chooses an 6 A7 Then we denote the payoff to player j for j 1 2 n by f 11 a2 an and call it the payoff function for player j The strategic form of a game is defined then by the three objects 1 the set N 1 2 n of players 2 the sequence A1 An of strategy sets of the players and 3 the sequence f1a1 an fna1 an of real valued payoff functions of the players A game in strategic form is said to be zerosum if the sum of the payoffs to the players is zero no matter what actions are chosen by the players That is the game is zero sum if Zfia1a2an 0 2391 for all 11 6 A1 12 6 A2 1 6 A7 1n the first four chapters of Part 11 we restrict attention to the strategic form of two person zero sum games Theoretically such games have clear cut solutions thanks to a fundamental mathematical result known as the mini max theorem Each such game has a value and both players have optimal strategies that guarantee the value 1n the last three chapters of Part 11 we treat two person zero sum games in extensive form and show the connection between the strategic and extensive forms of games 1n particular one of the methods of solving extensive form games is to solve the equivalent strategic form Here we give an introduction to Recursive Games and Stochastic Games an area of intense contemporary development see Filar and Vrieze 1997 Maitra and Sudderth 1996 and Sorin 2002 1n Part 111 the theory is extended to two person nonzerosum games Here the situation is more nebulous 1n general such games do not have values and players do not have optimal optimal strategies The theory breaks naturally into two parts There is the noncooperative theory in which the players if they may communicate may not form binding agreements This is the area of most interest to economists see Gibbons 1992 and Bierman and Fernandez 1993 for example 1n 1994 John Nash John Harsanyi and Reinhard Selten received the Nobel Prize in Economics for work in this area Such a theory is natural in negotiations between nations when there is no overseeing body to enforce agreements and in business dealings where companies are forbidden to enter into agreements by laws concerning constraint of trade The main concept replacing value and optimal strategy is the notion of a strategic equilibrium also called a Nash equilibrium This theory is treated in the first three chapters of Part 111 On the other hand in the cooperative theory the players are allowed to form binding agreements and so there is strong incentive to work together to receive the largest total payoff The problem then is how to split the total payoff between or among the players This theory also splits into two parts 1f the players measure utility of the payoff in the same units and there is a means of exchange of utility such as side payments we say the game has transferable utility otherwise nontransferable utility The last chapter of Part 111 treat these topics When the number of players grows large even the strategic form of a game though less detailed than the extensive form becomes too complex for analysis 1n the coalitional form of a game the notion of a strategy disappears the main features are those of a coalition and the value or worth of the coalition 1n many player games there is a tendency for the players to form coalitions to favor common interests 1t is assumed each coalition can guarantee its members a certain amount called the value of the coalition The coalitional form of a game is a part of cooperative game theory with transferable 5 utility so it is natural to assume that the grand coalition consisting of all the players will form and it is a question of how the payoff received by the grand coalition should be shared among the players We will treat the coalitional form of games in Part IV There we introduce the important concepts of the core of an economy The core is a set of payoffs to the players where each coalition receives at least its value An important example is two sided matching treated in Roth and Sotomayor 1990 We will also look for principles that lead to a unique way to split the payoff from the grand coalition such as the Shapley value and the nucleolus This will allow us to speak of the power of various members of legislatures We will also examine cost allocation problems how should the cost of a project be shared by persons who benefit unequally from it Related Texts There are many texts at the undergraduate level that treat various aspects of game theory Accessible texts that cover certain of the topics treated in this text are the books of Stra in 1993 Morris 1994 and Tijs 2003 The book of Owen 1982 is another undergraduate text at a slightly more advanced mathematical level The economics perspective is presented in the entertaining book of Binmore 1992 The New Palmgrave book on game theory Eatwell et al 1987 contains a collection of historical sketches essays and expositions on a wide variety of topics Older texts by Luce and Raiffa 1957 and Karlin 1959 were of such high quality and success that they have been reprinted in inexpensive Dover Publications editions The elementary and enjoyable book by Williams 1966 treats the two person zero sum part of the theory Also recommended are the lectures on game theory by Robert Aumann 1989 one of the leading scholars of the field And last but actually first there is the book by von Neumann and Morgenstern 1944 that started the whole field of game theory References Robert J Aumann 1989 Lectures on Game Theory Westview Press lnc Boulder Col orado R J Aumann and L S Shapley 1974 Values ofNon atomic Games Princeton University Press E R Berlekamp J H Conway and R K Guy 1982 Winning Ways for your Mathe matical Plays two volumes Academic Press London Eric Berne 1964 Games People Play Grove Press lnc New York H Scott Bierman and Luis Fernandez 1993 Game Theory with Economic Applications 2nd ed 1998 Addison Wesley Publishing Co Ken Binmore 1992 Fun and Games 7 A Text on Game Theory DC Heath Lexington Mass D Blackwell and M A Girshick 1954 Theory of Games and Statistical Decisions John Wiley 66 Sons New York V Chvatal 1983 Linear Programming W H Freeman New York 6 J H Conway 1976 On Numbers and Games Academic Press New York Lester E Dubins amd Leonard J Savage 1965 How to Gamble If You Must Inequal ities for Stochastic Processes McCraw Hill New York 2nd edition 1976 Dover Publications Inc New York J Eatwell M Milgate and P Newman Eds 1987 The New Palmgrave Game Theory W W Norton New York Thomas S Ferguson 1968 Mathematical Statistics 7 A decision Theoretic Approach Academic Press New York J Filar and K Vrieze 1997 Competitive Markov Decision Processes Springer Verlag New York Peter C Fishburn 1988 Nonlinear Preference and Utility Theory John Hopkins Univer sity Press Baltimore Robert Gibbons 1992 Game Theory for Applied Economists Princeton University Press Richard K Guy 1989 Fair Game COMAP Mathematical Exploration Series Samuel Karlin 1959 Mathematical Methods and Theory in Games Programming and Economics in two vols Reprinted 1992 Dover Publications lnc New York David M Kreps 1990 Game Theory and Economic Modeling Oxford University Press R D Luce and H Raiffa 1957 Games and Decisions 7 Introduction and Critical Survey reprinted 1989 Dover Publications lnc New York A P Maitra ans W D Sudderth 1996 Discrete Gambling and Stochastic Games Ap plications of Mathematics 32 Springer Peter Morris 1994 Introduction to Game Theory Springer Verlag New York Roger B Myerson 1991 Game Theory 7 Analysis of Con ict Harvard University Press Peter C Ordeshook 1986 Game Theory and Political Theory Cambridge University Press Guillermo Owen 1982 Game Theory 2nd Edition Academic Press Christos H Papadimitriou and Kenneth Steiglitz 1982 Combinatorial Optimization re printed 1998 Dover Publications lnc New York Alvin E Roth and Marilda A Oliveira Sotomayor 1990 Two Sided Matching 7 A Study in Game Theoretic Modeling and Analysis Cambridge University Press L J Savage 1954 The Foundations ofStatistics John Wiley amp Sons New York Martin Shubik 1982 Game Theory in the Social Sciences The MIT Press John Maynard Smith 1982 Evolution and the Theory of Games Cambridge University Press Analysis on Metric Spaces Summer School Lake Arrowhead September 09th September 14th 2007 Organizers John Garnett7 University of California7 Los Angeles Peter Petersen7 University of California7 Los Angeles Raanan Sohul7 University of California7 Los Angeles Christoph Thiele7 University of California7 Los Angeles supported by NSF grant DMS 0701302 Contents 1 D a Metric and geometric quasiconformality in Ahlfors regular Loewner spaces Jonas Azzam7 UCLA 6 Preliminaries 6 12 Main theorem and discussion 8 13 Details and techniques of proof 9 131 Outline 9 132 Quasiround sets and generalized packing measure 10 133 The generalized modulus and its role 11 Quasisymmetry and 2spheres 15 Michael Bateman7 Indiana University 15 21 Introduction and de nitions 15 22 Approximating metric spaces by graphs 16 23 Modulus in graphs 17 24 Proof of the main result 20 Quasisymmetric rigidity of Sierpinski carpets 21 S Zubin Gautam7 UCLA 21 31 Introduction 21 311 Carpets 21 312 Quasisymmetric maps 22 313 Main results 22 32 Carpet modulus 23 33 Preliminary results 24 331 Group like carpets 24 332 Assumed results and applications 24 34 A preferred pair of peripheral circles 26 35 Proof of Theorem 2 27 36 Weak tangents of Sp 27 37 Proof of Theorem 1 29 38 Proof of Theorem 3 30 Rigidity of Schottky sets 33 Hrant Hakobyan7 University of Toronto 33 41 Introduction and main results 51 00 42 Connectivity properties of Schottky sets 34 43 Schottky groups 34 44 Quasiconformal maps 36 45 Extension of quasisymmetric maps between Schottky sets 36 46 Proof of theorem 1 38 47 Proof of theorem 2 39 48 Proof of theorem 3 40 The Poincare Inequality is an Open Ended Condition 41 Colin Hinde7 UCLA 41 51 Introduction 41 511 De nitions 41 512 Remarks on Context and Consequences 42 52 Outline of Proof of Theorem 1 43 521 Technical Estimates 45 Distortion of Hausdorff measures and removability for quasiregular mappings 48 Vjekoslav Kovac7 UCLA 48 61 Notation and basic de nitions 48 62 Quasiconformal distortion of Hausdorff measures 49 63 Quasiconformal distortion of 1 recti able sets 52 64 A removability theorem for quasiregular mappings 52 The quasiconformal Jacobian problem 54 Peter Luthy7 Cornell 54 71 Introduction 54 72 Main Result 57 Separated nets in Euclidean space and J acobians of bi Lipschitz maps John Maki7 University of Illinois at Urbana Champaign 61 81 Introduction 61 82 Main results 62 821 The equivalence 63 822 The answer is no 65 83 Related results 67 831 The answer for Holder is yes 67 832 Prescribing the divergence of a vector eld 67 9 Newtonian spaces An extension of Sobolev spaces to metric measure spaces 69 Kenneth Maples7 UCLA 69 91 Introduction 69 92 Preliminaries 70 93 N147 and friends 71 94 N147 is Banach 72 95 Similarities between N147 and M147 73 96 Analogues of Sobolev space results 74 10 Two counterexamples in the plane 76 William Meyerson7 UCLA 76 101 Introduction 76 102 Proof of our rst theorem 77 103 Proof of our second theorem 80 11 Removability theorems for Sobolev functions and quasicon formal maps 85 Nicolae Tecu7 Yale 85 111 Introduction 85 112 De nitions and introductory remarks 85 113 Main results 87 114 Outline of proofs 88 12 Measurable Differentiable Structures and the Poincare In equality 94 Armen Vagharshakyan7 GaTech 94 121 Preliminary Notations 94 122 Main Theorem 94 123 Auxilary Results 97 13 Sobolev met Poincare 101 Marshall Williams7 University of Michigan 101 131 Introduction 101 132 Some preliminaries 103 133 Summary of results 105 134 Outline of proofs 107 1 Metric and geometric quasiconformality in Ahlfors regular Loewner spaces after Jeremy T Tyson A summary written by Jonas Azzam Abstract Here we describe Tyson7s theorem on the equivalence of geometric quasi conformality7 local quasisymmetry7 and a variant of metric quasiconformality on domains of metric spaces and give a summary of the main points and tech niques of its proof 11 Preliminaries Let Xd and Yd be metric spaces We denote dy and d zy by lx 7 yl when there is no confusion about which space and metric we are concerned with A map f X 7 Y is said to be quasisymmetrie or 77 quasisymmetrie if7 for all yz E X and for any i gt 07 whenever he 711 S We 7 2t We have We 7 MIN S 77tlfy 7 f2l 1 for some homeomorphism i7 000 7 000 We say that f is locally quasisymmetrie if there exists a homeomorphism 77 such that every point in X has a neighborhood in which 1 holds for yz in this neighborhood We write AgB if A S CB7 and A N B if AC S B 3 CA for some constant C 2 1 We refer to the constant C as the implied constant in the inequality A Borel measure M is Ahlfors Dayid Q regulaiquot if for any open ball Bz r centered at z of radius r in X7 mm m re lt2 We say that X is Q regulaiquot if there exists such a measure on X A measure M is locally Q regulaiquot if the above condition holds for all z for suf ciently small 7 gt 0 Let X be a metric space with a locally nite Borel measure a For a collection of curves P and a number p7 we de ne the p modulus of to be ModpPinf pde P X where the in mum is taken over all Borel functions p X a 000 such that fvp 2 1 by a curve7 we will mean a continuous nonconstant function y I a X for some interval possibly noncompact I Q R Such a p satisfying these conditions we call admissible for P The modulus satis es the following properties A ModpQ 0 1 ii i P Q P implies ModpP S ModpP Modp is countably subadditive lt ModpP S ModpP whenever every curve in P has a curve in P as a subcurve v If 1 lt p lt 00 and BL is an increasing sequence of curve families whose union is R then limiH00 Modan ModpP With X as above7 X is said to be a p Loeumeiquot space if there exists a nonincreasing positive function p 000 a 000 such that ModpE F 2 ltpt gt 0 whenever E and F are disjoint nondegenerate ie they are not singleton points continua in X such that distE F S tmindiamE diamF where EF denotes the collection of curves connecting E to F We call p a Loeumeiquot function of X A function f X a Y between two metric measure spaces each having a locally nite Bore measure is said to be geometrically K quasicoaformal if7 for every collection of curves P7 Modpl Modp KModpP De ne for r gt 0 LAW sup fw fyl i l96 yl S r and MW inflf96 M i l96 9l 2 r We say that f is metrically H quasiconformal if there exists H 2 1 such that L lim sup M lt H THO Zf7r 7 for alleX 12 Main theorem and discussion The problem considered in this paper is the equivalence of quasiconformality and quasisymmetry in metric spaces In particular the author has proved earlier see 3 theorem 14 that geometric quasiconformality and quasisym metry were equivalent for maps f between Q regular metric measure spaces Furthermore we know that metric quasiconformality is equivalent to local quasisymmetry for homeomorphisms between Q regular Loewner spaces see 2 and therefore that quasisymmetry and metric and geometric quasicon formality are equivalent for homeomorphisms between Q regular Loewner spaces The author raises the question of whether one can generalize these results to maps between subdomains of these spaces ie f U a U where U Q X and U Q Y where U and U need not be Loewner or Q regular The result is true for R and we ask how far we can extend it to arbitrary metric spaces In this context the author proves the following result Theorem 1 Let X d n and Y d V be locally compact Q regular spaces with Q gt 1 where X is Loewner and Y is locally linearly connected Let U Q X and U Q Y be open sets and f U a U a homeomorphism Then the following are equivalent i there epists t gt 1 and H lt 00 such that Lfx tr 1 f ltH ow for all z E X it for each t gt 1 there em39sts Ht lt 00 such that L t hmsup M ltHt THO fT for all z E X ill there em39sts a K lt 00 such that f is geometrically K quaslcorzformal iv there em39sts 77 000 a 0 00 such that f is locally n quaslsymmetrlc A metric space Y is C locally llrzearly connected if there exits C 2 1 such that for every point z and every r gt 0 any two points in Bzr can be connected by a curve lying in Bz Cr and every two points in Bz rc may be connected by a curve in BzrC Note that this is a weaker condition than Y being Loewner that is Loewner spaces are linearly locally connected see 1 theorem 823 Note also that the theorem does not achieve equiv alence with metric quasiconformality but a weaker variant described by the conditions and ii 13 Details and techniques of proof 131 Outline We brie y outline here the steps for proving the main result to put the later details in context 1 We rst develop a generalized form of modulus Modp which is con structed depending on data 039 that we specify We will sketch the construction of this in the section below 3 For locally Q regular metric spaces this new modulus is comparable to the traditional modulus C40 For this new modulus if condition is satis ed by a homeomorphism f then Modz39mF S Modz39mfl where the data 039 and U depend on the constants t and H Pulling this together we get that ModpPg mod I39ml S Modg lfP Modpr 3 Hence this shows implies 77hali 7 of iii that is one half of the inequality for quasiconformality 4 We then show that if this half of iii is satis ed then this implies quasisymmetry In summary we get implies half of iii which in turn implies iv It is not hard to show iv implies ii the only assumption needed here on X and Y is that Y is locally C linearly locally connected ii implies trivially and since the inverse of a quasisymmetric map f is also quasisymmetric the above discussion implies the reverse inequality in 3 so iv implies iii so all statements are equivalent Since the bulk of the proof lies in the rst three steps we will discuss those below and mention the result needed for step 4 in the end 132 Quasiround sets and generalized packing measure Here we introduce quasiround sets and describe how to construct the general ized packing measure needed to de ne the generalized modulus The author presents the construction in a more general setting working with quasiround balls instead of restricting all attention to just balls A normal ball under the maps we will consider may be morphed to some degree so by weakening our de nition of roundness these balls will fall within the same class of ob jects under these maps quasiround sets will be mapped to quasiround sets We can then develop the tools we have de ned for usual balls doubling set functions covering lemmas et cetera for this larger class Let A Q X be a closed set If k gt 1 we say A is k quasz39mund if there exists z E X such that Bxr1 Q A Q Bxr2 where 0 lt r1 lt r2 lt kn We call 71 and r2 the inner and outer radius respectively We say that a set A is Z quasimund if A Bzr for some z E X and r gt 0 We denote BAX to be the set of all quasiround sets 81X to be the set of all balls and 8X to be all quasiround sets Two sets A and A are an Z quasz39m39ng if there exist 71 lt r2 lt in lt 00 and z E X such that xr1 Q A Q A Q zr2 These are generalizations of balls and concentric balls allowing the boundary of the ball to vary by a factor Let E Q X and V a collection of quasiround subsets ofX We say that V is a ae or Vitali cover of E if every point in E is contained in a quasiround set in V of arbitrarily small outer radius We say a subcollection C Q V is V full if there is a function A E a 000 such that every element of V centered at z with outer radius at most Az is in C Two ne collections V and V are an Z admissible pair if there exists a map A gt gt A from V to V such that AA is an l quasiring This is the generalization of doubling of balls7 so for exarnple7 81X is admissible with itself via the doubling rnap Bzr gt gt Bz 27 For V Q BAX7 k 2 17 a ne cover of X7 C Q V a subcollection7 a set function 7 V a 0 oo7 and 6 E 0 oo7 de ne the variation V M sup 2 MAJ where the suprernurn is taken over all nite or countable collections Q C of pairwise disjoint elernents Then7 for a set E Q X7 de ne wild igf We where the in rnurn is over all V ne covers C of E This turns out to be a Borel regular measure on X Furthermore7 given the same 7 and V as above7 we can develop a gener alized notion of length For a curve 7 I a X7 de ne KT 7 lengthk supr 13131 1nf Z ibAl where the suprernurn is taken over subcurves of y by restricting to compact subintervals of I7 and the in rnurn is taken over nite or countable covers Al E V of the image of 70 with outer radii at most 6 For an l adrnissible pair V V 7 we say a set function 7 VUV a 0 00 is locally M blaaketed for some M gt 0 if the collection of sets A E V for which 104 S M WA is a V full cover of X 133 The generalized modulus and its role Now we may de ne the generalized modulus and begin the steps of the proof of the main theorem Let X be a metric space Let Z gt k 2 17 M lt 007 and 11 V V Q BAX be ne covers of X which form an Z admissible pair Let P be a family of curves We de ne the packing type generalized p modulus of P with respect to the data a kl M to be Modz39ml inf llz39kvX where the in mum is taken over all generalized packing measures lMy gen erated by set functions if where w V U V a 000 is a locally blanketed M function such that KI 7 length y 2 1 for all y E P Such a w satisfying these conditions is said to be admissible for F We may prove that this and the original modulus are comparable via the following lemma First7 for a metric space X7 we de ne the constant of uniform perfectness to be the constant c gt 0 such that diam z r 2 2cr for every x E X and r gt 07 and X is uniformly perfect if such a constant exists Lemma 2 Let X n be a locally Q regular metric measure space with Q gt 1 Lettgtlgt1andMltoo Then MonPgMod 1lMP for any family of curves P and ne collections V Q 81X with V tV If in addition X is locally compact then for ped h 2 1 Mod klMP MonP for all Z gt h and for any M 2 lc where the implied constants depends only on k Q and the implied constants in equation 2 and c is the constant of uniform perfectness for X Note that Q regular spaces are uniformly perfect Both inequalities are proven with a similar motif in mind In the rst inequality7 we let 2 V U V be any admissible function for P with respect to the generalized modulus and use it to construct a particular admissible Borel function p for the classical modulus that will help us achieve the bounds we need to prove the inequality Proving the second inequality follows similarly7 by picking any p admissible for the classical modulus and constructing a w admissible for the generalized modulus Next7 there is the task of showing step 3 in the proof Lemma 3 Let f X a Y be a homeomorphism between locally compact metric spaces satisfying condition in the main theorem Let Z gt t l k gt H and M lt 00 Let V be the ne collection of balls B such that the image of the l ring BtB under f is an l quasiring Then we have that for data 0 and 0 depending on t and H MOdlt1lM PSMOdkMfr where ModklM is computed relative to the l admissible pair fV ftV Condition of the theorem ensures that V is a ne cover of X We outline the main points of the proof let 7 fV U ftV be an admissible locally M blanketed function for the generalized modulus of fF and de ne p V a 0 00 as A WM for A E V U tV Then p will be locally M blanketed Next we show that 1 ltlgt length1w 2 ll lengthlfw for all y E P 2 ltlgt391vE S llglfVE for every Borel set E Q X Then p is admissible for the generalized modulus of P by 1 and 2 nishes the proof by letting E X and taking the in mum of both sides The proof is complete once we mention this last lemma which completes step 4 of the proof Lemma 4 Let X n be a Q regular Loewner space and let Y V be a Q regular C locally linearly connected space with Q gt 1 Let f U a U be a homeomorphism between open subsets of X and Y respectively which satis es MonPgMonfP for all curue families P in U Then f is locally quasisymmetric References 1 J Heinonen Lectures on analysis on metric spaces Springer New York NY 2001 2 J Heinonen and P Koskela Quasiconformal maps in metric spaces with controlled geometry Acta Math 181 1998 1 61 13 3 J T Tyson7 Metric and geometric quasiconfumality m Ahlfors regular Loewner spaces7 Conform Geom Dyn 5 20017 21 73 JONAS AZZAM UCLA email jonasazzam mathuclaedu 2 Quasisymmetry and 2spheres after Bonk and Kleiner A summary written by Michael Bateman Abstract Let Z be a 2 regular metric space homeomorphic to 52 Then Z is quasimobius to 52 iff Z is linearly locally connected 21 Introduction and de nitions We sketch a proof of one of the main theorems of Quasisymmetric parametriza tions of two dimensional metric spheres by Bonk and Kleiner7 stated here in the abstract The theorem proved there uses 7701uasisymmetric77 instead of 77quasimobius777 and linearly locally contractible77 instead of 77linearly locally connected These notions are equivalent in our setting7 and we focus on the version stated here A metric space Z is A linearly locally connected A LLC if my can be connected inside AB whenever Ly are in the same ball B7 and if my can be connected in ZB whenever my 6 ZB lf 1z2z3x4 E X are distinct7 de ne the modi ed cross ratio as mindz1 x4d27 31 lt9517952a9537954gt 1 A homeomorphism f Z1 a Z2 is n quasimobius if 77 Rt a RJV is a homeomorphism7 77t a 0 as t a 07 and ltf9517f9527f9537f954gt S 77lt95179527953a954gt 2 A continuum is a connected compact set If E F are continua7 de ne their relative distance to be distEF AE F W39 3 Proposition 1 Let Z be LLC39 Then for all 6 gt 0 we have A If lt1z2x3x4gt is small enough then there are continua E and F such that zhzg E E and zgz4 E F and AEF 2 B If there are continua EF such that zhxg E E and zgz4 E F and AEF is small enough then lt1z2x3z4gt lt 6 15 Proof We include the proof of B7 which is simple mindiamEdiamF 7 1 lt1234gt S i 4 D We end this section by remarking that if Dk Q X with XY compact7 and fk Dk a Y is a sequence of i7 quasimobius functions with the same control function where the Dk7s become arbitrarily dense in the limit7 then we may extract a convergent subsequence by using a standard diagonalization argument7 as long as we assume that there are and C 2gt 0 such that d Y and deer we gt lt5 This ensures that the fk are equicontinuous dxl gt 17 k diam X C 22 Approximating metric spaces by graphs Let G be a graph with vertex set V We write u N i to mean that there is an edge connecting u and i Let A Gprl1 represent a graph G7 a function r V a 0007 a function p V a Z7 and an open cover l1 thev of Z K E N de ne the K star of V to be StKi UkmwKK Uw7 where h denotes combinatorial distance in the graph Let Nsi w E V kwi S 5 De ne meshA suvaVrQI lf K E N we say A is a K approximation of Z if 1 The degree of every vertex i E V is bounded by K 2 Bpiri Q Uv Q BpiKri7 and NM Q StKi for all i 3 lfi w7 then Uv uw 31 07 and ri S KT LI lf Uv aw 31 07 then kiw lt K 4 If Ly E Uw7 then there is a curve 7 connecting z and y in StKi Conditions 1 and 3 give an upper bound on the number of neighborhoods or K stars that can cover the same point7 and Condition 5 is similar to the LLC condition The map p tells us the location of i in Z Proposition 2 IfZ is LLC doubling and homeomorphie to 82 then there is a K approrirnation A 0fZ with arbitrarily small mesh size such that the graph G ofA is the Z skeleton of a triangulation 0f 82 This is proved by taking a net in the space7 connecting points that are close together with the LLC condition7 and verifying the conditions for ap proximations Adding a few extra points and edges makes it into a triangu lation7 which is what we want Now given any graph that is a triangulation of 827 we can realize it as a circle packing with circles having centers at the vertices7 by a result of Andreev Koebe Thurston So we can construct an approximation A on S2 from our K approximation on Z by letting p o be the center of the circle associated to i 7 r o its radius7 and U the interiors of all triangles with a vertex at 1 Having the same graph approximate both Z and S2 gives us an obvious candidate for a quasimobius map between Z and SZ for z E pV Q Z7 let fp p p 1x We will construct such functions on a sequence of graphs with mesh size tending to zero and ap ply the convergence result stated above lt remains to nd conditions under which these maps are quasimobius The de nition of quasimobius is symmetric in the sense that if f is quasi mobius7 then f 1 is also quasimobius This is a consequence of the symmetry properties ofthe cross ratio and the fact that the control function 77 is a home omorphism with 77t a 0 as t a 0 The following proposition says that7 under favorable circumstances7 we can relax the condition on the control function 77 Proposition 3 Suppose A and A are K approcoimations ofX and Y with the same underlying graph Suppose f pV a p V is given by fp p p 1z Suppose X and Y are LLC39 doubling and homeomorphie to SZ and suppose there is a function 6 RJV a RJV such that for all 6 gt 0 we have zlz2x3z4l lt 6 whenever fz1fx2fz3fx4 lt 66 Then f is quasimobius 23 Modulus in graphs In analogy to the de nition of modulus in a metric measure space7 we de ne modulus in a graph G modgzJ A B inf Eva woQ7 where the inf is taken over functions w V a 0 00 such that Zuecw z 2 1 for any chain 0 of vertices connecting the set A and the set E Proposition 4 IfA Q NKA B Q NKA thenmodgA B g modgAB Proof Given an admissible function w for A B7 de ne 4121 ZueBmK Then the assumption on the degree of G yields an upper bound on the number 17 of vertices in BuK for any 1 so End7141062 g End714109 It remains to show that w is admissible if ul un is a chain connecting A to B then U1BuK contains a chain from A to B so 2 2 1 D For a set E Q Z let VE 1 E V U E 31 0 The next few results relate the modulus of sets VE VF in a graph to properties of the sets E and F in the metric space Z Proposition 5 Let Q 2 1 A a K approcclmatlorz of a Q regular metrlc measure space Z and let EF be such that dl5tVEVF is hot too small Thea monE F g modgVE VF The conditions on the separation of vertices here and in what follows should not cause alarrn 7 we will consider graph approxirnations with very small mesh size so most pairs of vertices will have large cornbinatorial dis tance in the graph The assumptions are needed for technical reasons To prove this proposition we take an admissible function w for VE VF and con struct a function p adrnissible for E F with a similar rnass bound Given a curve connecting E to F we nd the vertices that are close to the curve then add up the values of the function w for those vertices after scaling thern appropriately to re ect the density of vertices near the curve If we additionally assume that Z is Q Loewner in the proposition above ie if we assume Z is recti ably connected and that there is a decreasing function b R a R such that AE S monE F then we have ltAltE F modgwE VF 6 if dl5tVEVF is not too small Note that 52 is 2 Loewner The following proposition establishes the reverse inequality in certain situations Proposition 6 Let Q 2 1 and let A be a K approcclmatlorz of a Q regular metrlc measure space Z Then there em39sts 7 Rf a 0 00 with 7t a 0 as t a 0 such that modgwE VF 7AE F 7 We do not give any proof here since it is a bit technical but it is very similar in spirit to the proof of the analogous fact for the traditional rnod ulus in a Q regular space We de ne another notion of cross ratio this time for vertices in a graph lf Q 2 1 de ne the Ferrarzd cross ratio as 18 v1v2v3v4lQ inf modgAB where the inf is over all chains A 9 111113 and B 9 v2v4 The motivation for this de nition is that in a Q regular Q Loewner space 1 zgx3z4l is small i there are continua E 9 1 3 and F 9 zgx4 with small modulus by Proposition 1 Proposition 7 IfQ 2 1 and Z is LLC39 with a K approccimation A satisfy ing 7 then there is a61 RJV a RJV such that ife gt 0 then v1v2v3v4lQ lt 6 WWWUCquot MUNj 2 2K L and W01P 27P 37P 4l lt 516 Proof If pv1pv2pv3pv4 is small there are continua E 9 pv1pv3 and E 9 pv2pv4 with AE F large by Proposition 1 Cover E by neigh borhoods UM By 3 of the de nition of K approximation there is a chain A Q NKUw Q NKVE connecting v1 and 113 Similarly we nd a chain B connecting v2 and 114 Now apply assumption 7 with Proposition 4 to get modgAB g modgVEVp g 7AE But v1v2v3v4lQ is the inf over all such chains A and B and AEF being large makes 7AE small by Proposition 1 so we are done 1 Proposition 8 If Q 2 1 Z is LLC39 with a K approccimation A satis fying 6 then there is a function 62 R a RJV such that if E gt 0 then pv1pv2pv3pv4 lt 6 whenever kvvj 2 is not too small and U1121314Q lt 626 We skip the proof since it is similar to that of the previous proposition Now we combine these propositions to obtain conditions guaranteeing the existence of a quasimobius map between graph approximations Proposition 9 Let Q 2 1 Let X be connected with K approccimation A Gp 73L satisfying Let Y be LLC39 and doubling with K approccimation A Gp r l satisfying Note that the underlying graphs are the same Let W Q V be a set of vertices that have large combinatorial separa tion ie large enough to satisfy the separation conditions in 6 and De ne f pW a p W in the obvious way ie fp p p 1x Then f is quasimobius with control function depending quantitatively on the data The reason we want the control function to depend only on the data is so that we can take a sequence of graphs with smaller and smaller mesh approximations nd a quasimobius map on each graph and then apply the convergence result stated above Proof Our assumptions here allow us to apply Proposition 3 All we need to show is that 1z2x3x4 is small whenever fx1 zz fz3 f4 is small But this follows by applying the previous two propositions in suc cession D As a corollary7 we see that if Ak and A are sequences of graph approx imations with the same graphs Gk and if mesh4k and mesh4 go to zero7 then we can apply the convergence result as long as the condition 5 is satis ed 24 Proof of the main result Proof In Proposition 97 let X 27 and let Y be the 2 regular LLC space Z By Proposition 27 there is a K approximation of Z that is combinatorially equivalent to the 1 skeleton of a triangulation of 2 with meshA7 S By the remarks following the proposition7 we know that this gives us K approximations A of Y7 with the same underlying graphs Gj Since 52 satis es 67 and since Z satis es 77 we may apply Proposition 9 to nd7 for each j7 a quasimobius map from a p V with the same control function for all j by the remarks following the proposition Finally we remark that the condition 5 needed to apply the convergence result can be met by choosing three separated points 2122 23 E Z7 and using vertices Uf v that approach 21 22 23 7 respectively7 as j a 00 Then in the approximation ofthe sphere7 we may actually choose to make pvpv and pv be well separated on the sphere We can do this by applying a sequence of Mobius transformations to the sphere to send any three points to three equidistant points on a great circle This allows us to apply the convergence lemma7 so we are done D MICHAEL BATEMAN INDIANA UNIVERSITY email mdbatema indianaedu 20 3 Quasisymmetric rigidity of Sierpinski car pets after M Bank and S Mererzkov 5 A summary written by S Zubm Gautam Abstract We summarize the results of 57 where it is shown that the group of quasisymmetric self maps of the standard Sierpinski i carpet S17 is nite in particular7 the group of quasisymmetric self maps of S3 is the dihedral group D4 We also prove that any two distinct standard Sierpinski carpets are not quasisymmetrically equivalent 31 Introduction 311 Carpets For p gt 1 odd7 the standard Sierpiriskz39 i earpet S17 is a self similar subset of R2 obtained recursively by partitioning a square into p2 subsquares of equal size7 removing the open middle square7 repeating this procedure on the p2 7 1 remaining squares7 and continuing inde nitely1 The construction is analogous to that of the usual Cantor sets in R in the following7 we will view the carpets SZ7 as being constructed in the unit square 01 gtlt 01 unless otherwise noted More generally7 a carpet is a topological space homeomorphic to the stan dard Sierpinski carpet S3 or equivalently to any Sp Carpets embedded in the sphere 82 can be characterized as sets S SZ Dl7 where the Di are Jordan domains with pairwise disjoint closures7 such that S has empty interior and limlH00 diamDl 0 in the spherical metric In a general carpet S7 a peripheral circle is a closed Jordan curve 7 such that S y is connected For carpets in the sphere as above7 the peripheral circles are simply the boundaries dDi In the sequel7 we will glibly pass between the extended complex plane and the Riemann sphere 82 via the usual conformal stereographic projection We call a carpet S a carpet in K ifK is a closed Jordan domain7 S C K C 827 and 3K is a peripheral circle of S S is a square carpet in K if its peripheral 1It will probably be useful to the reader to sketch pictures throughout this summary 21 circles possibly excepting 3K bound geometric squares with sides parallel to the real and imaginary axes in C Similarly7 S C 82 is a round carpet if its peripheral circles are bona de geometric circles We also consider carpets in cylinders llDT7 where ll is the strip Pz7y lt l0 y 1 and T is a cyclic group of horizontal translations These carpets simply arise as projections of T invariant carpets in ll The bottom and top peripheral circles are the projections of y 0 and y 17 respectively 312 Quasisymmetric maps A homeomorphism f X a Y of metric spaces XdX and Y dy is 77 quasisymmetm39e if 77 0 00 a 000 is a homeomorphism and dyf96 fy dx9c y dyf96f2 S ndX2 for all distinct my 2 E X The homeomorphism 77 will usually be suppressed in the notation7 and X and Y are said to be quasisymmetm39eally equivalent if there exists a quasisymmetric homeomorphism f X a Y Heuristi cally7 if y and z are symmetric about 7 then f distorts the symmetry of the triple Luz by a bounded amount Quasisymmetric maps serve as a nat ural generalization of quasiconformal maps to the setting of general metric spaces in particular7 the quasisymmetric self maps of R are precisely the quasiconformal ones See 8 for more on the basic theory of quasisymmetric maps Much of the motivation for studying quasisymmetric mapping properties of carpets arises from geometric group theory To wit7 any quasi isometry of hyperbolic groups induces a quasisymmetric map between their bound aries at in nity7 and carpets arise naturally as boundaries of Kleinian groups associated with hyperbolic 3 orbifolds See 1 for a brief discussion of this connection 313 Main results The main results of 5 are the following theorems 22 Theorem 1 The group of quasisymmetric self maps of the standard Sierpinski carpet Sg is the dihedral group D4 consisting of rotations and reflections of the square Theorem 2 The group of quasisymmetric self maps of the standard Sierpinski carpet Sp is nite Theorem 3 The standard Sierpinski carpets Sp and 5 are not quasisym metrically equivalent for p 31 q 32 Carpet modulus The key tool used to prove the above theorems is a discrete analogue of the classical conformal modulus of a curve family in 82 see 8 the conformal modulus for curve families in 82 is the 2 modulus in the notation therein Let S be a carpet in 82 and let p be a mass distribution ie a nonneg ative function de ned on the set of its peripheral circles The p length of a curve 7 C 82 is m i Z Moi Y Ci0 and the total mass of p is M i 29 Given a curve family P in 82 a mass distribution p is P admissible if there exists a subfamily To C P of conformal modulus zero such that lp y 2 1 for all y E P To The modulus of with respect to S is MW A mass distribution p is ecctremal for P if Mp modSP if an extremal p exists it is unique by the usual 2 convexity argument In the sequel modulus77 will refer to this carpet modulus while conformal modulus77 will refer to the classical 2 modulus The conformal modulus is a conformal invariant but it is only quasi invariant under quasiconformal maps By contrast it is easy to check that the carpet modulus just de ned is genuinely invariant under quasiconformal self maps of 82 The following monotonicity property of the modulus is crucial and easily veri ed Write P j P if every curve 7 E P has a subcurve in 1quot Then T j P i mods S modSP mods inf p Feadmissible 23 33 Preliminary results 331 Grouplike carpets A closed Jordan curve C C 82 is a quasicircle if it is the image of a geometric circle under a quasi Mobius map the notion of a quasi Mobius map is similar to that of quasisymmetry for the sake of brevity we refer the reader to Curves in a family are uniform quasicircles if they are images of circles under quasi Mobius maps of uniformly bounded distortion The family is uniformly separated if there exists 6 gt 0 such that gt 6 mindiam0i diam0j T in the chordal metric on 82 A carpet embedded quasisymmetrically in 82 whose peripheral circles are uniformly separated uniform quasicircles is said to be group like The de ning properties are enjoyed by all carpets that arise as boundaries of hyperbolic groups7 whence the name Lemma 4 Any quasisymmetric image of a group like carpet is again a group like carpet Moreover any quasisymmetric map between group like car pets extends to a quasiconformal self map of 82 The rst claim follows from basic properties of quasi Mobius maps see 3 the second is proved via the classical Ahlfors Beurling extension theorem see 4 for details In fact7 the second claim holds even if one drops the uniform separation hypothesis on the peripheral circles By the invariance of the carpet modulus under quasiconformal homeo morphisms7 we see that mods is a quasisymmetric invariant for group like carpets S ie7 mods modf5fP for all quasisymmetric f 332 Assumed results and applications We will assume the following three foundational uniformization and rigidity results Theorem 5 Cylinder Uniformization Theorem Let S be a group like carpet with CO and 01 two distinct peripheral circles of S Then there epists a quasisymmetric map ofS onto a square carpet in some cylinder llDT such that CO and 01 are mapped to the bottom and top peripheral circles respectively 24 Moreover none of the peripheral circles in the square carpet is a point See 2 Theorem 6 Round Uniformization Theorem Euery group like carpet is quasisymmetrically equivalent to a round carpet in 82 See see also I for a sketch of the proof The Round Uniformization Theorem is an analogue of Koebe7s uniformiza tion theorem for circle domains in fact Koebe7s theorem is the basic ingre dient of its proof Theorem 7 Round Rigidity Theorem IfS is a round carpet in 82 ofHaus dorff measure 0 then every quasisymmetric map of S onto another round carpet is the restriction of a Mobius transformation See We now apply these three theorems to establish some preliminary results Lemma 8 Top bottom lemma Let S be a group like carpet of measure zero let CO and 01 be two distinct peripheral circles and let P be the curve family connecting CO and 01 Then the ecctremal mass distribution pp for P satis es ppC 60 fori 01 where Ci is the side length of the square in lPT corresponding to C under the Cylinder Uniformization Theorem Proof sketch After applying the Cylinder Uniformization map consider the curve family P of straight lines 7 t gtlt 01 in lPT connecting the top and bottom peripheral circles Say 0a gtlt 0 1 is a fundamental domain for llDT Let p be any P admissible mass distribution then 10 Z Jogt21 teioagtE Y Ci where E is a measure zero set accounting for the exceptional subfamily in the de nition of admissibility lntegrating over 0 a E yields a Zoo pm a MEMO 0MP by Cauchy Schwarz recall that the peripheral squares C have sides parallel to the coordinate axes SO Mp 2 a Mpp for all P admissible p and hence pp is extremal for P pp is also P admissible so the claim follows from the monotonicity property of the modulus since P j P 25 Lemma 9 Let S be a group like carpet of measure 0 arid CO 01 two pe ripheral circles Theri the group of orientation preserving quasisymmetric self maps ofS that cc CO arid 01 setwise is riite cyclic Proof sektch By Cylinder Uniformization7 it suf ces to assume S is a square carpet in a cylinder suppose f is such a self map f has a quasicon formal extension f by Lemma 4 By Lemma 8 and quasisymmetric invariance of the modulus7 we see that f maps peripheral squares to peripheral squares of the same size By considering vertical line segments connecting peripheral squares C to the top and bottom of the cylinder7 one can also show that fC has the same distance to the top and bottom as does 0 and that f maps vertices to vertices Thus there is a rotation of the cylinder taking 0 to fC using absolute continuity properties of f7 one can in fact glue these rotations together7 showing that f is itself a rotation of the cylinder Since there are only nitely many peripheral squares of a given side length7 we conclude that the group in question is nite cyclic Theorem 10 Three Circle Theorem Arty two orientation preserving qua sisymmetric self maps that act the same ori a triple of distirict peripheral circles or a triple of distirict poirits must be the same map This follows from the Round Uniformization and Round Rigidity Theo rems see 1 for details Lemma 11 Let x aridy be two distirict poirits ori a sirigle peripheral circle of a group like carpet S Theri the group of quasisymmetric self maps of S pirig z aridy is cyclic As above7 this follows from Round Uniformization and Round Rigidity 34 A preferred pair of peripheral circles The outer and middle squares of a standard Sierpinski carpet Sp are de ned in the obvious way Lemma 12 Every quasisymmetric self map of S17 preserves the outer arid middle squares as a pair Proof sketch Let P be the curve family connecting the outer and middle squares7 and let D be that connecting the peripheral squares Cl and 077 26 at least one of which is neither the middle nor the outer square By the self similarity of Sp7 every curve in D passes through a common copy77 of Sp draw a picture hence mod5pPij S mod5pP lf equality were to hold7 then by uniqueness of extremal mass distributions7 the extremal distribution for B would have to concentrate on the aforementioned copy of Sp7 which is a proper subset of Sp this contradicts the Top bottom77 lemma 35 Proof of Theorem 2 We now prove that the group of quasisymmetric self maps of Sp is nite By Lemma 127 every element ofthis group preserves the outer and middle squares as a pair By Lemma 97 the subgroup of orientation preserving elements that x these two squares is nite cyclic The composition of any two orientation reversing maps is orientation preserving7 and similarly the composition of two maps switching the outer and middle squares preserves them7 so the whole group of quasisymmetric self maps is in fact nite 36 Weak tangents of 310 A weak tangent space of a metric space X is a Gromov Hausdorff limit of pointed dilations ofX see 7 and 6 for more useful and in depth discussions Heuristically7 one blows up77 the in nitesimal structure of X near a chosen point We isolate three particular weak tangents of Sp 1 A g weak tangent of Sp is any metric space isometric to Wg U anp n0 equipped with the planar metric The vertem of the g weak tangent is the point corresponding to the origin This is a weak tangent space to Sp at any corner point of the outer square one can think of standing at the origin and blowing up77 the carpet Sp to ll up the entire rst quadrant of the plane 2 A n weak tangent of Sp is any metric space isometric to 1 W Uopqsp i E n 27 with the planar metric As above the vertem is the point corresponding to the origin in W and a 7T weak tangent is a weak tangent space to 5 at the midpoint of any side of any peripheral square 00 A weah tangent of Sp is any metric space isometric to W37 which is obtained by gluing77 three copies of Wg together along the boundary axes thus WaTW may be taken to live in the rst three quadrants of the plane The vertem is again the point corresponding to the origin and a weak tangent is a weak tangent space to 5 at any corner point of any non outer peripheral square These can all be seen to be group like carpets By applying Lemma 11 and completing each weak tangent by 00 in the Riemann sphere we see that the orientation preserving quasisymmetric self maps of any of these weak tangents xing the vertex form a cyclic group All of these groups contain all pth power multiplication maps in W so they are in nite cyclic Now let H be either the closed rst quadrant the closed upper half plane or the closed union of the rst three quadrants Let W be the corresponding weak tangent space of SF in H as described above If 7 is a quasisymmet ric self map of W xing the vertex Lemma 4 guarantees a quasiconformal extension 7 H a H Using Round Uniformization and Round Rigidity combined with Mobius transformations ofthe upper half plane one can show that the quotient space is homeomorphic to a cylinder with boundary with W projecting to a carpet in this cylinder strictly speaking one should omit the points 0 and 00 from The boundary of W is mapped to the top and bottom peripheral circles is strictly a topological carpet as there is no canonical metric on One de nes the modulus of a curve family similarly to the manner above but in the de nition of admissibility for mass distributions one takes an exceptional family To C P whose prez39mage under the quotient map H a has zero conformal modulus It is not hard to check that if 7 bk for some h E Z and P is the curve family connecting the top and bottom of such a cylinder then modWWP Lemma 13 There is no quasisymmetric map from Wg t0 WaTW sending the vertem t0 the vertex Proof sketch Let H1 denote the rst quadrant in the plane and let H2 denote the union of the rst three quadrants Suppose there is such a map 28 f Let 15 be a generator of the cyclic group G of quasisymmetric self maps of Wg xing the vertex7 and let 15 H1 a H1 be a quasiconformal extension of 15 Then the conjugate 15 f15f 1 generates the group G of quasisymmetric self maps of WaTW xing the vertex f admits a quasiconformal extension f H1 a H27 and 15 131513 1 is a quasiconformal extension of 15 to H2 Wg G and WaTWG are carpets in the cylinders and H2lt15 gt as described above let PG and Fax be the curve families connecting the tops and bottoms of the respective cylinders Then it is not too hard to check that modWaTWGPG mOClerzGPG Recall that WaTW consists of three glued together copies of ng let 71 E G agree with 15 on one of these copies and extend by Schwarz re ection to the other copies One checks that 1 1 0 lt mOClWigwqrwq S gmodwgGPG ngdWEEEGPGI This is impossible since 71 is an element of the cyclic group G 7 modW lt gtPlt zgt must be an integer multiple of modWaw mfg as remarked above 2 T 37 Proof of Theorem 1 We now prove that the group of quasisymmetric self maps of Sg is the dihedral group D4 View Sg as obtained by subdividing 01 gtlt 01 in the plane7 and let f be a quasisymmetric self map By Lemma 127 f preserves the outer and middle squares as a pair for now we assume f maps each of these squares to itself Let 00 be a peripheral square of size side length 3 such that the modulus of the curve family connecting CO to the outer square is maximal Then using the self similarity of Sg and monotonicity of the modulus as in the proof of Lemma 127 one can see that 00 must be mapped to another square of size The eight squares of size 3 are called corner or middle squares in the obvious manner If 00 and fCO are both corner squares7 then there is a rotation or re ection of Sg that acts the same as f on the outer square7 the middle square7 and CO The Three Circle Theorem then shows that f 6 D4 Now suppose CO is a corner square and fCO is a middle square Without loss of generality we may assume CO is the bottom left square and fCO is the bottom middle square If M is the re ection in the line y z and D is 29 the re ection in the line z 127 we see that f and MfD act the same on CO and the outer and middle squares Again by the Three Circle Theorem7 the two maps must coincide Considering the xed points of M and D7 we see that f00 120 or 1217 and f1313 1213 or 1223 A quick examination of the induced maps on the weak tangents to Sg at these points yields a contradiction to Lemma 13 If 00 is a middle square7 we can run the same argument backwards by considering f l To rule out the case where f interchanges the outer and middle squares7 we refer the reader to the following proof of Theorem 3 38 Proof of Theorem 3 To conclude7 we prove that Sp and Sq are not quasisymmetrically equivalent for p 31 q We will use the following lemma7 whose proof follows along the lines of those of Lemma 8 and Lemma 9 Lemma 14 Let Sl arid SQ be measure zero square carpets iri the uriit square K 01gtlt01 ff 81 a 2 is ari orientation preserving quasisymmetric map that takes the vertices ofK t0 the vertices arid preserves the origiri theri f is the identity map so iri particular 51 SQ Now suppose p 31 q and f Sp a Sq is a quasisymmetric homeomorphism By Lemma 127 f maps the outer square of Sp to either the outer or the middle square of Sq suppose rst that f maps the outer square to the outer square Let GP Gq denote the groups of quasisymmetric self maps of Sp and Sq preserving the outer square7 and let 0p and Oq denote the orbits of the origin under these respective groups Note that the proof of Theorem 2 shows that GP Gq g Zn gt4 Z2 for some n 4k Here Z2 can be taken to act by a re ection of the unit square7 and there is a subgroup of Zn gt4 Z2 isomorphic to D4 acting by rotations and re ections of the square Since f induces an isomorphism of GP and Gq7 we have that fOp is Gq invariant thus fOp is afortiori D4 invariant lf f0 0 E Oq7 then post composing with an element of Gq allows us to reduce to the case where f00 00 and f maps the vertices of the unit square to the vertices This is because f must conjugate a 90 degree rotation of the square to either itself or its inverse by the dihedral structure of GP Gq From here Lemma 14 yields a contradiction Now suppose f00 Oq7 so fOp Oq Q We claim that 120 Op to see this7 suppose that y 6 GP maps the origin to 120 Then by exploiting the structure of GP and applying the three point77 version of 30 the Three Circle Theorem7 we can show that y M yD as in the proof of Theorem 1 Just as in that proof7 the ensuing discussion of weak tangents yields a contradiction Now combining the fact that 120 017 with the fact that 017 contains the vertices of the unit square and is DAL invariant7 we obtain that Op fOp 8n 4 for some n 2 0 Exploiting the D4 invariance of fOp and noting that fOp cannot contain the vertices of the unit square by assumption7 we see that 12 0 E fOp But then we may as well assume f0 0 1207 and an analogous f MfD77 argument as above again yields a contradiction Similar arguments treat the case where f maps the outer square to the middle square one replaces Oq with the Gq orbit of the point This completes the proof of Theorem 3 Remark The same argument actually rules out the case of quasisymmetric self maps of Sp switching the outer and middle squares7 so as a corollary of the proof we obtain the following re nement of Theorem 2 Corollary 15 For allp the group of quasisymmetric self maps 0f the start dard Sierpiriskz39 carpet Sp is a semidz39reet product Zn gt4 Z2 for some n 4k Actually7 it is claimed in 1 that the group is dihedral7 but no proof is provided It is conjectured that in fact S 2 D4 for all p the p 3 case is of course Theorem 1 It is claimed in 5 that similar methods provide the result for p 5 the remaining cases are still open References 1 Bonk7 M 2006 Quasiconformal geometry of fractals Proc lnternat Congress Math Madrid7 2006 Zurich Europ Math Soc 1349 1373 2 Bonk7 M Uniformizing Sierpinski carpets In preparation 3 Bonk7 M and Kleiner7 B 2002 Quasisymmetric parametrizations of two dimensional metric spheres lnvent Math 1507 no 1 127 183 4 Bonk7 M Kleiner7 B and Merenkov7 S 2007 Rigidity of Schottky sets Preprint 5 Bonk7 M and Merenkov7 S 2006 Quasisymmetric rigidity of Sierpinski carpets Preprint 31 E Burago7 D Burago7 Yu Ivanov7 S 2001 A Course in Metric Geometry7 Graduate Studies in Mathematics7 33 Providence7 RI Amer Math Soc 3 David7 G and Sernrnes7 S 1997 Fractured Fractals and Broken Dreams7 Oxford Lecture Series in Mathematics and its Applications7 7 New York Clarendon Press7 Oxford UP E Heinonen7 J 2001 Lectures on Analysis in Metric Spaces New York Springer Verlag S ZUBIN GAUTAM UCLA email sgautam mathuclaedu V39ais39al39a7 J 1984 5 Quasi Mobius rnaps J Analyse Math 44 218 234 32 4 Rigidity of Schottky sets after M Barth B Kleirzer arid S Mererzhou I A summary writterz by Hrarzt Hahobyarz Abstract It is proven that every quasisymmetric homeomorphism of a Schot tky set of spherical measure zero to another Schottky set is the re striction of a Mobius transformation of S In the other direction it is shown that every Schottky set in 82 of positive measure admits non trivial quasisymmetric maps to other Schottky sets 41 Introduction and main results A subset S of the unit n sphere S is said to be a Schotthy set if its comple ment is a union of at least three disjoint open balls The metric on S is the metric induced from S A Schottky set S C S is said to be rigid if every quasi symmetric map of S onto another Schottky subset of S extends to a Mobius transformation of S Theorem 1 Every Schotthy set irz S n 2 2 of spherical measure zero is rigid Theorem 2 A schotthy set irz 82 is rigid if arid only if it has spherical measure zero Theorem 3 For each n 2 3 there is a Schotthy set irz S that has positive measure arid is rigid One of the motivations for studying Schottky sets in S comes from the fact that every Schottky set can be thought of as the boundary at in nity77 of a convex region bounded by disjoint planes in the Hyperbolic space lllln So the results above can be translated to the corresponding results about quasi isometric rigidity of such domains meaning every quasi isometry between such domains is actually a restriction of a hyperbolic isometry 33 42 Connectivity properties of Schottky sets First7 any Schottky set can be written as SS UBl 1 i39eI Where7 i 31 j7 implies Bl Bj Q and the index set I is either nite or is the set of natural numbers N The collection of the n 71 spheres dBi is the collection of the peripheral spheres of S The proofs of the following results are quite elementary Lemma 4 Let S C S n 2 2 be a Schotthy set andB an open or possibly degenerate closed ball in S Then S B is path connected or empty In particular S is path connected Proposition 5 Let E be a topological n71 sphere contained in a Schotthy set S C S n 2 2 Then S E is connected if and only ifE is a peripheral sphere of S Corollary 6 Every homeomorphism between Schotthy sets S and S in S n 2 2 maps peripheral spheres ofS to those of S 43 Schottky groups Let S be a Schottky set in S and let R S a S be the re ection in the peripheral sphere SE The subgroup of the group of Mobius transformations of S generated by the re ections R corresponding to S is denoted by P5 and will be called the Schotthy group associated to S strictly speaking P5 is a subgroup of the group generated by all re ections in spheres where Mobius transformations form an index 2 subgroup Its easy to see that this doesnt affect the results in the paper Note that for every element U 6 P5 there is a nite sequence i1 ik of indices from I7 called a reduced sequence in the paper7 such that ik 31 ik and URLloORk Proposition 7 The group P5 is discrete P5 has a presentation given by the generators Rhi E I and the only relations are R idgni E I 34 Proof For the discreteness of P5 it is enough to nd a 6 gt 0 st inf ltmaxlUz ixl 26 UEI sVd 163quot This is done by explicitly constructing 6 The second part of the statement follows basically from the same argu ment as discreteness D Next7 the notion of Hausdorff convergence of subsets of a metric space is required For that one rst introduces the following notion of the Hausdor distance between any two subsets A and B of a metric space X dX distHAB inf6 gt 0 A C N5X and B C N5X where N5A denotes the open 6 neighborhood of A in X Then a sequence Ak of subsets of X is said to converge to A C X if distHAk A a 0 This is written as Ak a A The following lemma is easy to see and is not proven in the paper Lemma 8 Ika a B C S where Bk s are closed balls andB is closed then B is a closed ball and aBk a 3B Ifz E intB then there is a 6 gt 0 st Bz 6 C intBk for large k Let us denote l1 UBiU E P5i E I Lemma 9 For every 6 gt 0 only nitely many of the balls inU have diameter 2 6 Proof The idea is to show that if the conclusion did not hold then P5 would be non discrete7 contradicting to Proposition 7 For the rest of the proof the following set would be very important to us See U US USPS Remark 10 As a side remark one may notice that ifS was a complement of nitely many disjoint balls then So0 would be the so called ordinary set of the Kleinian group P5 and the complement of SOD limit set APS in this case would be just a Cantor set 35 Lemma 11 For each point z E S So0 there epists a unique sequence Uk 6 U such that Uk1 C Uk and z E keN j Uk From the previous lemma it follows that diamUk a 0 and hence So0 is dense in S Proof This is done by using successive re ections in the boundary circles 1 44 Quasiconformal maps In this section authors recall the de nitions of the quasiconformal and quasi Mo39bius maps and state the main extension theorems which are used later Theorem 12 Ahlfors Beurling n 2 Tukia V39ais39al39a n 2 3 Letn 2 2 Every H qc map f R 1 a R 1 has an H qc eactension F R a R where H only depends on n and H Proposition 13 Let D 77 S and D 77 S be closed non degenerate balls in S n 2 2 and f 3D a 3D a homeomorphism i ff is 77 quasi Mo39bius then it can be eptended to an 77 quasi Mo39bius map F D a D where 77 only depends on n and 77 ii If each of the balls D and D is contained in a hemisphere and f is 77 quasisymmetric then f can be eptended to an 77 quasisymmetric map F D a D where 77 only depends on n and 77 45 Extension of quasisymmetric maps between Schot tky sets This is one of the important sections of the paper It is proved that every quasisymmetric map f S a 5 between two Schottky sets in S in fact ex tends to a global quasiconformal map of S which is equivariant with respect to the actions of P5 and P s Given an element U 6 P5 with the presentation U R71 0 RLk let U 6 P57 denote the element U Rgl o First they note the following result7 which is easy to prove Lemma 14 There epists a unique bijection foo See a Sf that eactends f equiuariantly that is fools f and foo 0 U U o foo VU 6 P5 36 To obtain the main result of this section the extension theorems of section 4 are used to obtain a non equivariant quasiconformal extension of f to S Proposition 15 Every quasysimmetrie map between Sehotthy sets in S n 2 2 extends to a quasieonformal homeomorphisn of S Next7 one needs to modify the obtained qc map to obtain an equivariant qc map This is done one step at a time as follows Suppose T C S n 2 27 is a Schottky set7 E a peripheral sphere of T7 and R the re ection in 2 Then T T U RT is also a Schottky set7 called the double ofT along 2 Let T be another Schottky set in Squot7 and F S a S be an H quasiconformal map with FT T Then 2 FE is a peripheral sphere of T Let R be the re ection in 2 and T be the double of T along 2 Denote by B the open ball with E 3B and B T Q We de ne a mapFS aS by ES B R oFoRz x63 Lemma 16 The map F is an H qe map with FlT F T and F o R R o F So if there is an H qc map of S mapping a Schottky set T to a Schottky set T 7 then there is a natural modi cation of this map to the correspond ing doubles of T and T which is also H qc Using the previous Lemma inductively one obtains the following result Lemma 17 SH 2 1 Sehotthy sets Sk and S in S and H qe maps Fk S a S for h E No with the following properties i F0 ESQ 556 5 ii E No iii Sk1 D Sk is a double ofSk and Sig D S is the corresponding double of 81 for all h E No W Fklsk foolsk for k 6 N0 U UkENOSk See 37 Proposition 18 The quaslsymmerle map f S a S has an equivarlant quasleonforrnal eatenslon F S a S Proof Since the maps Fk of the previous Lemma are uniformly quasiconfor mal7 they are also uniformly quasi Mobius Since Fklsk foolsk it follows that foo is a quasi Mobius map from So0 to SIX And since the latter sets are dense foo extends to a quasi Mobius map F S a S which is then also equivariant since foo is 1 46 Proof of theorem 1 Lemma 19 Let g R a R n E N be a map that is differentiable at 0 Suppose there is a sequence Dk of non degenerate closed balls in R with dlamDk a 0 st 0 E Dk and D 9Dk is a ball for all h E N Then the derivater Dg0 ofg at 0 is a possibly degenerate or orientation reversing conformal map 239e Dg0 AT where A 2 0 and T R a R is a linear lsometry Proof Assume 90 0 Let D ka7 where rk is the radius of Dk Then subconverges to a closed ball D of radius 1 Since rk a 0 the maps gk R a R de ned as gkx rglgrkz converge to the linear map L Dg0 locally uniformly on R Since LD is a Hausdorff limit of the closed balls rg lDQ rglgDk 9Dk7 it follows that D is also a ball and hence L is conformal D Proof of Theorem 1 By Proposition 18 there is an equivariant quasicon formal extension F of f to S We need to show that F is in fact conformal For that all we need to show is that F is conformal almost everywhere7 which is derived in the next paragraph using the previous Lemma 19 lndeed7 from the analytic de nition of quasiconformal maps it then follows that F is 1 quasiconformal and hence conformal Since 8 0 and So0 is a countable union of copies of S under Mobuis maps it follows that lSool 0 and therefore S So0 has full measure On the other hand every quasiconformal map F is differentiable7 with invertible derivative almost everywhere So ae point z E S is a point in S Soo where F is differentiable Therefore to nish the proof we need to show that at all such points the conditions of Lemma 19 are satis ed But as shown before7 every point in S So0 is contained in an in nite sequence of balls Dk with diamDk a 0 st every ball Dk is an image of a complementary ball of S 38 under some transformation from P5 Since F maps peripheral spheres of S to peripheral spheres of S7 and is equivariant7 it follows that FDk is also a ball for every h E N D 47 Proof of theorem 2 The main result of this section is a consequence of the Measurable Riemann Mapping Theorem which is only available in dimension 27 which says that given any measurable function M such that Hull lt 1 there is a unique homeomorphic solution to the complex differential equation it nwz the Beltrami equation The solution is in fact a quasiconformal map Here uniqueness is understood in the sense that if there are two solutions they differ by a post composition with a conformal map So if one is looking for a solution of the Beltrami equation in C with prescribed values at three points then the solution is unique On the other hand for every qc map f of the plane there is a unique Beltrami coef cient of 6 L00 with Hull lt 1 One says that the Beltrami coef cient M is invariant under a group of Mobius transformations P if 7a M2 V7 6 P and ae z E C Lemma 20 If is a group of Mobius transformations and F C a C is a go map with a P invariant Mp then 1quot FoPoF l is also a group of Mobius transformations Lemma 21 Let S be a Sehotthy set in C and F C a C a go map with a P5 invariant Mp Then S FS is a Sehotthy set Proof of Theorem 2 Suppose lSl gt 0 Let V be any nontrivial Beltrami co ef cient on S7 say V 12 on S and V 0 on the complement of S Next de ne M by spreading around V in a P5 invariant fashion7 ie let M be de ned on S00 as follows 7a Vz for every 7 6 P5 and every 2 E S Declare M 0 on the complement of S00 From the previous two lemmas it follows then that the qc solution F of the Beltrami equation with the Beltrami coef cient Mp M ae maps S onto a another Schottky set S and since Mp is 12 31 0 on S00 which has positive measure if follows that its not a Mobius transformation D 39 48 Proof of theorem 3 The complement of at least three disjoint Open balls in a domain D C S is called a relative Schottky set in D A relative Schottky set T in D is called locally porous at z E T if there is a neighborhood of z and constants C 2 1 and p gt 0 such that Vy E t UVr E 0p there is a subset B in the complement of T in D with Byr B 31 Q and rC S diamB 3 Cr If T is locally porous at every point then its just locally porous Note that a locally porous sets have zero measure even though they may have Hausdorff dimension 2 unlike porous sets Theorem 3 follows from the following two facts the proofs of which we omit Lemma 22 Let n 6 N71 2 3 T and T be relative Schottky sets in regions DD C S respectively and 7 T a T a quasisyrnrnetric map IfT is locally porous then i is the restriction of a Mobius transformation Lemma 23 Every region D C S contains a locally porous relative Schottky set Proof of Theorem 3 Let D be a complement of a Cantor set C C S Of positive measure and let T C D be a locally porous relative Schottky set in D given by the Lemma above Then TUC is a Schottky set in S of positive measure Every quasisymmetric map f OfTUC onto another Schottky set in S restricts to a quasisymmetric map of T onto another relative Schottky set7 which is a restriction of a MObius transformation by the rst assumption to D Since D is dense in S it follows that f is the restriction of a MObius transformation to T U C D References 1 M Bonk7 B Kleiner and S Merenkov Rigidity of Schottky sets HRANT HAKOBYAN U OF TORONTOSUNY AT STONY BROOK email hhakob mathsunysbedu 40 5 The Poincare Inequality is an Open Ended Condition after S Keith arid X Zhorig 4 A summary written by Coliri Hiride Abstract We prove a result of Keith and Zhong 4 stated succinctly in the title 51 Introduction Theorem 24 Let Xd u arid complete metric measure space with u Borel arid doubling that admits a 110 P0iricare inequality withp gt 1 Theri there erists E gt 0 such that Xdu admits a 1q P0iricare iriequality for arty q gt 10 i 6 We begin by reviewing the necessary de nitions to understand the state ment of the theorem7 and provide some additional context and consequences of this result In section 2 we outline a proof of the main theorem Through out this chapter 0 gt 1 is an abused constant its value may vary from use to use7 but will always depend only on the values of the constants associated to the hypotheses of the main theorem 511 De nitions Xdu throughout denotes a metric measure space with a regular Borel measure For E C X measurable diamE and denote the diameter and u measure of E The metric ball Bzr y E X dxy lt r and tBxr Bxtr u is doubling if there exists and constant C such that O Brl 2 l2Bzrl for any ball For a measurable function u we denote its mean value over on a set by uA i fAudu fAudu Note that although the doubling condition is only de ned in terms of a concentric ball with double the radius it implies more general volume com parison For any ball BxR7 y E BR and 0 lt r lt R lBRl S CRr lByrl 2 where the value of oz depends on the doubling constant of X 41 For a Lipschitz function u X a R we de ne Lip luff Now we are ready to de ne the Poincare inequality De nition 25 X d u a metric measure space admits a 1p P0incaiquote inequality p 2 1 with constants C 2 10 lt t lt 1 if every ball in X has nite non zero measure and l utBldM S Cf Lip uPdM1p 3 t3 B for any ball B and Lipschitz function u Notice that by Ho39lder7s inequality a p Poincare inequality implies q Poincare for q gt p While for any q lt p one can construct a space with 1 7p Poincare but not 1q According to 2 one way to achieve this is by gluing two copies of Euclidean R together along an appropriate Cantor set Of course Ho39lder7s inequality does not help us improve the left hand side so we close this section with the remark that restricting our attention to 17p Poincare instead of considering 11 is not as restrictive as it at rst appears A result by Haslasz and Koskela 1 demonstrates that for metric spaces with a doubling Borel regular measure 1p Poincare inequality with p 2 1 implies the qp inequality with q gt 1 512 Remarks on Context and Consequences We can think of metric spaces with a doubling measure also called homoge nous spaces to be the most general class of spaces that admit a zero order calculus Then the addition of a Poincare inequality will allow us to use many of the tools of rst order calculus such as results from second order partial di erential equations and Sobolev spaces and the di erentiability of Lipschitz functions In this realm the small improvement seen in the main theorem here can prove vital Several problems from nonlinear potential theory can be solved given the hypothesis that one is working on R with a q admissible measure equiva lently one that veri es the 17q Poincare inequality where q lt p some crit ical dimension For example in this scenario quasiminimizers of p Dirichlet 42 integrals satisfy Harnack7s inequality the strong maximum principle and are Ho39lder continuous In another section of this conference we were introduced to various alter native de nitions of Sobolev spaces Shanmugalingam showed that Cheeger7s Hajlasz7s and his own de nitions for 1p Sobolev space yield isometrically equivalent Banach spaces when we are in the realm of a homogenous metric space with a 1 7 q Poincare inequality q lt p Again we may now improve the statement of this result to include the equality case We stress here that this open ended property is a property of the space itself not just of the particular functions involved We say that a pair of functions ug E LPX satisfy the 1p Poincare inequality if inequality 2 holds with 9 replacing Lip u in the right hand side Given Theorem 1 we could hope that such a pair would also satisfy a 11 7 6 inequality regardless of whether or not the underlying space satis es 1p Poincare Unfortunately this is not the case Keith constructs an counter example on a Cantor space It has been noted by several authors that a Poincare inequality and hence a rst order calculus is in some way tied to the existence of a suf cient quantity of recti able curves in a space eg 2 In an earlier paper 3 Keith made this connection explicit by proving that a Poincare inequality is equivalent to a bound on the modulus of the families of curves connecting separate points in the space Although the proof here does not make use of this fact perhaps it can aide one7s intuition in understanding the nature of the open endedness of the condition by considering the modulus inequality as well 52 Outline of Proof of Theorem 1 We begin by making simplifying assumption that X d M is in fact a geodesic metric space This can be done without loss of generality because any ho mogenous metric space that admits a Poincare inequality is bi Lipschitz equivalent to a geodesic space This assumption is simplifying beyond en suring the existence of geodesics because in a geodesic metric space the parameter t in the Poincare inequality can be taken to be 1 by perhaps increasing the constant 0 Also we can now control the measure of a thin outer shell of a ball by lBxrBxr76rl S lBxrl 4 43 where or depends only on the doubling constant The proof of the main theorem will require the use of some variants of the sharp fractional maximal operator For Lipschitz u 1 Mt 7 7 d 5 W sgpdiamw u um ltgt where the supremum is taken over all balls that contain p It is a straightforward but useful fact that we can control Lip u on sets where Mun is bounded For the variants we shall encounter later7 statements close to the following hold Proposition 26 Xdu a metric measure space with u doubling andu a Lipschitz function Then there epists C gt 0 depending only on the doubling constant such that for all r gt 0y E X and z E Byr we 7 ugw Oerum lt6 which implies that the restriction ofu to the set x E X M ux S A is 20A Lipschitz The proof of the main theorem is quite long and technical The presenta tion here is backwards from the original with hopes that the reader will better appreciate the implications of some of the preliminary technical estimates by the time they are mentioned The proof is completed by integrating an estimate of the measure of the super level sets of a constrained version of the sharp fractional maximal operator Fix a ball E C X For t 2 1 we have 1 Mrum sgp u e ugldu lt7 for any Lipschitz u and z E B the supremum is taken over all balls such that tB C B and z E B Notice that this de nition yields 1 Ti M43u95 2 mil l uB ldM 8 E where 40 is some rather arbitrary large number7 and B i 44 Let U x E E M 3ux gt A We can show using local estimates that for any 04 E N there exists kg 6 N such that for all integer k 2 kg and every A gt 0 we have lUA 2kP D lngAl 8kp lU kl 10z e E Lip ugt10 kAl 9 Next we integrate both sides of the 04 3 version of this inequality from 0 to 00 against the measure dA 57 where 0 lt E lt p 7 1 is chosen so that 8 lt 2 The proof is then essentially complete one only needs to translate the integration with respect to the parameter A into integration on X and apply the observation of inequality 7 to nish In the nal subsection we will sketch the techniques used to arrive at the local estimates and7 thus7 estimate 521 Technical Estimates The local estimate is based on another small variant of the sharp fractional maximal operator Fix a ball X1 C X and let X 2i 1X1 Given a Lipschitz function u7 let 1 Mi 7 i d 10 MM Sigp diam BJilu UBl M be de ned for any x E Xi with the supremum taken over all balls B C X14L that contain x We will let U x 6 X4 gt A denote the super level sets The estimate is as follows Proposition 27 Let 04 E N There em39sts k1 E N that depends only on C and 04 such that for all integer k 2 k1 and every A gt 0 with 1 7 d A 11 diale llu uxmgt lt gt we have pm 2kp lU2kl 8kp lU Al 8kltPlgtHz 6 x5 Lip uz gt 8 kAl 12 We can simplify the exposition of the proof by rescaling ud7 and n to assume without loss of generality that A diamX1 lel 1 The proof then is by contradiction we assume that the proposition does not hold and that therefore for large enough values of h Uzk lt 24W 7 Usk lt 84W 7 and Hx 6 X5 Lipux gt S kH lt 8 k 1 What follows is a sequence of lemmas to nd lower bounds for such a h eventually leading to a contradiction 45 Lemma 28 We have lu euXWidu 210 X2U2k Loosely speaking it has some oscillation outside of Ugh The proof uses little more than assumed contradiction the above mentioned properties of geodesic doubling spaces and the cleverly de ned set of balls that overlap with Ugh on between 14 and 34 of their total volume Next we can preserve these large scale oscillations while smoothing on the small scale by taking advantage of the Lipschitz property of sub level sets of maximal type operators Lemma 29 There CCElStS a CSk Lipschitz extension f ofulXawgk to X3 such that MSW who for every x 6 X2 Ugh Now nally getting to use the fact that we are in a 17p Poincare space we nd that that the p norm of Lipf is small outside of the superlevel sets of u Meanwhile we de ne another new function h via Lipschitz extensions of f and demonstrate a lower bound for the p norm of Liph These Poincare type estimates lead to a contradiction in which it is crucial that p gt 1 thus proving Proposition 4 References l H Haslasz P and Koskela 13 Soboleu met Poincare Mem Amer Math Soc 145 688 2000 B Heinonen J Nonsmooth Calculus Bull Amer Math Soc 44 2007 163 232 E Keith S Modulus and Poincare inequality on metric measure spaces Math Zeitschrift 245 2003 no 2 255 292 4 Keith S and Zhong X The Poincare Inequality is an Open Ended Condition To appear Ann Math 46 emmes in mg curves on genera spaees mug quail i a we opo 5 S 7S7F39d39 l th h t39t t39 t l ogy with applications to Sobolev and Poincare inequalities Selecta Math NS 2 19967 no 27155 295 COLIN HINDE UCLA email chinde mathuclaedu 47 6 Distortion of Hausdorff measures and removability for quasiregular mappings after K Astala A Clop J Mateu J Orobitg and I Uriarte Tuero I A summary written by Vjekoslau Kouac Abstract We study absolute continuity properties of pull backs of Hausdorff measures under K quasiconformal mappings especially at the criti cal dimension We also consider the analogue of the Painleve problem for bounded and BMO K quasiregular mappings 61 Notation and basic de nitions Let us rst introduce all concepts that appear in the text The main purpose of the paper 1 is to prove several new results relating those notions and compare them to previously known results Throughout this text H denotes the a dimensional Hausdor measure and M denotes the a dimensional Hausdor content More generally for every continuous non decreasing function h 0 00 a 0 00 such that h0 0 any such function is called a measure function and for every F C C we can de ne MW mfg1a where the in mum is taken over all countable coverings of F by disks with diameters 67 Then M becomes a special case for the choice ht t We also denote by fa the class of all measure functions such that ht S t and liming 0 Finally the lower a dimensional Hausdor content of F is de ned to be sup hefa We say that f is a K quasiregular mapping in a domain 9 C C if it belongs to the Sobolev space and satis es the distortion inequality max ldafl S K m 046C lal1 046C in ldafl ae in Q lal1 Here daf denotes the directional derivative of f taken in the sense of distri butions 48 A K quasiconformal mapping is a homeomorphism p Q a 9 between two planar domains 9 9 C C that is also K quasiregular In order to nor malize quasiconformal mappings p C a C7 we say that p is a principal K quasiconformal mapping if it is conformal outside a compact set and sat is es ltpz 7 z as a 00 Now we introduce several notions of removability A compact set E is said to be removable for bounded resp BMO analytic functions if for any open neighborhood 9 of E7 every bounded resp BMOC analytic function on Q E resp C E has an analytic extension to Q We say that a set E is removable for bounded resp BMO K quasiregular mappings if every K quasiregular mapping in CE which is in L OC resp BMOC admits a K quasiregular extension to C A quantitive notion of removability is called capacity 7 again we have several versions of it Remember that for a function f holomorphic in a neighborhood of 00 E C we de ne f oo lim z H00 z 7 foo Let E C C be a compact set The analytic capacity of E is WE suplf 00l i f E H E W LOOCE7 f00 0 llflloo 1 The BMO analytic capacity of E and the VMO analytic capacity of E are respectively de ned to be ME suplf 00l i f E H E W BMOC7 f00 0 llfllBMo 1 ME suplf 00l i f E H E W VMOC7 f00 0 llfllBMo 1 For any pair 04 gt 07 p gt 1 with 0 lt Dip lt 27 one de nes the Riesz capacity of E by 0047037 supME u where the supremum runs over all positive measures a supported on E and such that H101 gtk MW 3 1 Here 042 Mia and 1 1 62 Quasiconformal distortion of Hausdorff measures It is well known that quasiconformal mappings preserve sets of zero Hausdorff dimension However7 they do not preserve Hausdorff dimension in general A result by Astala states that for every compact set E with dimension t and for every K quasiconformal mapping p we have tltt gt W7 Kltti l 49 Moreover7 both equalities may occur7 so the above bounds are optimal Some of the questions we study here are analogous estimates at the level of Hausdor measures Ht This leads us to the following conjecture Conjecture 1 Let E C C be a compact set and let p C a C be a K quasiconformal mapping a For any t 6 02 denote t Hfgim We ask if it is true that H E 0 e H ltpE 0 b Ifltp is also conformal on C E is it true that for t E 0 2 we have MMMM with constants depending only on K and t Let us remark that classical results of Ahlfors and Mori assert that the statement a is true in extremal cases t 0 and t 2 As the main result of this section we prove that statement for t Ki t 1 Theorem 2 Let E be a compact set and let p C a C be K quasiconformal Kil 2K wammow m WH WP0 HWWWw c Ifltp is also principal and conformal on C E then M1ltltPE N M1037 Remarh Part b is what we call absolute continuity of Hausdorff mea sures HD Observe that it is an immediate consequence of part a7 since M and H have the same zero sets Sketch of the proof We rst prove part c by restating it in terms of the BMO analytic capacity Because of Verdera7s result M1E 70E it suf ces to show VOltpE 70E but this follows immediately from the de nition of 70 To prove part a we may WLOG assume that E is a subset of the unit disk ll7 and that p is principal Then we decompose p ho p17 where h and 50 p1 are principal K quasiconforrnal rnaps7 p1 is conformal in QUCD and h is conformal outside p1 for some appropriately small open neighborhood 9 of the set E Finally we use part c applied to h to obtain M1holtp1E S CM1ltp1Q and then some rough area estimates to control M1ltp19 I Another interesting result in the same direction is Theorem 3 Let E C C be a compact set and let p C a C be a K quasiconformal map a IfHKLHE is o nite then H1ltpE is also o nite b ma a 0 than MiltltpE 0 Sketch of the proof From the result of Sion and Sjerve we know that 0 if and only if F is a countable union of sets with nite oz dirnensional Hausdorff measure Therefore parts a and b are equivalent We may also assume that p is principal Now we pass to the VMO capacity using another Verdera7s result7 74E and observe that yiltpE 74E The rest of the proof is similar to the proof of the previous theorern I The last result of this type is the quasiconforrnal invariance of the Riesz capacities Theorem 4 Let p C a C be aprincipalK quasiconformal mapping which is also conformal on C E Let 1 lt p lt 2 anda i 7 1 Then OapltPE N 0042037 with constants depending only on K and p Sketch of the proof The main idea is to use the following result about Riesz capacities CapE1 N suplf 00l i f E HUME f00 0 llfllwlew 31 Here Wl gtpC denotes the hornogenous Sobolev space Therefore it suf ces to show that ylwwz S CK ylap E and the latter follows from the fact that every K quasiconforrnal rnap p induces a bounded linear operator on Wl uhp C given by f gt gt f o p I 51 63 Quasiconformal distortion of 1recti able sets Recall that a set E C C is 1 rect239 able if there exists a set E0 of zero length such that EE0 is contained in a countable union of Lipschitz curves In this section we basically show that up to sets of zero length a K quasiconformal image of any 1 recti able set has dimension strictly larger than Theorem 5 Suppose that E is a 1 rect239 able set and let p C a C be a K quaslconforrnal mapping Then there em39sts a subset E0 C E of zero length such that dimltpE E0 gt Sketch of the proof We may assume that E is a subset of the unit circle 3D with dimE 1 Then we recall from a result by Lehto and Virtanen that whenever p is K quasiconformal and K K1K2 Kn for some K gt 17 then p can be factored as p can 0 0 W 0 p17 where each p is Kj quasiconformal This allows us to assume that K from the statement of the theorem can be as close to 1 as we wish Finally7 the theorem follows from this technical lemma Lemma 6 There ecclst constants c0 c1 c2 gt 0 such that if E C 3D 0 lt 8 lt co and f C a C is 1 6 quaslconforrnal then dimE 2 17 c1 82 i dimltpE 2 17 c2 82 64 A removability theorem for quasiregular mappings The classical Palnleu problem consists of giving metric and geometric charac terizations of those sets E that are removable for bounded analytic functions7 or equivalently yE 0 The Painlev theorem states that H1E 0 im plies yE 0 ln particular7 all sets E with Hausdorff dimension strictly smaller than 1 have yE 07 while Ahlfors showed that sets E with di mension strictly larger than 1 satisfy yE gt 0 However7 the exact char acterization of sets with zero analytic capacity is more complicated David and Tolsa proved that among the 1 dimensional sets with o nite length we have yE 0 if and only if E is purely unrecti able7 ie H1E P for all recti able curves P If we consider the removability for BMO analytic functions7 the charac terising property is simply H1E 07 as is proved by Kaufman Here we consider removability for boundedBMO K quasiregular map 2 pings7 and now the dimension m turns out to be critical 52 Theorem 7 Let E be a compact set in the plane and let K gt 1 a IfHKLHE 0 then E is removable for BMO K quasiregular maps b IfHKLHE is o nite then E is removable for bounded K quasiregular mappings c For K quasiconformal map p the image ltpE is purely unrecti able Sketch of the proof For the proof of part a7 take f E BMOC which is also K quasiregular on CE We can decompose it as f Fo p7 where p is K quasiconformal and F E BMOCC is holomorphic on C From the theorem 2 we know that H1ltpE 0 Thus7 by Kaufman7s result ltpE is removable for BMO analytic functions In particular F extends to an entire function and f extends to a K quasiregular map on the whole C For the part b we use the same decomposition f Fog and then apply theorem 3 to the map p Since ltpE has o nite length7 from Besicovitch7s result it can be decomposed as ltpE U Un U B 7 where Rn are 1 recti able sets7 Un are purely 1 unrecti able sets and sets Bn have zero length Finally we use countable semiadditivity of analytic capacity7 the result by David and Tolsa which implies 7Un 0 and theorem 5 which implies 7Rn 0 Therefore yltpE 0 and we are done Part c is an immediate consequence of I We end this text with the following conjecture7 which would imply that the result in part a above is sharp Conjecture 8 For every K 2 1 there epists a compact set E with 0 lt 2 HK1 lt 00 such that E is not removable for some K quasiregular map ping in BMOC References 1 Astala K7 Clop A7 Mateu l7 Orobitg J7 Uriarte Tuero l7 Distortion of Hausdor measures and improved Painleve removability for quasiregular mappings7 arXivmath0609327v1 mathCV VJEKOSLAV KOVAC UCLA email Vjekovac mathuclaedu 53 7 The quasiconformal Jacobian problem after M Bonk J Heiriorieri arid E Saksmari I A summary written by Peter Luthy Abstract Quasiconformal maps have been a fundamental part of complex analysis since the 1930s Their applications stretch far beyond strictly analytic problems Despite the longevity of the idea7 it is unknown in an analytic sense which functions are comparable to the Jacobian of such a map We begin by exploring an interesting relationship between metric doubling measures and bi Lipschitz maps on R2 From this7 we will produce a rather broad class of functions comparable to the Jacobian of a quasiconformal map 71 Introduction As stated in the abstract7 quasiconformal mappings of the plane have been in the mathematical vernacular for quite some time That they survived into the modern era is due in no small part to their critical role in proving big theorems Their creation7 however7 resulted from a rather small problem The Riemann mapping theorem guarantees the existence of a holomorphic map from a square to a rectangle not a square However7 this map does not carry vertices to vertices ln fact7 no such holomorphic map exists Grotzsch asked whether there was a best almost holomorphic77 map which had this desired property In doing so7 he produced an early de nition of quasiconformal maps The quasiconformal Jacobian problem asks the following question which functions it are ae comparable to the Jacobian of a quasiconformal map f By this we mean to determine for which functions it there is a quasiconformal map f R2 a R2 satisfying the estimate ww S Jfz S Cwx for ae z E R2 for some 0 gt 0 To proceed with our discussion we shall need some termi nology De nition 1 The local 172 Sobolev space lVli C2R2 is the class offurietioris iri LIZOCGRZ whose rst order weak derivatives are in LIZOCUR 54 De nition 2 A homeomorphism f R2 a R2 is said to be quasiconformal if each component off belongs to lVli C2R2 and iff satis es the inequality lDfzl2 KJfz locally in Baez where Dfltgt difj and Jf det Dfz Note The smallest such K is referred to as the dilatation of f The locally in L1R2 condition is deliberately used over an ae con dition because7 for instance7 lines have measure zero and Cantor Lebesgue type functions do not satisfy the integration by parts formula It is not dif cult to show that using de nition 2 implies that any continuously differen tiable quasiconformal map takes in nitesimal circles to in nitesimal ellipses of bounded eccentricity Proving the converse is somewhat more involved De nition 3 A Borel measure M on R2 is a doubling measure ifll is non trivial and there is a constant 0 independent ofx and r so that MB z r S CllBd 2r for any ball Bz r centered at z of radius r Any doubling measure M may be associated with a quasimetric dM by the relation M96711 6MB WBW lit 7 ill U Bu lit 7 MW We see that a typical dM only satis es the following weak triangle inequality dMl S KdMl 2 dMZl and so generally dM will not be a metric This motivates the following de nition De nition 4 A metric doubling measure M is a doubling measure for which the quasimetric dM de ned above is comparable to a metric 6M We may now observe that if f R2 a R2 is quasiconformal7 then M Jfddm2x is a metric doubling measure7 where m2 is the standard Lebesgue measure on the plane lndeed7 we may choose 6M 7fyl To see why this selection works7 we7ll need the following useful properties of quasiconformal maps taken from De ne hs M For any 3 points in the plane abc7 let T denote the triangle with those vertices and de ne skewa b c to be the ratio of the longest side to the shortest side of T 55 Theorem 5 ff R2 a R2 is quasiconformal then for any potntszy in the plane there are neighborhoods D1 and Dy so that for any triples a be C D1 and a b c C Dy we have SkeWUW fbf0 S MSkSWWabaC 1 skewf 1a f 1bf 1e S hskewa b c Conformal maps take in nitesimal equilateral triangles to equilateral tri angles since those maps preserve angles lnequality 1 says that quasicon formal maps can7t do all that much worse Moreover7 inequality 1 allows us to prove that quasiconformal maps do not distort space too much First7 we are able to prove that for all small equilateral triangles T7 diamfT g 7712fT l27 where the constant depends only on how small we require the perimeter to be The isoperimetric inequality gives the op posite inequality so that the two quantities are in fact comparable A similar triangles argument can extend to arbitrary equilateral triangles T diamfT 2 m2fT12 It is not too hard to then extend the theorem to balls or other compact7 convex sets Second7 one can show via inequality 1 that being K quasiconformal is equivalent to the existence of a homeo morphism 77 0 00 a 0 00 with the property that WC 7 f y f 95 i f 2 Using this identity7 it is not hard to show that if B is de ned as above7 lf96 MIN 2 diam BzyD Putting this discussion all together7 we deduce that iy ziz 12 MW 2 diam BmyD 2 mzfBzy12 Jf96dm296gt any Note that the right side is precisely dMxy7 and so our choice of 6 works perfectly well David and Semmes proved in 2 that every metric doubling measure has a density in the so called Ac0 class At that time7 it was already known that A00 included the quasiconformal Jacobians David and Semmes called the class of densities of metric doubling measures strong A00 It should come 56 as no surprise that they asked whether the strong A00 functions were all comparable to quasiconformal Jacobians Unfortunately7 Laasko showed the answer is negative It follows immediately from the properties used above7 however7 that if f R2 a R2 is quasiconformal7 then for any metric 6 comparable to dM we have f R26 a R2 is a bi Lipschitz equivalence The converse to this question is af rmative Speci cally7 Theorem 6 Suppose that u is a metric doublirig measure so that dM is comparable to a metric 6p Ariy homeomorphism f Ra a R2 which is bi Lipschitz has the additiorial following properties 0 There is a riori riegatiue furictiori w E LfOCR2 so that do wdmg where dmg is the staridard Lebesgue measure ori R2 Iri fact w 6 A00 0 f R2 a R2 is quasiconformal o Jf 2 w for ae z E R2 Moreover the comparability coristarit arid the dilatatiori off deperid orily ori data associated with u The theorem follows quickly from results derived above 72 Main Result Theorem 6 creates a mechanism for producing quasiconformal Jacobians 0 Select a suitable ie at least A00 function w 0 Show that do wdmg is a metric doubling measure with associated to some metric 6p 0 Produce a bi Lipschitz equivalence Ra a R2 We now state our main result Theorem 7 Main Theorem Suppose that u E L1 R2 with distribu loc tiorial gradierit Wu 6 L2R2 Theri there is a quasiconformal map so that Jfx 2 ezum for ae z E R2 The comparability coristarit C arid the dilatatiori off deperid orily ori lqullL2R2 57 Proof of the Main Theorem We will follow the usual proof strategy of rst establishing the theorem is true for u E 08 R2 then reducing the gen eral case to this one by approximation In the case of a smooth compactly supported it we have that exp is strictly positive and bounded above and below hence showing that do exp 2uxdm2 is a doubling mea sure is easy Likewise showing that the quasimetric dM is comparable to expuldzl is trivial The existence of a bi Lipschitz equivalence is all that remains This is of course easier said than done however our work is made substantially easier by the following theorem of Fu 3 Theorem 8 Les X be a complete Riemannian 2 manifold that is homeo morphic to R2 There are absolute constants 60 gt 0 and L0 gt 0 with the following property if the integral curvature ofX does not ewceed 60 then X is bi Lipschitz equivalent to R2 with bi Lipschitz constant L0 In this spirit consider the Riemannian 2 manifold Xu R2e ldzl Because it is smooth and compactly supported exp is strictly positive so the identity map is a homeomorphism Observe that if b E 08 R2 then Xkb R2e m bmldxl is A bi Lipschitz equivalent to R2e mldzl by the identity map where A exp Now whenever we scale the Euclidean metric by a smooth positive func tion h we have the Liouville curvature equation Alog h iKhZ In our case h e hence the Gaussian curvature of Xu is given by K ie zsAu The total Gaussian curvature of Xu is then given by lKle2 dm2 lAuldmg XS n2 In order for Fu7s theorem to be useful to us we need to modify it to control the Ll norm of Au without losing the bi Lipschitz equivalence stated above The following lemma 7 which follows entirely from classical theorems 7 gives us the power to choose b correctly so that 7 bl l1 is small 58 Lemma 9 Let u R a R be a smooth function of compact support Then for every 6 gt 0 there epists a decomposition u s b into two compactly supported smooth functions such that HASlll SE and 13 Hblloo lt HWHS Using lemma 9 pick a decomposition u s b using 6 602 where 60 is the absolute constant from Fu7s theorem The total Gaussian curvature of Xkb is now smaller than 602 By Fu7s Theorem7 Xkb is bi Lipschitz equiv alent to R2 Thus Xu is bi Lipschitz equivalent to the plane with coef cient 26 LoexpllWll Hence for all compactly supported smooth functions7 our theorem is true Now7 suppose that u is locally integrable with Wu 6 L2R2 By 6 we may select a sequence iii 6 08 R2 so that W a u in L1206R2 and Vul a Vu in L2R2 Then by the work above7 we may nd a sequence of Ki quasiconformal maps fl R2 a R2 so that in 2 exp2ui 2 Our choice of constant in 2 depends only on euimdz and We shall show below that exp2ul converges locally in L1R2 to exp2u so that we may assume our constants of comparability are independent of i For the same reason7 each fl will be K quasiconformal for some suf ciently large xed K Using Trudinger7s inequality and Bunyakovsky7s inequality we conclude that e mdmg AOm2B 3 B where B is an arbitrary ball in the plane This in turn implies exp a exp 2n in L0CR2 Now we proceed to show that some subsequence of fl converges on all compact sets to a K quasiconformal map f Conjugation of fl by a conformal mapping changes neither the Jacobian nor the dilatation Thus we may assume without loss of generality that fi0 0 for all i By 87 we conclude 59 that is a normal family and thus some subsequence converges locally uniformly to f By 77 we deduce that f is K oluasiconformal7 and7 moreover7 we have weak convergence of in to Jf A calculation involving 2 and 3 allows us to deduce that Jf is comparable to exp The constant of comparability depends only on the constant of comparability from 2 and hence only on The same holds for the dilatation of f Thus our proof is complete D References ll 2 Bl M Bonk7 J Heinonen7 and E Saksman7 The quasiconformal Jacobian problem Contemp Math 355 20047 pp 77796 G David and S Semmes7 Strong A00 weights Soboleu inequalities and quasiconformal mappings7 in Analysis and partial di erential equations7 Lecture Notes in Pure and Appl Math 1227 Dekker7 New York7 19907 pp1017111 J Fu7 Bi Lipschitz rough normal coordinates for surfaces with an L1 curvature bound Indiana Univ Math J 47 19987 pp4397453 J Hubbard7 Teichmuller Theory Volume 1 Teichmuller Theory Matrix Editions7 lthaca7 2006 TJ Laasko7 Plane with AGO weighted metric not bilipschitz embeddable to R27 Bull London Math Soc 34 20027 pp6677676 J Maly and WP Ziemer7 Fine regularity of solutions of elliptic par tial di erential equations Mathematical Surveys and Monographs 517 American Mathematical Society7 Providence7 RL 1997 YG Reshetnyak7 Space mappings with bounded distortion Translations of Mathematical Monographs 737 American Mathematical Society7 Prov idence7 R7 1989 I 1 mal LEC J V39ais39al39a7 Lectures on n 139 H Berlin Heidelberg New ture Notes in Mathematics 2297 Springer Verlag7 York7 1971 PETER LUTHY CORNELL UNIVERSITY email pml25 cornelledu 60 8 Separated nets in Euclidean space and Ja cobians of bi Lipschitz maps after 0 McMallen and D Burago and B Kleiner I A summary written by John Mahi Abstract Are all separated nets in R bi Lipschitz equivalent to the integer lattice Z Also is each non negative L OR function which is also bounded away from zero the Jacobian of a bi Lipschitz mapping from R to R We rst show that the questions are equivalent and then we show the common answer to both is no demonstrated directly by a counterexample for the second question Additionally some related problems are considered 81 Introduction We begin with a de nition De nition 1 A subset X of a metric space Y is a separated net if there eccist two constants a b gt 0 such that dzx gt a for every pair a x E X and dyX lt b for every y E Y We note two facts Every metric space contains separated nets and two spaces are quasi isometric if and only if they contain bi Lipschitz equivalent separated nets If all separated nets in R are bi Lipschitz equivalent to Z this simpli es the task of determining if two spaces are quasi isometric this clearly motivates the following question Q1 Are all separated nets in R bi Lipschitz equivalent to the integer lat tice Z This was posed in its present context by Gromov The answer to the corresponding question is yes if R is replaced by any non amenable space with slight conditions on local geometry or by hyperbolic space H with n 2 2 Also it is easy to verify the answer is yes in R1 A di erent question of some interest is that of attempting to prescribe the Jacobian of a homeomorphism Speci cally for a given type of homeo morphism can we identify a class of functions which coincides with the set of 61 Jacobians of those homeomorphisms Recall Rademacher7s theorem7 which guarantees that a Lipschitz map is differentiable ae With this in mind7 the particular version of this question which we consider is Q2 Given a function f E L R such that inff gt 07 will f necessarily be the Jacobian of some bi Lipschitz mapping from R to R Variations on this question include starting with Jacobians in L or Sobolev spaces 77 87 or Holder classes 00gt 82 Main results The primary result of the two papers resolves Q1 and Q2 The rst step of the proof shows the questions are equivalent7 while the second answers the second question in the negative Thus7 we produce the desired theorems Theorem 2 There erists a separated net in R which is not bi Lipschitz equivalent to the integer lattice Theorem 3 Let I 01 Given c gt 0 there is a continuous function p I a 11 c such that there is no bi Lipschitz map f I a R with Jacf DetDf p ae The answer to the latter theorem also resolves a similar question in quasi conformal mappings7 namely that there is a continuous function which is not the Jacobian of a quasiconformal mapping in R72 since any quasiconformal mapping with Jacobian bounded away from zero and in nity must necessar ily be bi Lipschitz ln general7 the effort to characterize the functions which correspond to Jacobians of such maps is called the quasiconformal Jacobian problem see 4 also7 the subject of lectures by Peter Luthy Notably7 not equality but rather equivalence between the function and Jacobian is pursued in that program Similar in spirit is the work of Bonk and Kleiner in their description of properties which identify when a metric space will be quasisymmetrically equivalent to a 2 sphere see Theorem 2 shows that a natural conjecture regarding the classi cation of subsets of R which are bi Lipschitz equivalent to Z is in fact false 62 821 The equivalence In this section we prove the equivalence of questions Q1 and Q2 Theorem 4 The following statements are equivalent A Every function p E L R such that infp gt 0 is the Jaeobz39an of some bi Lz39psehz39tz mapping from R to R B Every separated net Y C R is bi Lipsehitz to Z Proof A gt Let X C R be a separated net The task here is to match members of X to members of Z and the matching is ultimately guaranteed by set theoretic machinery To prepare for that endgame we need to relate members of X with members of Z such that any subset of X relates to a subset of Z which is just as large or larger and vice versa The bi Lipschitz property will derive from choosing nearby members for the relations For each y E Y let 0 denote the Voronoi cell 0 x x 7m lt in for all y in Yy De ne 1 pa 2 vole39 116 y Since Y is separated then inf vole gt 0 which means p E L OR Also since Y is a net then sup diam 0 lt 00 and we nd infp gt 0 Applying A we get a bi Lipschitz homeomorphism f R a R with Jacf p Denote fCy by Dy note that vol Dy 1 For 2 E Z let E1 denote the unit cube centered at 2 De ne the relation RCYXZ asthesetRyzyEYzEZ and Dy Ezy QJ Let A C Y be a nite subset and let RA z E Z yz E R for some y E A Since the cubes labeled by RA cover the cells Dy labeled by A and since volDy EZ 1 we get RA 2 Similarly R 1B 2 B for any nite set E C Z By the trans nite form of Hall7s marriage theorem R contains the graph of an injective map 151 Y a Z Similarly R l contains the graph of an injective map 152 Z a Y The Schroder Bernstein theorem guarantees the existence of a bijection in the presence of two such injective maps let o Y a Z be this bijection 63 Since diam Dy and diam E are bounded then the distance 7 z is also bounded for all yz E B As y is chosen among the z E Z for which yz E R we get sup 7 lt 00 and so the map b Y a Z is bi Lipschitz too B gt Let p E L OR2 with infp gt 0 Let I 01 Without loss of generality we may assume p 2 a 11 c We begin with the set up which produces a separated net that allows us to approximate p in a particular way Let 8521 be a disjoint collection of squares in R2 which have vertices with integer coordinates and sides which are parallel to coordinate axes Further we require that the side length lk of 5 tends to in nity as k a 00 Let bk 2 a 5 be the unique af ne homeomorphism with scalar linear part and de ne pk Sk a 11 c by translating the values of p to Sk pk p 0 ti Choose a sequence mk E Z4r with limH00 mk 00 and limH00 mklk 0 Subdivide 5 into mi equal subsquares of side length lkmk Let this 2 Wk collection be 7 TMha Finally we subdivide each TM into equal subsquares UM where 71 is the integer part of 4 kai pkd Note that the integral UM pd is approximately 1 We now choose a separated net X C R2 by placing one point at the center of each square UM and one point at the center of each integer square not contained in USk The points which are not in USg are added to satisfy the de nition of net the separated77 requirement is naturally satis ed since we are choosing centers of integer squares throughout The real business however will use the points of X inside of the Sk7s By hypothesis there is an L bi Lipschitz homeomorphism g X a Z2 Let X 1X c 2 and de ne fk X a R2 by W lt9 o W e 9 o we where M is some basepoint chosen in Xk The image of fk lies in an L gtlt L square which contains the origin hence the fk7s are a uniformly bounded collection of L bi Lipschitz maps From the proof of the Arzela Ascoli the orem there exists a subsequence of fk which converges uniformly to a bi Lipschitz map f 2 a R2 The counting measure on Xk normalized by 1li converges weakly to p times the Lebesgue measure since the subsquares of 5 were chosen to give X a local average density77 of approximately p1 in Sk The normalized 64 counting measure on fkXk converges weakly to Lebesgue measure from which we can conclude f1p f12 Recalling that the area of the image of a set is equal to the integral of the Jacobian over this set we get that p Jacf D 822 The answer is no Proof of Theorem 3 For convenience we will prove the theorem in the case of n 2 the proof for higher values of n follows with only minor modi ca tions Fix two constants L and c gt 1 and let I 01 We will construct a continuous function p 2 a 11 c which is not the Jacobian of a bi Lipschitz homeomorphism Ultimately we succeed in driving a wedge between p and such Jacobians using the following de nition De nition 5 We say that two points w y E 2 are A stretehed under a map f 12 a W Z39fdf96 MO 2 Admy We de ne a sequence of functions pN which force a self improving stretching property For N E N let RN be the rectangle 01 gtlt 01N and de ne a checkerboard function pN RN a 11 0 by pNxy 1 if N is even and 1 0 otherwise Lemma 6 There are k gt 0 M n and No such that lfN 2 N0 6 S nNZ then the following holds if the pair of points 00 and 1 0 is A stretehed under an L bl Llpsehltz map f RN a R2 whose Jaeoblan dz ers from pN on a set of area less than 6 then at least one pair of points of the form NSMWNSM is 1 hA stretehed where p and q are integers between 0 and NM and s is an integer between 0 and M Sketch of proof of Lemma 6 The proof is by contradiction Assume that all such pairs of points are less than 1 hA stretched Since the pair of points 00 and 10 are A stretched it is reasonable by the triangle inequality that most of the targeted pairs are approximately A stretched Stating that more precisely de ne subsquares S gtlt 0 and points 2 where 239 1 N and p and q are integers between 0 and M It can be shown that there is at least one square S which 65 has the property that all of its points of the form pin are at least 1 7 lA stretched when paired with their corresponding neighbor point 2451 for any 1 6 01 Given the density of the points pin in 5 the image of S under f is ap proximately a translate of the image of 511 under f since we have bounded the relative stretch of these adjacent squares above and below by values close to A However p is very close to the Jacobian of f which means that the areas of the images of S and 511 will be expanded by factors very close to 1 and 1 c This is in competition with the way in which the images of the sets are approximate translates of each other and ultimately leads to a contradiction D We may use a smooth change of coordinates to derive the following lemma from Lemma 6 Lemma 7 There epists a constant k gt 0 such that given any segment C 2 and any neighborhood C U C 2 there is a measurable function p U a 11 c E gt 0 and a nite collection of non intersecting segments C U with the following property if the pair x y is A stretched by an L bi Lipschitz map f U a R2 whose Jacobian differs from p on a set of area lt 6 then for some j the pair lj rj is 1 hA stretched by f The function p may be chosen to have nite image Finally we see how this self improving stretch can lead to a crisis for an L bi Lipschitz map as we repeat the construction on smaller scales Lemma 8 For each integeri there is a measurable function p 2 a 11 c a nite collection 8 of non intersecting segments C 2 and 6 gt 0 with the following property For every L bi Lipschitz map f 2 a R2 whose Jacobian differs from p on a set of area lt 6 at least one segment from S will have its endpoints g stretched by f Proof of Lemma 8 We proceed by induction The case i 0 is trivial Assume that the lemma is true for i 7 1 where 84 Let Uj be a disjoint collection of open sets with C Uj and with total area lt 6142 For eachj apply the previous lemma to Uj to get a function j U a 11c j gt 0 and a disjoint collection 811 of segments Now de ne p 12 a 11 c by for z E Uj and pi1p otherwise Let S USAF and 6 min j These will now exhibit the desired property 66 This lemma shows that certain measurable functions are not possible as the Jacobians of L bi Lipschitz maps7 for a given xed value of L7 but it further shows that we may iterate at smaller scales to increase the range of L values which will not work As we desire p Jacf ae7 the lemma will apply for all 6139 Finally7 we approximate the resulting sequence of measurable functions by a corresponding sequence of continuous functions7 differing on a set of small measure tending to zero The desired continuous function p will be the limit of this sequence of continuous functions D 83 Related results 831 The answer for Holder is yes If the bi Lipschitz requirement is relaxed to a homogeneous Ho39lder condition7 we nd the modi ed Q1 and Q2 have positive answers see De nition 9 We say o R a R is a homogeneous Holder map if there are constants K 2 0 and 0 lt 04 S 1 such that for M S R we have Wit yl S KRI l96 7 ill The purpose of this de nition for Holder is to make it a scale invariant property if Mp is homogeneous Holder7 then so is c zc for any c gt 0 Theorem 10 Fip n 2 1 Then 1 For any f E L OR with inff gt 0 there is a homogeneous bi Ho39lder homeomorphism o R a R ie 15 1 is also homogeneous Holder such that volltltzgtltEgtgt fltzgtdz E for all bounded open sets E C R 2 For any separated net Y C R there is a homogeneous bi Ho39lder bijec tion 7 Y a Z 832 Prescribing the divergence of a vector eld An in nitesimal form of the problem of constructing a map with prescribed volume distortion is to consider whether it is possible to prescribe the di vergence of a Lipschitz or quasiconformal vector eld i using a function f E L R We have the following result from 3 67 Theorem 11 For any n gt 1 there is an f E L R which is not the divergence of any Lipschitz or even quasiconformal vector eld References ll 2 E 3 Bl Burago D and Kleiner 13 Separated nets in Euclidean spaces and Jacobians of bi Lipschitz maps Geom Funct Anal 8 1998 no 2 2737282 Gromov M Asymptotic invariants for in nite groups In Geometric group theory London Math Soc 1993 MCMullen C T Lipschitz maps and nets in Euclidean space Geom Funct Anal 8 1998 no 2 3047314 Bonk M Heinonen J and Saksrnan E The quasiconformal Jacobian problem Comtemp Math 355 2004 77796 Bonk M and Kleiner 13 Quasisymmetric parametrizations of two di mensional metric spheres Inventiones Math 150 2002 no 1 127 183 Dacorogna B and Moser J On a partial di erential equation involving the Jacobian determinant Ann Inst H Poincar Anal Non Lin aire 7 1990 no 1 1726 Riviera T and Ye D Resolutions of the prescribed volume form equa tion Nonlinear Di erential Equations Appl 3 1996 no 3 25728 Ye D Prescribing the Jacobian determinant in Sobolev spaces Ann Inst H Poincar Anal Non Lin aire 11 1994 no 3 2757296 JOHN MAKI UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN email johnmaki mathuiucedu 68 9 Newtonian spaces An extension of Sobolev spaces to metric measure spaces after N Shanmugalmgam A summary written by Kenneth Maples Abstract We extend the de nition of Sobolev spaces to metric measure spaces and prove corresponding results to the classical embedding the orems 91 Introduction We would like to extend the de nition of Sobolev spaces W147 to more general spaces than R such as Riemannian manifolds This is so that we can de ne PDE in greater generality However7 the de nition of Sobolev space depends on the distributional derivatives of the function this concept may not be de ned on a general metric measure space Tthe de nition of Sobolev space must be replaced by an equivalent de nition that only depends on the metric and measure of the underlying space A previous attempt by Hajlasz noted that u E W147 if and only if there was a p integrable g 2 0 such that W 7 uyl S lz 7 Maw 99 for almost all x y E R In 3 Shanmugalingam replaces this inequality with one based on upper gradients77 of the function7 Which generalizes the idea of a primitive of a vector eld Instead7 we let u E N147 if there is ap integrable p 2 0 such that W e uyl p for all recti able compact paths 7 except for a set of paths of go modulus zero7 This paper de nes the Newtonian also known as Newton Sobolev spaces N147 and proves several fundamental theorems about them Notably7 N147 is shown to be a complete subspace of L 7 and several analogues of classical Sobolev embedding theorems are proven For a motivation of the techniques used see 69 92 Preliminaries Let Xdi denote a space X with metric d and measure M Let H1 denote Hausdorff one dimensional measure A path is a continuous map 7 I a X where I is some interval in R The image is denoted by M and the length is denoted by l y We have the following standard notations for path families Treat Non constant recti able paths with compact image equivalently I is compact PA Paths in Treat that intersect the set A PX Paths in Treat that intersect the set A on a set of positive one dimensional Hausdorff measure ie H1l yl A gt 0 We will only consider subfamilies of Trent we will therefore assume that paths are parameterized by arc length We de ne the p modulus of a family of paths P as follows 7 17 ModpP 312 Mm We write p P if p is admissible77 for the family P this means that p 2 0 is Borel measurable and fypdx 2 l for all y E P There are several general properties of p modulus that we will omit here a general and highly readable treatment is in lf Q is a statement about paths7 then we say that Q holds p ae if the family of paths where Q is false has zero p modulus A function u is called ACCp if it is absolutely continuous on p ae path In other words7 u 07 is absolutely continuous on 0 l y for p ae y 6 Treat Likewise7 if X is a domain in R we say that a function u is ACL if it is ab solutely continuous on every line perpendicular to the axes in other words7 if u0 y is absolutely continuous on 0 l y for Hn1 ae line 7 parallel to the co ordinate axes Note that an ACL function has directional derivatives almost everywhere We call an ACL function ACLp if the directional derivatives are in L Given u X a R we call a Borel measurable p 2 0 an uppeiquot gradient of u if for all y 6 Treat rum uyl pds 70 where z and y are the endpoints of y We will also write p gt u or u lt p for this If the inequality only holds on p ae path7 then p is called a p weak upper gradient of u and we will write p gtgtp u or u ltltp p We say that X supports a 1p P0ineaiquote inequality if there exists a uniform constant C gt 0 such that for all open balls B in X and pairs of functions ug with u ltlt 9 and u E Ll7 Bluiugl CdiamltBgtlt ng1P39 If X supports a 1p Poincare inequality7 then it supports a 1q Poincare inequality for all q gt p 93 N140 and friends Let Q be a domain in R For 1 S p lt 00 we can de ne the SoboleV space W147 as the space of functions on Q with nite W147 norm7 where 71 Hulw Hulle Z Halum k1 Here 31 is the distributional derivative in the ith coordinate The goal of this paper is to extend the de nition of SoboleV spaces to more general spaces than R One earlier attempt noted that u E W147 if and only if there exists some 9 2 07 g 6 LP such that for ae L y 6 R72 M96 7 uyl S lz 7 yl99 9a We will call such functions 9 a Hajtasz gradient of ii We de ne M1gtpX to be functions u 6 LP along with a g 2 07 g 6 LP such that for n ae my 6 X7 M96 7 uyl S dy9 9a This space carries the norm Hulle 7 Mn 1 Halle where the in mum is taken over all Hajlasz gradients ofu M1gtpX is Banach with this norm 71 The space introduced in this paper exploits the concept of upper gradi ents7 de ned in the rst section Let N1gtpX be the space of all u E LPX R such that u ltltp g for some 9 2 07 g 6 L N147 is a vector space it is also a lattice We can de ne a seminorm on N147 in the following way u u inf ll Him H Hmultltpgll9llm This seminorm is hot positive de nite However7 the relation u N 1 if 7 UHNW 0 is an equivalence relation and partitions N147 into equivalence classes We will denote the space of these equivalence classes by N1gtpX with the norm induced by the seminorm above Another extension of Sobolev spaces based on upper gradients was devel oped by Cheeger We de ne H1pX to be the subspace of LP such that m mm giggly Hmin is nite The limit in mum is taken over all upper gradients gi of the func tions where a f in L 94 N140 is Banach To show that N147 is complete7 we must introduce the capacity of a set We can de ne the p capacity of a set E Q X as CappE 52 Hull w where u E means that u is admissible77 for the set E ie u E N147 such that u 2 1 on E Other de nitions for the capacity of a set are possible7 but the di erences appear to be immaterial The following lemma is key to the argument Lemma 1 IfF Q X with CappF 0 then ModpPF 0 Theorem 2 N147 is Banach Proof We follow the same procedure as for showing that L is complete We choose a Cauchy sequence in N1gtpX and7 after noting that it suf ces to examine subsequences7 select a subsequence with rapid convergence We 72 can force upper gradients of the di erence of consecutive terms ie gm gtgt uk 7 uk1 to have similarly shrinking LP norm We then construct Ek x E X 7 uk1zl 2 27k As has capacity rapidly converging to 07 we also construct F lim sup Ek On X F7 converges pointwise to a function u By the previous lemma7 because CappF 07 ModpPF 0 Hence on every Treat PF we can sum up the di erence upper gradients gm to form an upper gradient of u 7 uk for each h This gives an upper gradient for u and similarly shows that 7 u in N147 D 95 Similarities between N140 and M17 First we verify that N1gtpX is a genuine extension to W147 Theorem 3 IfX Q is a domain in R dxy lx 7yl and u is Lebesgue n measure then N1gtpX W1gtp9 as Banach spaces Proof This theorem follows from considering the absolute continuity prop erties of u E N147 1 Lemma 4 The set of equivalence classes of continuous functions u E M147 embeds into N147 with Hulle S 4llullM1m Proof We can choose u to have a Hajlasz gradient 9 such that the Hajlasz inequality holds everywhere If y connects my 6 X7 consider Lg If the integral is 00 then the upper gradient inequality holds If it is nite7 we can partition 7 into n pieces 71 of equal length Choose xi 6 such that S H1l yil 1 Li 9 then we can convert the Hajlasz inequality into the upper gradient inequality W950 95M E Z 51 95i1939 995i1 349 Y Letting n 7 00 we have the upper gradient inequality for the endpoints of y The embedding is well de ned D 73 From this lemma7 Theorem 5 The Hajlasz space M1gtpX continuously embeds into the N1gtpX Theorem 6 If X is a doubling space and X supports a 1q Poincaiquote inequality foiquot some q 6 hp then N1gtpX M1gtpX isomorphically as Banach spaces Finally7 the two de nitions of Sobolev type spaces based on upper gradi ents are nearly the same7 as the following theorem shows Theorem 7 H1pX is isometrically equivalent to N1gtpX when p gt 1 96 Analogues of Sobolev space results Theorem 8 IfX is a doubling space that supports a 1p Poincaiquote inequal ity then Lipschitz functions are dense in N1gtpX Recall the following classical embeddings for W1gtpR W1gtPX H ann l if p lt n W1gtPX g 004 if p gt n Theorem 9 Let Q gt 0 IfX is a doubling space satisfying MltBltwgtgt 2 W with C unifopm overs 0 lt r lt 2 diam X and ifX supports a 1p Poincaiquote gliyugliiy foiquot some p gt Q then N1gtpX continuously embeds into the space Equivalently7 every equivalence class in N147 has a Holder continuous representative with exponent 1 7 Qp with bounded Ho39lder norm Proof We de ne two sequences of balls that converge to z and y7 respectively7 with initial radii ddy such that the radii shrink in half at each step If x and y are Lebesgue points of u7 then the average values of u on the balls converges to the function values after some calculation7 we derive M96 uyl S CQapllPlledWwV Qp 74 Because M is doubling7 the non Lebesgue points form a set of measure zero Therefore7 u restricted to the Lebesgue points is Holder continuous and can be extended to a Ho39lder continuous function I on all of X By excluding the zero p modulus families P2 and P0 y uo y is not absolutely continuous7 we have u I on p ae path7 hence I E N147 of the same equivalence class as u D References 1 Fuglede7 l37 Extremal length and functional completion Acta Math 98 19577 1717218 2 Heinonen7 J7 Lectures on Analysis on Metric Spaces Springer Verlag7 2000 3 Shanmugalingam7 N7 Newtonian spaces An extension of SoboleV spaces to metric measure spaces Rev Mat Ibemam 16 2000 no 2 2437279 KENNETH MAPLES UCLA email maples mathuclaedu 75 10 Two counterexamples in the plane after 0 Bishop 1 and T Laahso A summary written by William Meyerson Abstract We equip the standard Euclidean plane with two pathological weights an A1 weight which is not comparable to a quasiconfor mal Jacobian and a strong Ac0 weight whose induced metric is not biLipschitz equivalent to Euclidean distance 101 Introduction De nition 1 An A1 weight on some Euclidean space E is a locally integrable w E a 0 00 such that whenever B is a ball the average value ofw on B is bounded up to a multiplicative constant 0 independent of B by its essential in mum on B De nition 2 A strong Ac0 weight onE is a locally integrablew E a 000 such that the measure induced by fwdz where dx is Lebesgue measure is doubling and the distance function sending a pair my to the integral of w over the ball centered at x y2 with radius x 7 y2 is biLipschitz equivalent to some metric lf w is A1 then clearly w j Mw almost everywhere where M refers to the Hardy Littlewood maximal function by Lebesgue differentiation however7 because we need only consider those balls with rational centers and radii when looking at the maximal function and for such a ball7the average value of w on B is bounded above by Cw almost everywhere7 we therefore have that Mw j w almost everywhere as well Further7 A1 weights are well known by a straightforward geometric argu ment to be strong A00 weights by a 1993 theorem of Semmes 37 E equipped with the distance function induced by some A1 weight is biLipschitZ equiv alent to a subset of some larger Euclidean space However7 as will be shown below by a counterexample due to Laakso 2 in the plane7 this does not hold for strong A00 weights Further7 we shall also introduce a reasonable conditions which strong Ac0 maps and even A1 maps do not necessarily satisfy7 even in R2 To do this7 we need more de nitions 76 De nition 3 A continuous function f R2 7 R2 is said to be quasi conformal if there epists some constant C such that for each x E R2 the limit superior of map z gy w as r goes to zero is bounded above by Remark 4 Note that this is a generaliztion of conformal maps in BBRZ for these maps 0 1 and the theory is well known as conformal maps of the plane can be viewed as complecc ualued functions in this setting they are either analyticor conjugate analytic De nition 5 We say that a strong A00 w is comparable to a quasiconformal Jacobian if there epists some quasiconformal f from R2 to R2 such that Jf N w ae where Jf is the Jacobian off With all the machinery set up7 we can nally state the following two results Theorem 6 Laakso There eccist a strong Ac0 weight w on R2 which is not comparable to a quasiconformal Jacobian Theorem 7 Bishop There eccist an A1 weight w on R2 whose induced metric is not bilipschit L H L in any F quot1 space and therefore w is not comparable to a quasiconformal Jacobian 102 Proof of our rst theorem We shall proceed following Via Laakso7s construction of a strong Ac0 weight in R2 such that when R2 is given the induced distance metric7 it is not bi Lipschitz embeddable to any Euclidean space In doing so7 we shall assume the following 1996 theorem of Semmes 4 Theorem 8 Suppose that Xd is a metric space and 0 lt t lt 1 such that the snow ake space X dt can be embedded biLipschitzly into some Euclidean space E Then there epists a strong A00 weight w in E such that Xd can be embedded biLipschitzly into E equipped with the induced metric from w Fixing a E 341 and h E 07T6 there exists some b E 014 such that a be 1 7 be we de ne the ve similarity mappings 1 S5 on the Euclidean plane as follows 812 a27 822 a beikz7 832 a 7 be lkz7 842 1 7 be lkz7 and 852 1 7 be ikz 77 Our planar set Z shall be de ned as the closure of the images of 0 under repeated iterations of the similarity mappings as 0 810 we have that Z is the union of its rst order similarity parts SlZ S5Z here we say that an nth order similarity part is the image of Z under n mappings in 1 S5 Clearly SlZ intersects SZZ and 32 at 17 while S4Z and S5Z intersect at 1 further7 SZZ and S4Z intersect at a be while 532 and S5Z intersect at a 7 be To show that these are the only intersection and boundary points of rst order pieces one notes this requires a bit of a geometric argument7 which we omit here that there exists a minimal angle 6 such that if SjZSkZ is a pair of similarity parts whose intersection point 2 was listed above7 and z 31 zy 31 2 are in SjZSkZ respectively then the angle zzy has argument at least p We next choose some number s gt 1 such that as 2b5 1 note here 1 a 2b cos k lt a 2b Letting Zn be the set of boundary points of nth order similarity parts ie the elements achieved when applying n similarity transforms to either 0 or 1 we can de ne a metric dn on Gn as follows dnxy is the in mum of lag 7 1115 lam 7 amjlS where the 11 are a sequence of adjacent ie any two consecutive 11 form the boundary points for some nth order similarity part elements of Zn such that al z and am y Clearly this is a metric and Zn 3 Zk for each n gt k To de ne a metric on the union of the Zn7s whose closure is Z itself it therefore suf ces to show that dn1 is an extension of dW for each n To see this we note that if z FO and y F1 are the bound ary points of an nth order similarity part FZ corresponding to zero and one7 respectively7 under the appropriate composition F of similarities then dn1xy lx 7 yls we can see this by noting that an optimal path goes from FO to Fa to Fabeik to F1 considering Z to be a zeroth order similarity part this argument even works for 0 and 1 Consequently7 if z and y lie in Zn then an optimal chain of adjacent boundary points in Zn joining z and y can be made by taking the optimal chain of boundary points in Zn from x to y and between each consecutive pair of points in this chain which can be written as FO and F1 for some nth order similarity inserting the subsequence FaFa 196k This chain therefore witnesses that dn1xy dnzy so we can indeed de ne d to be the union of the dn on the union of the Zn To extend d to a metric on all of Z it suf ces to show that d is continuous 78 with respect to Euclidean distance we shall do this by showing that d is biLipschitZ equivalent to the quasinorm To do this rst note that if z SlZ then dx0 and lx 7 0 9 are both in a51 so d and the quasinorm are comparable for these points by apply ing the inverse map of 1 to an arbitrary nonzero z E Z we get the same comparability ie d0 lx 7 0 9 Similarly7 if y S4Z U S5Z then dy 1 ly 7 1 5 iterating inverse images of S4 and S5 gives the same result for arbitrary y different from 1 If x is a boundary point of some similarity part FZ and y is an interior point of FZ then applying F 1 and the preceding paragraph yields dy lx 7 yls as F 1 is zero or one Further7 if z and y are in different similarity parts which do not share a boundary point7 then as dy and lx 7 yls both lie in b2517 dy 196 7 119 lf z and y lie in adjacent similarity parts with common boundary point 2 then as lx 7 yl lx 7 21 12 7 yl because of the lower bound on the angle xzy so lx 7 yls lx 7 2 9 12 7 yls and dxy dz dzy7 CHM 196 7 119 Finally7 if z and y lie in the same similarity part7 then by repeatedly applying inverse similarity transformations we can reduce to an earlier case so no matter what7 dy lx 7 yls as desired we can therefore suppose d is de ned on all of Z To show that Z equipped with the metric d does not embed biLips chitzly into any Euclidean space7 we shall use the following 7rounded ball7 property of Euclidean spaces which is an easy consequence of the parallelo gram law for Hilbert spaces Proposition 9 For each 6 gt 0 there em39sts 6 gt 0 such that ifs 31 y in some Euclidean space E theh B lx yl1 62 By l yl1 62 has diameter less than 61 7 Consequently we suppose for a contradiction that f Zd 7 E is bi Lipschitz with coef cient L where E is Euclidean For any unequal pair of points my 6 Z we use A to refer to W S A S L Letting z a be and y a be m7 we have that da dx 1 d1y dya b5 while da 1 dy 2b5 Therefore lf7fyl 2 L Zlfa7f1l so by the rounded ball property7 if fx and fy are both within 7 of fa and f1 then there exists 6 gt 0 79 such that r gt 1 6lfa 7 f1l2 This implies that either AM7 AM7 AW7 or AW all of which are boundary points of a similarity part is at least 1 6Aa1 We derive a contradiction as follows because AgL 2 there exists n 2 0 such that Aan THrl 2 which implies that there exist some pair st of boundary points of a similarity part one will be either a or an the other will be either x or any with A5 2 By repeating this argument for this similarity part applying the appropriate tranformation we get a pair 5 t of boundary points of a similarity part contained in the similarity part bounded by s and t such that Aggy 2 72 lterating this procedure recursively implies that the Lipschitz coef cient of f is at least 116 for each n letting 71 go to in nity contradicts that f is Lipschitz By our theorem of Semmes there exists a strong A00 weight w on R2 such that Zd is bi Lipschitz equivalent to a subset of R2 with distance metric induced by w To prove our nal result we now supose f is a quasiconformal and there fore quasisymmetric map from R2 to itself such that w is comparable to Jf further7 we use m to denote Lebesgue measure If x and y are unequal points in R2 we use B to denote the ball centered at x y2 with radius lx 7 ylQ therefore quasisymmetry gives that 7 fyl2 As f is absolutely continuous and the standard change of variables rules apply we note that letting w be the volume derivate of f ie the limit of W7 which equals lJfl ae yields that xgwyltzgtwltzgtdz 7 XBMltf1ltagtgtwltf1ltagtgtJf1ltagtda 7 xmwm da 7 mummy so 7 dy where d is the metric induced by w7 producing a contradiction as f is therefore biLipschitZ and proving the theorem as desired 103 Proof of our second theorem Following 17 we begin by constructing a descending sequence E0 3 E1 3 quotEn of subsets of the unit square 01 x 01 in the plane their inter section E will be a set of measure zero on which our desired weight should be in nite 80 To do this one needs to introduce the notion of LM N pieces where LMN are positive integers such that LN are odd and greater than or equal to three De nition 10 In this setting an L M N piece of a square Q is the subset produced as follows we rst divide Q into L2 equal subsquares to be called the type 1 subsquares of Q with center square Q0 nept we divide Q0 into M2 equal subsquares to be called the type 2 subsquares of Q and nally divide these type 2 subsquares into N2 equal subsquares to be called the type 3 subsquares the LM N piece on is de ned to be equal to Q with the open central type 3 subsquare of each type 1 subsquare removed Supposing that Ln7 Mn7 and Nn are sequences of this type7 we construct a sequence of compact sets as follows for notational purposes we let tn LnMnan lL and 5 t1 gtk gtk tn We begin by letting E0 be the unit square for n 2 1 we take each partition Elkl into disjoint squares of size 5W1 these are the size of the type 3 subsquares we call them the n 7 1 generation squares and replace each such square with the corresponding Lm Mm Nn piece For the remainder of this talk we assume that EN2 ooEN3 lt ooEM2 lt 00 while the LL are constant for concreteness we can set Ln 77 Mn n7 N 3 2 From these assumptions we know that area of E is zero in fact7 if S is an nth generation square and h gt n then lEk 5 j Further7 if Q is an nth generation square of side length 5 the second outerrnost layer of type 1 subsquares are of size Len7 This layer forms a ring separating the central type 1 subsquare Q of Q from the boundary of Q also7 any point on this ring is at least Len7 away from both Q and 6Q De nition 11 In this setting we call these squares the ring squares WQ 0f By the fact that quasiconforrnal maps on R are quasi syrnrnetric7 we have that if f is K quasiconforrnal and xes 0 and 007 dfW5fQ U fQ 5K diamfQ 81 Armed with these sets7 we can now construct our weight w We begin by setting Fn to be a sn neighborhood of En for each n this serves to ll in the type 3 subsquares removed from Ewl to get to En and adding if necessary a ring of size 5 around each n 7 1st generation square Clearly this comes nowhere close to doubling the area of En further7 dFm F571 2 iswl because the points in Fn are no more than 5 away from En1 in horizontal and vertical distance Letting An be an increasing sequence of positive numbers such that A0 1 and An approaches in nity with n we set w 1 outside of F1 and w An on FnFn1 further7 for n 2 1 we can set an An 7 An1 Making the additional assumption that there exists some A lt xE so that Agn 3 AA which implies An 0 and the bookkeeping assumption that an S 1 for each n we bound the average value of w on some nth genera tion square S as follows this will irnply local integrability noting that this average is up to a constant bounded above by nEzinaf we have 7122 S 0An 71 by a computation which implies letting N go to in nity that the average value of w over a dyadic square S is bounded above up to a multiplicative constant by its essential in rnurn w on that square To show that w is A1 we therefore need only consider a ball B Bz r of radius r letting n be the largest positive integer with 5 gt 47quot we let k be such that Ak is the in rnurn of w on B we can do this because w is in nite only on a set of measure zero if k S n then as 47quot 3 5k we have that 2B is outside Ek1 as points within 47quot of Ek1 lie in Fk1gt and therefore B lies outside Fk2 otherwise we could move by 8n2 ltlt 7 to get a point in Ek2 which lies in 2B so w is bounded above by Ak2 S 3Ak on B However7 if k gt n then as r 2 isk we can cover B by disjoint k 1 generation squares of size Sk1gt which are all contained in 2B therefore7 the average of w over B is at most four times the largest average of w over one of these square which is bounded above by up to a multiplicative constant 4Ak1 S SAk and therefore w is A1 Now7 if z is not in En then x is as far from En as from E because edges of nth generation squares in En are never removed in any later stage Also7 if z is in EnE then x was removed at the kth stage for some k gt n therefore7 z is at most sk2 S sn112 frorn Ek and thus is that distance away from E so 82 x 5 S dx E lt 5 x 5 S dx En lt 5 FnFn1 Therefore7 we can write it to be the square of some function of dx E in doing so we can suppose f 1 if r 2 r07 f is non increasing7 and 1 3 62 3 17 where 739 can be made arbitrarily small because An can be made to grow arbitrarily slowly We nish by sketching the proof that there exists no quasiconformal map f on the plane such that Jf N w on the plane where J refers to the Jaco bian To do this it suf ces to show that if f is a quasiconformal map on the plane with Jf N it then fE contains a recti able curve 7 as y blows up on E7 this would imply that f 1 has Jacobian vanishing on 7 so f 1 y is a point which contradicts that f is a homeomorphism The main result we will use is the following 7good path7 limit7 whose proof a geometric argument depending on various properties of quasiconformal maps we omit for lack of time Lemma 12 There epists n0 6 N such that ifh 2 n0 and y is a good path for fEk ie a polygonal path contained in fEk whose uertices lie on the boundaries of the images of nth generation squares then there is a good path 7 for fEk1 such that each vertep of y is also a uertep of y and W1 3 1711 1 where ClCgn depend only on the quasiconformal map f Armed with this lemma7 we obtain the recti able curve in fE by a com pactness argument extending a line segment in fEnO gives us a good path for fEnO and iteratively applying the good path lemma gives us a sequence mg of good paths in fEk of uniformly bounded length parametrizing these paths appropriately and extracting a uniformly convergent subsequence yields in limit a good path contained in fEk This produces our desired contradiction so it is indeed comparable to no quasiconformal Jacobian References 1 Bishop7 CJ 77An A1 weight not comparable to any quasiconformal Jacobian Ahlfors Bers Colloquium proceedings Ann Arbor7 2005 83 2 Laasko7 Tomi 77Plane with AGO Weighted Metric not Bilipschitz Embeddable t0 Rm 7 Bull London Math 500 London Oxford University Press7 2002 3 Sen1n1es7 Stephen7 77Bi Lipschitz mappings and strong A00 weights 7 Ann Acad Sci Fenn Sen Helsinki Academia Scientiarum Fennica7 18211 2487 1993 4 Sen1n1es7 Stephen7 77On the nonexistence of bilipschitz paran1etrizati0ns 7 Re m39sz39ta Matematz39ca Iberoamen39cana Madrid Real Sociedad Maten1atica Es panola7 12 337 4107 1996 84 11 Removability theorems for Sobolev func tions and quasiconformal maps after P W Jones and S K Smiinou 4 and P Koshela A summary written by Nicolae Tecu Abstract We give suf cient conditions sets to be Sobolev and quasiconfor mally removable 1 11 Introduction We start with presenting the main results in 4 on quasiconformal remov ability and continue with those in The paper is organized as follows we will rst introduce several necessary notions and make a few introductory remarks We will continue with a presentation of the main theorems and7 in the end7 outline the proofs 112 De nitions and introductory remarks The setting of all the theorems is R Q is an open set In the follow ing W1gtp9 will denote the functions in LPG whose distributional par tial derivatives are also functions in L This means that u E W1gtp9 if u E LPG and there are functions dju E L Qj 1 n such that u6jidz 7 iodjudp 1 n n for all test functions 7 6 0309 and all 1 S j S n De nition 1 A compact set K C U is quasiconfopmally removable inside domain U if any homeomorphism ofU which is quasiconfopmal on UK is quasiconfopmal on U De nition 2 A compact set K C U is called W1gtp iquotemouable for continuous functions inside domain U if any function continuous in U and in W1gtpU K belongs to W1gtpU De nition 3 A closed set E C U is called W1gtp iquotemouable 07quot p pemouable if W1gtPU W1gtPU 85 Note the difference between the last two de nitions in de nition 2 the Sobolev function is assumed to be continuous in U7 while in de nition 3 this doesn7t happen A few observations are in order E is removable for W147 if and only if for each x E E there is r gt 0 such that W1gtpBxr W1gtpBzr as sets Secondly7 as smooth functions are dense in W1gtp 2 E7 it is suf cient to verify relation 1 for u 6 035 E W1gtp 2 E and w 6 0809 In the following K will be a compact set which is the boundary of a connected domain 9 We may assume it is bounded In this context we consider the Whitney decomposition of the domain W We will denote by lQ the side length of cube Q and by fQ the mean value of f in that cube lf Q is any cube7 aQ denotes the concentric cube with the side length a times that of Q De nition 4 Fip a family P of curves starting at the ped point 20 E Q and accumulating to 39 such that their accumulation sets cover 39 The shadow SHQ of a cube Q is the closure in 39 of the union of all curves 7 E P starting at 20 and passing through Q Denote by 5Q the diameter of SHQ Remark 5 Ifb is small enough any curve in P will pass though at least one Whitney cube of that size Thus 39 is covered by shadows of nitely many cubes of su ciently small size This in turn implies that the Lebesgue volume of 39 is zero De nition 6 Given a domain 9 we can consider the metric on it with the W volume element idis z n This will be denoted distqh and called quasi hyperbolic metric The quasihyperbolic metric is geodesic for more information see De nition 7 We call a set E C R 1 p porous 1 lt p lt n 7 1 iffor each Hn l ae z E E there is a sequence and a constant 01 such that rl 7 0 and each n 7 1 dimensional ball Bri contains a ball B C Bri E of radius no less than Cmrlln wm p For n 7 1 S p lt n we replace the balls B by continua E of diameters no less than Cmrlln wm p Forp n E is p porous if the diameter ofE is no less than Cyriezp71Cmri 86 113 Main results We now state the theorems Theorem 8 Iffor some p 2 1 a domain 9 C R satis es 28QZQpl 1lQl lt 00 2 QeW where 11 1p 1 then K 39 is W1gtp removable for continuous func tions In particular7 for p n this gives the following Corollary 9 If satis es 2 8Q lt 00 3 QEW then K 39 is quasiconformally removable It is suf cient to include in the sum above only the cubes Q which are in a neighborhood of K and so there is no loss of generality if we restrict ourselves to bounded domains Q If condition 3 holds7 every curve in P starting at 20 has exactly one landing point and each point on 39 is a landing point of such a curve lf 2 E SHQ there is a such a curve passing through Q Theorem 10 Iffor some red 20 E Q where Q C R is a domain satis es distqh20 E L QKm 4 then K 39 is quasiconformally removable and W1gt removable for contin uous functions Here 9K is a neighborhood of K in 9only integration near K is needed Theorem 11 A totally disconnected closed set E C R 1 is W1gtp removable for p gt n Theorem 12 fa set E C R 1 is p porous 1 lt p S n then E is removable for W147 in R In addition for each 1 lt p S ii there is a p iporaus set E C R 1 that is not removable for W1 for any q lt p 87 114 Outline of proofs Theorem 8 follows from the following proposition Proposition 13 IfQ satis es 2 then any continuous function f which belongs to W147 for bounded subsets of K0 is ACME Any continuous function f E W1gtpKC is by the proposition in ACME Since K has Lebesque measure 0 this implies f E W1gtpR see 67 theorem 214 Any homeomorphism f which is quasiconformal in the complement of K7 is also an element on and thus ACMR by the proposition Since K has volume zero7 f EACLGR implies that f is quasiconformal in R72 which is what we wanted see 57 Section 34 Outline of proof of proposition 13 Consider a bounded domain U which con tains K and let l be a line parallel to coordinate axis A The main part of the proof consists in proving that the total variation of f on l U equals the total variation of f on l U K If this holds then one can prove f is absolutely continuous on l f is in W1gtpU K and this means that the partial derivatives of f exist almost everywhere and are in L U K However7 U is a bounded set and hence the partial derivatives are also in L1 U K Fubini7s theorem implies that 6A 6 EU U K for almost every line l parallel to A Thus we see that the total variation of f on l UK is fmmK ldAfl for almost every line l in that particular direction Denote the total variation of f on if U by fmU ldAfl In order to prove that the total variations are the same7 we will prove that laAfl laAfl 5 U AK where the double integrals are to be understood as integrals over all lines parallel to A of the total variations Fubini7s theorem gives us the desired result To prove 5 one starts by approximating fmU ldAfl by expressions of the type 7 6 6 Ej W967 mm WUjmjyjl Afl where zjyj cover l K7 with xjyj E l K Fix now one of these intervals Since the shadows of the Whitney cubes of size A picked at the end cover 88 K we can nd a partition of this intervals such that uo zbun y and for any other 239 we either have uhulH C K0 or uhui belong to the same shadow SHQl In the rst case we have the estimate lfltmgt 7 fowl lam lt7 uigtui1l In the second case we can nd curves 711in E P which connect ui and ui1 to Q respectively From these two one can obtain a curve which connects ul to ui1 and passes through Qi call it 71 With some care one then gets lfltuigtefltui1gtl2 Z laxflQlQ Q yi where the sum is taken over all Whitney cubes intersecting 71 All the Whit ney cubes in the sum above are of size at most A and can be chosen to be in U Adding up everything and making a few more reductions one obtains Warm p l5 fl 2 l5 lQlQ lyjlK A Z A 5HQ l1jgtyjl0 This implies lama Z laxflQlQ lam U SHQ l7 0 WK where the Whitney cubes appearing in the relation are of size at most A By Fubini we get laxfl 2 ZlaxflQlQ8Q 1 le By Hoelder7s inequality 117 Z laifiltQgtzltQgtsltQgtH ZlaxflQplQlgtlp28QZQpl 1lQlgt The cubes that appear in the sums are those of size less than A While the rst sum is nite by the fact that f E W1gtpU K7 condition 2 implies the second surn goes to zero as A a 07 which proves the proposition D 89 Outline of proof of theorem 10 One considers the center of some Whitney cube Q20 Now join the centers of any two adjacent cubes by intervals and de ne the function q qQ the number of intervals in the shortest chain joining the centers of Q20 and Q One can remove redundant intervals such that qQ is preserved and the collection of intervals is a tree We then have qQ X distqhQ20 for all Q E W This transforms 4 into 2 qQ lQ oo lt8 QEW The next step consists in de ning a collection of curves P take all the chains of intervals in the tree above that contain an in nite number of inter vals One can prove that each point in K is a landing point of exactly one of these curves Then7 for each cube Q one can nd a curve from P which has length comparable to 5Q Finally7 one can prove that 2 8Q s 0 Z 1Q QQ lt9 QEW QEW The proof is complete by applying Theorem 8 with p n D Outline ofproof of theorem 1 The theorem is proved rst for n 2 and p gt 2 The set E is closed Since removability is a local property7 we may assume E is compact and further that it is a subset of 01 One can show that p removability is equivalent to the Hl ae equality of u and u on E7 where ux lim0lttno ux1tz x10 E lV7 and u z is de ned analogously Next7 one can use the Sobolev embedding theorem to argue that u is uniformly Hoelder continuous both in the upper and lower halves of the ball B0 2 Since E is totally disconnected the only connected components are points7 it has empty interior and a series of approximations gives that the difference of ux and u x can be made as small as one likes For larger 71 one uses this base case and proceeds by induction using also integration by parts Outline of proof of theorem 12 Consider rst the case n 2 We may make the same assumptions as in the previous proof The key idea is to prove that uz u x for Hliae z E E One assumes the opposite and uses the fundamental theorem of calculus and Hoelder7s inequality for p lt 90 2 or replaces u by a harmonic function as these minimize the 77energy77 LBW lV lzdz and uses capacity estimates for p 2 to prove that qulde 2 cm 10 Bm239ri for a sequence which decreases to zero This is a contradiction to the fact that hml qulz dx 0 11 ZTOT Bmr for Hliae z E B02see 67 p118 Since the last fact is true for all 717 one uses the same strategy for n gt 2 One may assume E C 01 1 and u E W1gtp 2 There exists a p harmonic function 1 with the Dirichlet data given by u Then u 7 1 6 W347 E and it suf ces to show that E is removable for 1 As 1 is p harmonic and me lelpdz lt 007 1 has upper and lower non tangential limits As before7 it is enough to show that lVUlpdz 2 CT 1 12 Bmri for z E E where the non tangential limits do not coincide and where the porosity condition holds This is accomplished by a series of reductions and capacity estimates Both for n 2 and n gt 2 the porosity provides sets of 77considerable size on which the function u or 1 take 77well separated77 values Then one uses averages or sequences of averages combined with the Sobolev Poincare inequalityor Trudinger if n p to get a lower bound for the gradient In the end one uses some covering argumentssee eg 1 To nish theorem 12 we use the following result Theorem 14 Suppose that W1 E UfilQi where I 01 and each Qi is an open cube in In l If1ltp lt n and Z diamQlquot p lt 00 and Hn 1I 1Uf12Qi gt 0 13 i1 91 then E is not p remooable If 2 log1diamQ1 lt 00 and Hn 1I 1 U ldiamQi 12Qi gt 0 i1 14 then E is not n remooable It is now suf cient to construct a p porous compact E C In 1 such that E is not removable in R for q lt p We only consider the case p lt n The set is going to be a Sierpinski type set One starts by removing a cube Q1 of side length s2 A where 14 S s S 12 and A n 7 10 from the center of In l The remainder can be divided in two types of cubes Some of them are going to be translates of Q1 on the coordinate directions and the others are going to have side length l1 1 7 s2 A2 and are going to be 2 1 of them Denote the collection of all these cubes by W1 Next step consists in deleting a cube of side length 52 for an appropriate 5 from the center of each cube in W1 which has side length at least l12 Subdivide the remainder of each cube as above and get a collection W2 Let l2 be the largest side length of a cube in W2 Continue in this fashion with the deleting and subdividing process to get a sequence of collections Wi Set E WW Then E will be p porous and condition 13 will be satis ed for 1 lt q lt p The previous theorem shows that this E is not q removable for all 1 lt q lt p References 1 Heinonen J and Koskela P Quasiconformal maps in metric spaces with controlled geometry Acta Math 1811998 1 61 B Koskela Pekka Old and new on the quasihyperbolic metric in Qua siconformal Mappings and AnalysisDuren 13 Heinonen J Osgood BPalka B eds pp 205 219 Springer Verlag New York 1998 E Koskela Pekka Removable sets for Sobolev spaces Ark Mat37 1999 pp 291 304 E Jones Peter W and Smirnov Stanislav S Removability theorems for Sobolev functions and quasiconformal maps Ark Mat 38 2000 pp 263 279 92 5 V39Eiis39zil39zi7 J7 Lectures on m 139 39 39 f mal r 39 Lec ture notes in mathematics7 2297 Springer VerIag7 Berlin eidelberg7 1971 6 Ziemer7 William P7 Weakly dz erentz39able functions Grad Texts in Math 1207 Springer Verlag7 New York7 1989 NICOLAE TECU YALE UNIVERSITY email nicolaetecuyaleedu 93 12 Measurable Differentiable Structures and the Poincare Inequality after Stephen Keith A summary written by Arman Vagharshakytm Abstract The techniques developed in 2 are applied to improve the dif ferentiable structure presented in It is shown that the coordinate functions of a differentiable structure can be taken to be distance func tions During the proof7 the differential of a function contained in H is described in terms of approximate limits 121 Preliminary Notations A metric measure space lt Xdu gt is a set X equipped with a metric d and a Borel measure it on it Note the measures that we consider will be af nite We say that the measure it is doubling if there exists a con stant C gt 0 st uBx2r S CMBzr for any x E X and r gt 0 Balls Bzr y dyx lt r are always assumed to have positive radius 0 lt r lt 00 For a function f X a R de ne LIPf supmayW Then LIPX f X a R LIPf lt 00 is the space of Lipschitz func tions Let LIP0X be the space of Lipschitz functions with compact sup port Also de ne lz pfx liminfrnosupdw krw and Lipfz limsuprnosupdw krw Write aplz39mynm y A iff for any 6 gt 0 we have hmrao Bgt7 lfmli l4lgt5l 039 We say that the metric measure space lt Xdu gt admits p Poincare inequality for some p 2 1 iff 1 balls have positive measure 0 lt uBr lt 007 for any x E X and r gt 0 2 for any f E LIPX and any ball Br fBzr lf fBltmgtl 3 LT 393mm Z Pf9 pd gt 117 122 Main Theorem Cheeger see demonstrated that metric measure spaces satisfying the Poincare inequality admit a 77differentiable structure7 with which Lipschitz functions can be differentiated almost everywhere To be precise 94 Theorem 1 Cheeger see Let lt X du gt be a metric mesure space which admits a p Poincare inequality with complete metric d and doubling measure it then there epists a sequence lt Xm u gtD called a strong mea surable differentiable structure where X0 s are measurable subsets of X and M s are uector functions de ned on X so that 1 X UXD 2 170 vim yw where each of coordinate functions p0 is in LIPX and the number of coordinate functions is unformly bounded 1 S Na S N lt oo 3 For each f E LIPX and each X0 there epists ae unique unique up to a set of zero measure function A a X0 a BM so that y f96 X u 77049 7 MM 0dyz aE z 6 X0 The main result of this article is that we can take all 1 to be distance functions ie Theorem 2 Main Theorem Let lt X du gt be a metric mesure space which admits a p Poincare inequality with complete metric d and doubling measure it then there epists a strong measurable differentiable structure lt Xa 170 gt where all 2 are distance functiuons ie ddz for some E X In order to prove the Main Theorem we need to establish some properties of Sobolev spaces over given metric spaces which admit a strong measurable differentiable structure So lets introduce Sobolev spaces right now Sobolev Space Consider the following norm on LIP0X llfll1pllfllpllLipfllp Denote the completion of LIP0X with respect to this norm by Sobolev space H Remark It is essentially shown in 1 that the space Hm is re exive for p gt 1 Also denote DX hdxx1 dn h E 0 0R R h has compact support n E N It turns out that DX is everywhere dense in Hm see rroposition 4 Cotangent Space TiXm For x 6 X0 consider X 7 Z7DzXERNa 95 This is an Na dimensional space of functions for ae z 6 X0 Of course the de nition depends on the choice of a If x 6 X0 X then we have 2 generally speaking different linear spaces A1 17049 170M x1 and A2 17199 5 x2 Let7s identify X17170z with 75AM i their di erence is ody This identi cation is a linear one to one correspondence for ae z 6 X0 And TXm will now be de ned to be the linear space of functions with the precision of that identi cation Note the di erential dfz Xf w 7170z z 6 X0 is a well de ned element of TXm SKeith see generalized Cheeger7s theorem for any subset of LIPX Applying this generalization for the set of distance functions we get Proposition 3 There epists a natural number No st Every set A C X with positive measure has a subset W C A with positive measure so that W satis es the following properties Property 1 There epists a vector function p1zpNp 1 S N S No whose components are distance functions dzz st the functions are independent in TXm for ae z E W and for any distance function p we have dpp E Spandpx for ae z E W Property 2 W C X0 for some 04 Property 3 the functions span TXm for ae z E W indeed it turns out that on the one hand ifu E DX then E Spandpx for ae z E X see property 1 above and proposition 9 on the other hand for each function iv x we can nd a sequence on E DX st iv x in TXm for ae z E W see consequence of proposition 4 So the system dpi is a basis for the space TXm Hence combin ing this fact with Cheeger s theorem we get Property 4 For every function f E LIPX we have y f96 WON 5049 MM Own796 for ae x E W Property 5 The function f w mentioned in Property 4 is ae unique as39 Afap is ae unique on W and the functions dpi constitute a basis for TXm for ae z E W 96 Proof of Main Theorem De ne the property PNO to hold for W E X if the set W satis es Properties 1 through 5 We know that every set A of positive measure has a subset W which satis es PNO7 hence the set X can be represented in a countable decomposition XOWnUZ 1 where the set Z has zero measure whereas all sets Wn are disjoint7 have positive measure7 and satisfy PNO Let be the vector function of distance functions provided by Property 1 for the set Wl then will be a strong measurable differentiable structure for lt Xd M gt st each p is a distance function 123 Auxilary Results We construct a canonical collection of norms on spaces TXm7 de ned for almost every x E X in the following way Norm on the space TXm is de ned in the following way X my 7 m m Lz39pd my 7 mltzgtgtgtltzgt It is well de ned on TXm and it is a norm on TXm for ae z E Xa Note f E LIPX then Ltpf Description of Sobolev Space We can rede ne the Sobolev space Hm to be the completion of LIP0X with respect to llfllm llfllp llldf96lmllp Now we can describe Hm Elements of Hm are pairs lt u w gt where u 6 L177 wz E TX7 6 LP for whom exists a sequence un E LIP0X st llUn ullp 0 and llldUn96 wlzllp 0 Proposition 4 DX is dense in Hm 97 Proof of proposition 4 l splitted the proof into two steps Step 1 Suppose7 we show that for any u E LIP0X there exists a sequence un E DX st 1 un u in LP and 2 HunHLP lt M lt 00 then this will imply that DX is dense in Hm lndeed7 Hm is re exive see 17 so a ball in Hm is weakly compact7 so after passing to a subsequence7 we can assume that un alt f g gt weakly in Hm for somelt fg gt6 Hm ln particular7 well have un alt fg gt weakly in LP Hence f for ae z E X This implies see proposition 6 9a for ae m 6 X7 so un a u weakly in Hm Therefore LIP0X is in weak closure of DX Hence DX is dense in Hm Step 2 For a given function u E LIP0X we want to construct the sequence un E DX which is demanded in step 1 of proposition 4 We will need a collection of well behaved balls7 which is provided by the following lemma Lemma 5 see Ifu is a doubling measure then for each n E N there eaists a family of balls Bl Bxi1n st for every h gt 1 every ball hBZ intersects with at most Mk balls kBj where Mk depends on the doubling constant of the measure u and the number k only Now consider the following functions 7 nd 9a Z nd 96739 where t R a 01 is any continuously differentiable function that satis es x1for0ltzlt1and x0forzgt2 Then the sequence M95 we 2 WM will be the desired one Consequence First of all7 for each one may construct a new function st c E LIP0X and and deg will be the same element of TiXm when x 6 X0 Then7 according to proposition 47 we can choose a sequence un in DX7 st llunicgllm a 0 And7 after passing to a subsequence7 well have 7 a 0 for ae z 6 X0 98 Proposition 6 If lt uw gt6 H1pX then w is determined uniquely by u The proof goes as follows Step 1 If lt uw gt6 H1pX then ifmm 6 6 6 0 asr60ae 6X Step 2 If lt uw gt6 H1pX then apltmyan 6 0 ae z 6 X Step 3 If lt uw1 gt6 H1pX and lt uw2 gt6 H1pX then M7712an 0 ae z 6 X therefore lt uw1 gtlt uw2 gt Proof of step 1 Denote My 6 6 and 1y 6 6 As we should prove the statement for almost every point z 6 X we are free to assume that z is a Lebesgue point for the function 1 Then using that M is doubling we may obtain that fwm Md 3 020 flaw2739 l Mm2M 01 as N 00 So a good upper estimate for an expression like I fa l1 6 03mm is needed Note we cant apply the Poincare inequality explicitly here as we might not have 1 6 LIPX yet we can apply Poincare inequality for a function 1 6 LIPX and note that 1 6 1 in LP to get I Ls w my 6 dwyyl dulp this last expression will have an estimate which will be admissible for us as soon as we assume that z 6 X is a Lebesgue point for the function 1 and a density point for one of the sets Xa and as it was noted above we are allowed to make such assumptions For the other two steps note that if the measure is doubling and p Poincare inequality holds then it cant be too much concentrated in the center of the ball namely Proposition 7 Ifn is doubling then there em39sts a constant 0 lt a lt 1 st for any x 6 Xr lt dz39amX and h 6 N we have MltBltmkrgtgt lt16 agtmltBltzrgtgt and in general the measure cant be too much concentrated on a part of a ball This is expressed quantitatively in the following way 99 Proposition 8 Eccists an absolute constant C gt 0 st ifz E Bz0r0 and r 3 r0 then qu gt 1 MBWOWOD 7 C We use proposition 7 to prove step 2 of our proposition 6 In order to show step 3 we prove that if it E LIPX then aplimy m 52 To 0 implies limynmdifJZQEL 0 The proof uses proposition 8 Proposition 9 Letu E DX then du E Spandp for ae z E W p is a dist function Indeed if u hp1zp2z where dxz are some distance functions then one can prove dh d i M H for ae z E X References 1 J Cheeger Di erentiability of Lipschitz functions on metric measure spaces Geom Funct Anal 9 1999 no 3 4287517 2 S Keith A di erentiable structure for metric measure spaces Preprint 2002 3 S Semrnes nding curves in general spaces through quantitative topol ogy with applications to Soboleu and Poincare inequalities Selecta Math New Series 2 1996 no 2 155 295 ARMEN VAGHARSHAKYAN GEORGIA TECH email armenv mathgatechedu 100 13 Sobolev met Poincar after Hajlasz and Koskela A summary written by Marshall Williams Abstract We summarize a number of results from the rst half of 37 which generalizes the notion of Sobolev functions to the setting of metric spaces with a doubling measure By taking the View that a func tion and its gradient should consist of a pair of functions satisfying a Poincare inequality7 the authors are able to extend a number of classical results to this more general setting 131 Introduction Notation We mainly follow the notation of Throughout this summary7 the term function is reserved exclusively for real valued functions 07 with or without a subscript7 will always be a positive constant7 as will 039 2 1 X XCll will denote a metric measure space7 with metric d and measure a Q Q X will always be an open subset of X Unless otherwise stated7 the measure a is assumed to satisfy a doubling condition on the volume of balls7 M23 0MB lt1 where B is a ball7 and AB is a concentric ball with A times the radius of B When A Q X has positive measure7 we set fdMMA 1Afda Motivation The main purpose of this paper is to generalize the theory of Sobolev spaces beyond the classical setting of R The authors develop this generalization in the setting of metric measure spaces7 subject only to the doubling condi tion This allows one to do analysis in very general settings7 including topological manifolds7 graphs7 and fractals7 as well as Carnot Caratheodory 101 Spaces There are a number of di ferent approaches to de ning Sobolev functions in general metric spaces An earlier approach by Hajlasz in 2 de ned the spaces M1gtpX to be the spaces of functions u E LPX such that for some nonnegative g E L X the inequality W uyl S d96y995 99 2 holds almost everywhere i e for all Ly E XE with ME 0 Here M need only be a Borel rneasure not necessarily doubling Another approach is based on the notion of an upper gradient introduced by Heinonen and Koskela in We say that a Borel function g is an upper gradient of a function u if the inequality W e uyl gds lt3 7 is satis ed for every recti able curve 7 joining any two points s and y In fact upper gradients have been used by Cheeger 1 and Shanrnu galingarn 6 to de ne Sobolev spaces in a general setting However one downside is that the theory becornes trivial in spaces with few recti able curves such as graphs or certain fractals To get a theory with applica tions to these latter types of spaces the authors take the perspective that a Sobolev function and its gradient ought to be related by a Poincare inequality We recall that the classical Poincare inequality in R states that for Lip schitz function u R a R we have i lu 7 uBlde1p 0npr qulpdz 117 4 The idea in this paper will be to use a generalization of 4 to extend the notion of a Sobolev function to a general metric space We will expand on this below and will see that there is a close connection with Hajlasz7s earlier approach in 102 132 Some preliminaries First7 motivated by 47 we make the following de nition De nition 1 Let Q Q X be open A pair of measurable functions u E LOCQ g 2 0 on Q is said to satisfy a p7P0incar inequality in Q or simply a p7Poincare inequality ifQ X if for every ball B with TB Q 9 we have 117 flu 7uB do 3 Cpr gpdu 5 B OB where Op gt 0 and o 2 l are ped constants We will sometimes add quotwith data Opp to be speci c We will also need to make use of the following truncation property7 which generalizes a property of gradients in R De nition 2 A pair of functions u g satisfying a p7Poincare inequality is said to also satisfy the truncation property if for every b E R 0 lt t1 lt t2 lt 00 and E i1 the pairs 6u 7 b fg satisfy the same p7 Poincare inequality ie with the same constants CF and o Here 1 minmax0y 7 t1t2 7 t1 One of the main questions explored in 3 is to what extent a p7Poincar inequality implies the following Sobolev inequality De nition 3 We say a pair ug satis es a global Sobolev inequality for Q with data pqC if 1q 117 inf u 7 clqdu S C gpdu 6 cElR Q Q We also say the pair satis es a weak Sobolev inequality with data qp C 0 iffor all balls B of radius r 1q 117 inf u 7 clqdu 3 Cr gpdu 7 cElR B OB We will want to investigate the relationship between the Sobolev inequal ities in 3 and the following weaker inequalities 103 De nition 4 We say that a pair u 9 satis es a global Marcinkiewicz Sobolev inequality on Q with data qp C if 117 inf supuz E Q 7 cl gt ttq S C gpdu 8 c lR tZO Q We also say the pair satis es a weak 7 39 S A l L 1 quot with data p q C o iffor each ball B of radius r i z7 a m e B W 7 cl gt tW f inf su lt Org pd 9 25 MB VB 9 t lt Finally7 we recall a few miscellaneous de nitions and theorems which will come in handy De nition 5 Marcinkiewicz spaces A functionfis in the Marcinkiewicz space LfUX if there is some in gt 0 such that for all t gt 0 NM gt a s w De nition 6 For f E Loc9 and R gt 0 we de ne the restricted HardyiLittlewood maximal function of f on 9 MdeW 10 1 sup f do 0ltrsR MBWaW mmm IfR 00 we write MQfX which we refer to as simply the maximal function off on Q We omit the subscript Q in the case that Q X We will also need a version of maximal theorem of Hardy7 Littlewood7 and Wiener7 adapted for doubling metric spaces For a proof7 see 47 chapter 2 Theorem 7 Maximal Theorem 1413 in Let u be doubling Then for t gt 0 we have up 6 Q Mguz gt t crl M do 11 n Thus M9 maps L1 continuously into Also for 1 lt p S 00 HMnullmn S Ollullmm 12 0 depends only on the doubling constant Cd and the eccponentp 104 We will need an elementary fact about doubling measures7 namely that for some 0175 depending only on the doubling constant 0 17 MB 5 m 2 Cbrr0 13 133 Summary of results We prove a number of results for pairs U7 9 satisfying the Poincare inequality 5 First7 we show that if we assume our pair also has the truncation property7 then a Marcinkiewiczisobolev inequality implies a Sobolev inequality That is7 Theorem 8 21 and 23 in Suppose MS lt 00 Suppose also that every pair ug satisfying a piPoincare inequality in Q with data Opp also satis es the global MarcinhiewicziSobolev inequality 8 with data qp 01 Then any such pair that also has the truncation property will in fact satisfy the global Sobolev inequality 6 with data qp02 where OZ 8 4001 Similarly if every pair u g satisfying the piPoincare inequality with data Op 0 satis es 9 with data qp 010 then every such pair with the trun cation property will in fact satisfy 7 with data qp 020 Next7 we examine the relationship between pairs satisfying a pilZ oincare inequality and the spaces M1gtpX see the Introduction From now on we assume it is doubling We prove Theorem 9 31 in Let 1 lt p lt 00 Then afunction u is in M1gtpX if and only ifu E LPX and there is a g E LPX such that the pair u 9 satis es a q Poincare inequality for some 0 lt q lt p We investigate spaces where pairs U7 9 with 9 an upper gradient of u7 satisfy a pPoincare inequality We look at an example Example 10 42 in 3 The cone X xl E R x xfl1 S equipped with Euclidean metric and Lebesque measure has the prop erty that every continuous u with and upper gradient 9 satisfy apiPoincare inequality if and only ifp gt n 105 We also show a geometric consequence of a Poincare inequality for upper gradients Theorem 11 44 in Suppose X is proper and path connected and dou bling andp 2 1 If each pair of a continuous function and its upper gradient satis es a p Poincare inequality with cced data then X is quasiconyecc ie every two points z and y can be joined by a path of length at most Cdxy Finally we prove a kind of Sobolev embedding theorem We have Theorem 12 51 in Suppose the pair ug satis es a p7Poincare in equality and suppose further that s is as in Let 10 sps 7 p when p lt s and similarly q sqs 7 q Then the following hold for all balls B of radius r 0 Up lt s then i 1 m e B me 7am gt W S M f 9W SUB Forpltqlts we have 1a 1q f u 7 med 3 Or quM B SUB If the pair has the truncation property 2 as well then 1P 117 f u 7 we at 3 Or gPdM B SUB 0 Up s then fexp lt01MB1siu uBl dM lt 02 B HgHLS5aB 7 0 Up gt s then u is almost everywhere equal to a Holder continuous function and sup 7 Ugi 3 Cr dashint5aggpdn1p 63 For Ly 6 B0 we have WW 7 u S OrgPdxy17sp 117 9 do 5030 All constants depend only on the data p q 5 Cd 0 Op and 05 106 134 Outline of proofs We very brie y outline the proofs for some of the results above Proof of 8 Here the idea is to pick b exactly so that u 7 b is nonnegative on at least half of the mass of Q and also nonpositive on half the mass We estimate the positive and negative parts of u 7 b separately labeling each one 1 The integral f9 11 du can be broken into pieces where the integrand is controlled by powers of 2 The truncation property says that these pieces and the corresponding restrictions of the gradient still satisfy a p7Poincar inequality so by hypothesis we can use the Marcinkiewicz inequality 8 on each piece Summing gives the desired estimate The same idea works for the weak inequalities where we restrict to balls D Proof of 9 i This direction is simple Integrating 2 over a ball B with respect to z and then y gives the desired result To see this we need to show that if u g satisfy a pPoincar inequality then we have WW My S Cdmy Mzadltzygt9 9 1 M20dltmygt9py1p 14 for almost every my 6 X This is 32 in Indeed if we show this then assuming g satis es a q7Poincar inequality replacing p with q in the above result and applying the maximal theorem 7 to 9 7 E Lpq gives the desired result All we have to do is show 14 holds almost everywhere To do this we let x be a Lebesque point of u and estimate 7 uBmdmyl with a series of successively smaller balls using the doubling property to transform each term into an average that looks like the left hand side of a Poincar inequality Applying Poincar and summing will give us our estimate We must estimate lqu wy 7 uBQ mym as well but this is easily done using the doubling property and Poincare This completes the proof We should point out that a converse to the fact that Poincar implies 14 is true as well That is ifu E LOCX0 S g E LfocX and l lt p lt 00 then if 14 holds for almost all my 6 X then the pair u 9 satis es a p7Poincar inequality with data depending only on the constants in 14 This is 33 in The way to see this is to average 14 over z and y in a ball B with radius r then apply Cavalieri7s principle and after truncating the resulting 107 integral7 apply the maximal theorem 7 to gl ng 6 L17 where B is a slightly larger ball Truncating the integral judiciously gives us the desired estimate Veri cation of example 10 To see that the cone in 10 satis es a Poincare inequality if and only ifp gt 717 we rst note that loglloglzll E W1gt B0 12 and can be truncated to construct functions whose gradients have arbitrarily small L norm7 yet which are constantly 1 on the lower cone7 and constantly 0 on most of the upper cone7 and thus have mean oscillation bounded away from 07 prohibiting an nipoincare inequality To show 5 holds for p gt 717 the trick is to use the classical Sobolev embedding theorem see7 eg 4 chapter 3 to get a pointwise estimate as in 14 for 7 and then invoke the remark above that such an estimate implies a Poincare inequality D Proof of I The details of this argument are a bit technical7 but the basic idea is fairly simple For any pair of points my 6 X7 we need to construct a path from x to y of length at most Cdxy For each k 6 N7 we construct a path W from x to y We then de ne7 for each k7 a kind of pseudoilength77 function l y7 and pick our paths yk so that lk yk is minimized up to a constant We de ne our pseudo length in such a way that g E 1 is an upper gradient of each function inf lk y where the in mum is taken over curves 7 joining z and z This lets us use 5 to estimate S Cdxy We thus have a sequence of curves with bounded pseudo length Because of the way we de ne this pseudo length7 we are able to modify our space a little bit to get a family of curves with bounded length Fortunately7 after we use a compactness argument to pass to a limiting curve7 we can show that this limiting curve lies in our original7 unmodi ed space X Proof of 12 This is also quite technical7 and in the interests of space7 we won7t go into detail here The main idea7 however7 is to split it into two steps First7 we estimate lu 7 uBl S CJi fg7 where the right hand side is a kind of generalized Riesz potential This is 52 in We then use a fractional integration theorem 53 in to control Jiffy Roughly speaking7 we need to show that Ji f maps LP into L5 continuously when p lt 57 and maps Lq into L f for p lt q lt s Also7 for p 57 we need to show Jiffy satis es 108 Hyperbolic Geometric Flow Kefeng Liu Zhejiang University UCLA Outline Introduction Hyperbolic geometric flow Local existence and nonlinear stability Wave character of metrics and curvatures Exact solutions and Birkhoff theorem Dissipative hyperbolic geometric flow ltgtltgtltgtltgtltgtltgt Open problems 1 Introduction Ricci flow Structure of manifolds Singularities in manifold and spacetime Einstein equations and Penrose conjecture Wave character of metrics and curvatures Applications of hyperbolic PDEs to differential geometry J Hong D Christodoulou S Klainerman M Dafermos l Rodnianski H Lindblad N Ziper Kong et al Comm Math Phys J Math Phys 2006 2 Hyperbolic Geometric Flow Let 11917 be ndimensional complete Riemannian manifold The LeviCivita connection 1 M 3921 39239 392739 r r 7 Z 2 823139 Bwj 821 The Riemannian curvature tensors k kl arl39el k k 7 Rijl I ipI I ijfl Ri e gkpjol The Ricci tensor Rik gleijkl The scalar curvature R ginij Hyperbolic geometric flow HGF 82917 at for a family of Riemannian metrics g jt on 1 4 1 General version of HGF 82917 8t2 2R I llt 89gt 0 2 J 1 9981 DeXing Kong and Kefeng Liu Wave Character of Metrics and Hyperbolic Geometric Flow 2006 Physical background 0 Relation between Einstein equations and HGF Consider the Lorentzian metric d32 dt2 l 91739 cc tdaidaj Einstein equations in vacuum ie CH 2 0 become 829739 1 8909 8939 89k 1 2 pq W P4pq W 40 3 at R 29 8t 8t 9 8t 8t This is a special example of general version 2 of HGF Neglecting the terms of first order gives the HGF 1 3 is named as Einstein s hyperbolic geometric flow 0 Applications to cosmology singularity of universe Geometric background Structure of manifolds Singularities in manifolds Wave character of metrics and curvatures Longtime behavior and stability of manifolds Laplace equation heat equation and wave equation Laplace equation elliptic equations Au0 Heat equation parabolic equations ut Au20 Wave equation hyperbolic equations Nit AUZO Einstein manifold Ricci flow hyperbolic geometric flow Einstein manifold elliptic equations Rij A917 Ricci flow parabolic equations 891739 8t Hyperbolic geometric flow hyperbolic equations Laplace equation heat equation and wave equation on manifolds in the Ricci sense Geometric flows at ij at Vijgz39j 2Rz39j 0 where aZj ijnij are certain smooth functions on 1 which may depend on t aij In particular I i i ii la zj n 0 hyperbolic geometric flow m CW Oa ij 19 0 Ricci flow Cw Oa ij 0717 const Einstein manifold Birkhoff Theorem holds for geometric flows FuWen Shu and YouGen Shen Geometric flows and black holes arXiv grqc0610030 All of the known explicit solutions of the Einstein solu tions such as the Schwartzchild solution Kerr solution satisfy HGF At least for short time solutions there should be a 1 1 correspondence between solutions of HGF and the Ein stein equation Complex geometric flows If the underlying manifold 11 is a complex manifold and the metric is K ihler 2 7 7 8 gij bij 891739 8t at where aijbZjcZj are certain smooth functions on 1 which may also depend on t Cijgij 0 1H 3 Local Existence and Nonlinear Stability Local existence theorem Dai Kong and Liu 2006 Let 1 gfjm be a compact Riemannian manifold Then there exists a constant h gt 0 such that the initial value problem 3291quot 8t2 7t 2Rijt 0 o 891739 0 ko 921 37 gij7 at 7 ijw7 has a unique smooth solution gijwt on J x 0 h where kgja is a symmetric tensor on 1 W Dai D Kong and K Liu Hyperbolic geometric flow I short time existence and nonlinear stability Method of proof 0 Strict hyperbolicity Suppose gij 13 t is a solution of the hyperbolic geometric flow 1 and 1m J gt J is a family of diffeomorphisms of 1 Let 9ij967 t W Qz a t be the pullback metrics The evolution equations for the metrics 91196 t are strictly hyperbolic o Symmetrization of hyperbolic geometric flow Introducing the new unknowns 39 39 gij hij 7 gijJe 17 8t 82 we have a gij 8t hi kl agijvk kl ahi 8t amk 3h 39 k 1 9M7 Haw 8t 82 Rewrite it as Bu Bu 0 J A 107 A iniaa Bu where the matrices A M are symmetric Nonlinear stability Let 1 be a ndimensional complete Riemannian manifold Given symmetric tensors 92 and 9 on 1 we consider 329a at 91739 w 0 EM 692196 t 2Rijt 392739 at w 0 ambush where e gt 0 is a small parameter De nition The Ricci flat Riemannian metric yijwc possesses the locally nonlinear stability with respect to 921912 if there exists a positive constant 50 50gfjg j such that for any a 6 050 the above initial value problem has a unique local smooth solution gijt ac ijw is said to be locally nonlinear stable if it possesses the locally nonlinear stability with respect to arbitrary symmetric tensors 92196 and 996 with compact support Nonlinear stability theorem Dai Kong and Liu 2006 The flat metric gij 617 of the Euclidean space Rquot with n 2 5 is nonlinearly stable Remark The above theorem gives the nonlinear stability of the hyperbolic geometric flow on the Euclidean space with dimen sion larger than 4 The situation for the 3 4dimensional Eu clidean spaces is very different This is a little similar to the proof of the Poincar conjecture the proof for the three dimen sional case and n 2 5 dimensional case are very different while the four dimensional smooth case is still open Method of proof Define a 2tensor h 9ij967t 617 hij967t Choose the elliptic coordinates 95quot around the origin in Rquot It suffices to prove that the following Cauchy problem has a unique global smooth solution 32m quot 32m Bhkl 92l t h 7 m2 w gawkawk kl78w1 78wi 8wq ah hij 7 0 92j7 7 0 Egz39lj 8t Einstein s hyperbolic geometric ow 329739 1 3939 3939 89k 2R1 Pqu Iqu at 129 8t 8t 9 8t 8t satisfy the null condition 0 Global existence and nonlinear stability for small initial data Dai Kong and Liu 4 Wave Nature of Curvatures Under the hyperbolic geometric flow 1 the curvature tensors satisfy the following nonlinear wave equations 82Rz jkl W ARijkl lower order terms 8212 8t2 2 Ala lower order terms 82R W 2 AR lower order terms where A is the Laplacian with respect to the evolving metric the lower order terms only contain lower order derivatives of the curvatures Evolution equation for Riemannian curvature tensor Under the hyperbolic geometric flow 1 the Riemannian curva ture tensor Rim satisfies the evolution equation 82 Rijkl ARz jkl 2 Bijkl Bijlk Biljk Bum 9pq Rpjkqui Ripkquj Rijleqk Rijkqul 8 8 8 8 29pq grilth arzarga v where Bijkl gPqustiquTksl and A is the Laplacian with respect to the evolving metric Evolution equation for Ricci curvature tensor Under the hyperbolic geometric flow 1 the Ricci curvature ten sor satisfies 82 Rik ARM 139 Zyprgqupiqurs 2gqupquk a a a a l P 1 P 1 29 gm Wank 5315 89 a 89 89m 2 7p lq iRi 2 JP rq sl iRi 9 9 at at gkl g 9 9 at at gkl Evolution equation for scalar curvature Under the hyperbolic geometric flow 1 the scalar curvature satisfies 82 2 R AR2RIC 9 9 9 9 2k l p q p q 29 9191 Ma ank 89 8 2 1k JP lq 171 9 9 9 at at gkl 3k agpq 897 s 29 pgquT74Rikgngg at at at 5 Exact Solutions and Birkhott theorem 51 Exact solutions with the Einstein initial metrics De nition Einstein metric and manifold A Fliemannian metric 91739 is called Einstein if Bi 2 Agij for some constant A A smooth manifold M with an Einstein metric is called an Einstein mani fold If the initial metric 9170 96 is Ricci flat ie Rij07113 0 then gijt w gij0 w is obviously a solution to the evolution equation 1 Therefore any Ricci flat metric is a steady solution of the hyperbolic geometric flow 1 If the initial metric is Einstein that is for some constant A it holds Rij0n Agij0w V a 6 then the evolving metric under the hyperbolic geometric flow 1 will be steady state or will expand homothetically for all time or will shrink in a finite time Let 92701796 Ptgz39j07 By the defintion of the Ricci tensor one obtains Rijt Rij0 Agij0 Equation 1 becomes 8t2 2gij 0 This gives an ODE of second order 77 i i 7 4 l39 39 CM mm 2 as dt2 One of the initial conditions is p0 1 another one is assumed as p 0 v The solution is given by pt 2 At2 vt 1 General solution formula is gij t w 2 t2 vt cgij0 2 Remark This is different with the Ricci flow Case The initial metric is Ricci flat ie A 0 In this case pt 2 wt 1 4 If v 0 then gijtw gij0m This shows that gijtw gij0m is stationary If v gt 0 then gjtw 1 rutg39j0m This means that the evolving metric gjt w ptgj0 2 exists and expands homothetically for all time and the curvature will fall back to zero like Notice that the evolving metric gijt as only goes back in time to v1 when the metric explodes out of a single point in a big bang If v lt 0 then gjtw 1 rutgj0w Thus the evolving metric gjtw shrinks homothetically to a point as t T0 Note that when t To the scalar curvature is asymptotic to Tait This phenomenon corresponds to the black hole in physics Conclusion For the Ricci at initial metric if the initial velocity is zero then the evolving metric gij is stationary if the initial velocity is positive then the evolving metric gij exists and expands homothetically for all time if the intial velocity is negative then the evolving metric gij shrinks homo thetically to a point in a nite time Case II The initial metric has positive scalar curvature ie A gt 0 In this case the evolving metric will shrink if v lt 0 or first expands then shrink if v gt 0 under the hyperbolic flow by a timedependent factor p p 1 l T r 39 ND i t 1 t Case 39L ltj I Case t gt O Case Ill The initial metric has a negative scalar curvature ie A lt 0 In this case we devide into three cases to discuss Case1 v2 4A gt 0 a v lt 0 the evolving metric will shrink in a finite time under the hyperbolic flow by a timedependent factor b v gt 0 the evolving metric gijt ptgij0 exists and expands homothetically for all time and the curvature will fall back to zero like P P Case 391 lt2 I Case n gt U Case 2 v2 4A lt 0 In this case the evolving metric 9110296 2 ptgij0 2 exists and expands homothetically if v gt 0 or first shrinks then ex pands homothetically if v lt 0 for all time The scalar curvature will fall back to zero like tiz Case 3 v2 4A 2 0 If v gt 0 then evolving metric gijtw ptgij0 exists and expands homothetically for all time In this case the scalar curvature will fall back to zero like If v lt 0 then the evolving metric gijtw shrinks homothetically to a point as t Tk 1 gt 0 and the scalar curvature Is asymptotic to th Remark A typical example of the Einstein metric is 1 d32 2 dr2 7quot2dl5l2 r2 sin2 0dltp2 1 HT2 where n is a constant taking its value 1 0 or 1 We can prove that 1 ds2 R2 t dr2 7quot2dl5l2 r2 sin2 0dltp2 1 HT2 is a solution of the hyperbolic geometric flow 1 where R2t 2Iltt2 clt 62 in which c1 and c2 are two constants This metric plays an im portant role in cosmology 52 Exact solutions with axial symmetry Consider t 9 z where f g are smooth functions with respect to variables als2 ft7 zdz2 dw Mt zdy2 92t7 2dy2 7 Since the coordinates 1 and y do not appear in the preceding metric formula the coordinate vector fields 8i and 8 are Killing vector fields The flow 8m resp 8 consists of the coordinate translations that send 1 to 1 Aw resp y to y Ay leaving the other coordinates fixed Roughly speaking these isometries express the winvariance resp yinvariance of the model The winvariance and yinvariance show that the model possesses the zaxial symmetry HGF gives St Ht 0 1 2 2 1 2 f gz Hz EHJClt 02 where gz and Z satisfy 99 ggzuzzuf g H 0 Birkhoff Theorem holds for axialsymmetric solutions Angle speed u is independent of t 6 Dissipative hyperbolic geometric flow Let 1 be an ndimensional complete Riemannian manifold with Riemannian metric gij Consider the hyperbolic geometric flow 829739 3939 3939 89 89quot 1 2RZ 2 quJ d2 pq m 1 at J 9 8t at 9 8t 8t 1 9qu 2 1 9qu 9qu PG 7 ltcn lg 8t n 1 8t 8t 9 for a family of Riemannian metrics gijt on 1 where c and d are arbitrary constants By calculations we obtain the following evolution equation of the scalar curvature R with respect to the metric 9mm t 82R 8t2 69 8R AR 2R39 2 d2pq m 7 zcl g at at 1 9qu 2 1 9qu 9qu 7 mi R ltcn lltg 8t n 1 8t at 81 erg or or 8t 8t 8t at q p p q ik grip arkq 89m grip arkq 8 ik argq argk 8t 8t 8t 8t 8t 8t Zgikgjlgpq Zgikgjlgpq 8g Introduce y 3 9 qu Tr 89m 8t 8t and 2 8 T 8 s 8 Z 2 gpqgrs 9 gq 917 8t 8t 8t By 1 we have g 2B n 2 2 d 2 7 2 cu at n 1y y n 1 Global stability of Euclidean metric Dai Kong and Liu 2006 7 Open Problems 9 Yau has several conjectures about asymptotically flat mani folds with nonnegative scalar curvature HGF supplies a promis ing way to approach these conjectures Q Penrose cosmic censorship conjecture Given initial metric 92 and symmetric tensor kij study the singularity of the HGF with these initial data These problems are from general relativity and Einstein equa tion in which HGF has root 0 Global existence and singularity HGF has global solution for small initial data 9 HGF flow and minimal hypersurface Study HGF with initial data given by initial metric and second fundamental form hij There was an approach of geometrization by using Einstein equation which is too complicated to use HGF may simplify and even complete the approach Relative meanings and other unexpected applications of the synonymy calculus Yiannis N Moschovakis UCLA and MHAA Athens Deduction in semantics workshop 10 October 2007 Referential uses of definite descriptions Maureen Dowd in a recent opinion column in the New York Times uses President Bush Mr Bush W the President the Texas President he him to refer to the same person Claim In the context of the Dowd column these phrases are synonymous gt I would use them interchangeably in a report of the column gt I would use them interchangeably in a translation of the column to Greek gt In a translation of a Greek column to English I would translate nkowmo tpxng literally planet master as the US president or Busch Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 125 Donnellan 1966 on Linsky s example Her husband is kind to her asserted on seeing a man Smith treating kindly a spinster in the mistaken perhaps shared belief that he is her husband Donnellan It seems to me that we shall on the one hand want to hold that the speaker said something true but be reluctant to express this by quotIt is true that her husband is kind to herquot This shows I think a dif culty in speaking simply about quotthe statementquot When definite descriptions are used referentialy Claim In this utterrance her husband is synonymous With Smith and so the utterance is true Yiannis N Moschovakis Reiative meanings and other unexpected appiications of the Synonymy caicuius 225 Outline 1 Formal Fregean semantics and there will be remarks on them throughout 2 Referential intension theory 3 The significance and use of truth values 4 Meaning and synonymy relative to linguistic conventions 1 Sense and denotation as algorithm and Value 1994 2 A logical calculus of meaning and synonymy 2006 3 with E Kalyvianaki Two aspects of situated meaning submitted These are posted on my homepage httpwwwmathuclaeduynm Yiannis N Moschovakis Relative meanings and other unexpected applications of the synonymy calculus 325 Frege on sense the sense ofa sign may be the common property of many people Meanings are public abstract objects The sense of a proper name is grasped Wird erfasst by everyone Who is sufficiently familiar With the language Comprehensive knowledge of the thing denoted we never attain Speakers of the language know the meanings of terms The same sense has different expressions in different languages or even in the same language The difference between a translation and the original text should properly not overstep the level of the idea Faithful translation should preserve meaning Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 425 The methodology of formal Fregean semantics gt An interpreted formal language L is selected gt The rendering operation on a fragment of English English expression l informal context render a formal expression l state gt Semantic values denotations meanings etc are defined rigorously for the formal expressions of L and assigned to English expressions via the rendering operation gt Claim L should be a programming language Slogan English as a programming language Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 525 The typed A Calculus with recursion Li K An extension of the typed cacuus into which Montague39s Language of Intensional Logic LIL can be easily interpreted by Gallin Basic types b E e i t i s entities truth values states Types 0 IE b i 01 a 02 Abbreviation 01 x 02 a 739 E 01 a 02 a 7 Every non basic type is uniquely of the form O39EO391gtltgtlt039ngtb eveb O eve01 x x an a b maxevez717 evean 1 Yiannis N Moschovakis Reiative meanings and other unexpected apphcations of the Synonymy caicuius 625 Li K syntax Pure Variables v67 vf for each type a v a Recursive variables pg pf for each type a p a State parameters a for each state a for convenience only Constants A finite set K of typed constants Terms with assumed type restrictions and assigned types A a sziaipiciBiciiAiva iAo wherep1 A17 pn An CIG39BI039gtT gt BCIT vaBT gt VBI039gtT A0a gt Aowherep1A1pnAna Abbreviation AB7 C7 D EABCD Yiannis N Moschovakis Reiative meanings and other unexpected appiications of the Synonymy caicuius 725 Li K denotational semantics c We are given basic sets ifsFe and Tr Q Te for the basic types THAT the set of all functions f TO A T7 le Tb U L the flat poset of Tb Page the set of all functions f TO a PT Each P0 is a complete poset with the pointwise ordering c We are given an object c E Pa for each constant c a gt Pure variables of type a vary over TU recursive ones over R gt If A a and 7139 is a typerespecting assignment to the variables then denA7r 6 R gt Recursive terms are interpreted by the taking of least fixed points Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 825 Rendering natural language in Li K render Abelard loves Eloise gt lovesAbelardEloise 1 Bush is the president m eqBushthepresident E liar m p where p p t render truthteller gt p where p p t E s a 1 type of Carnap intensions e E s a 9 type of individual concepts Abelard7 Eloise7 Bush president 5 a ieq e x e a faithe 5 He deniar dentruthteler L Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 925 Can we say nonsense In LK Yes In particular we have variables over states so we can explicitly refer to the state even to two states in one term LlL does not allow this because we cannot do this in English Consider also the term A E rapidlytallJohn E John is rapidly tall John talls rapidly only A is already a LlL term gt Distinct grammatical categories are mapped onto the same type both in UL and in and so we can say nonsense in both formal languages and there is nothing wrong with this Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1025 Rendering natural language in Li K John stumbled and fell m Xltstumbledx amp fellxgt John predication after coordination This is in Montague39s LIL the Language of lntensional Logic as it is interpreted in LK John stumbled and he fell m stumbledj amp fellj where j John conjunction after co indexing The logical form of this sentence cannot be captured faithfully in LlL recursion models co indexing preserving logical form Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1125 Meaning in Li K V V V p p In slogan form The meaning ofa term A is faithfully modeled by an algorithm intA Which computes denA7r for every assignment 7139 The referential intension intA is compositionally determined from A intA is an abstract not necessarily implementable recursive algorithm which can be defined in LK Referential synonymy A z B ltgt intA N intA where N is a natural isomorphism relation between abstract recursive algorithms Claim Meanings are faithfully modeled Claim Synonymy is captured defined Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1225 Is this notion of meaning Fregean Evans in a discussion of Dummett39s similar computational interpretations of Frege39s sense quotThis leads Dummett to think generally that the sense of an expression is not a way of thinking about its denotation but a method or procedure for determining its denotation So someone Who grasps the sense ofa sentence Will be possessed of some method for determining the sentence39s truth value ideal verificationism there is scant evidence for attributing it to Fregequot Converse question If you posses a method for determining the truth value of a sentence A do you then grasp the sense of A Sounds more like Davidson rather than Frege Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1325 The Reduction Calculus Canonical Forms gt A reduction relation A a B is defined on terms of LK gt Each term A is effectively reducible to a unique up to congruence irreducible recursive term its canonical form A cfA E A0 wherep1 A17 pn An gt lintA denA0 denA1 denAnl gt The parts A07 An of A are irreducible explicit terms the truth conditions of A gt Claim cfA is the logical form ofA gt Synonymy Theorem A z B if and only if B cfB E B0 wherep1 B17pm Bm so that n m and for i g n denA denB Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1425 Utterances and local meaning gt A sentence is a closed parameter free term 5 t which denotes a Carnap intension ie a function from states to truth values gt An utterance is a pair 57a of a sentence 5 t and a state a it is expressed in LK by the term 5a t The local meaning of a sentence 5 at a state a is inta the referential intension of the utterance Local meanings are the objects of knowledge belief etc Every term of pure type 5 t is synonymous with an utterance 9a so that mathematical claims can be known believed etc Kalyvianaki introduces the factual content of a sentence 5 at a state a another semantic value which captures what 5 says about the world at state a V V V Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1525 Presuppositions and errors the King of France is bald m BKFE baldtheking of France gt What is the truth value of BKFa when a is today39s state b Frege would leave it undefined gt Russell would make it false gt Executed as a program BKF would return an error using the definition the unique X such that pb l gt Xa7 thepa if one such X exists7 er7 otherwise where er is a truth value signifying false presupposition39 Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1625 Meaning under presupposition BKF baldk where k thep7 p king off7 f France Execution of the algorithm intBKF successively computes 1 f denFrance so that for every state b fb France 2 p denking of France so that for every state b pb ltgt pb is king of France 3 k dentheking of France so that for every state b kb theking of Franceb 4 denBKF denbaldka baldka er Claim Knowing this algorithm is tantamount to understanding the utterance BKF and the truth value which is returned is of some but little significance Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1725 Truth values galore gt denthe king of France is balda er there is no king of France gt denhis wife is beautiful a er he has two wives gt denis snow white 3 7true gt denis the king of France balda 7er gt Perhaps also true if a b true a but b false er by which A but B 99 A and B for any A B if a true b false otherwise Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1825 Meaning and entailment Consider the following statement at the current state If Hamlet is bald then snow is black Is it true or false gt The entailment is problematic gt The meaning as algorithm is clear Claim Entailment is a poor guide to meaning and in many cases it is irrelevant The logic of natural language is certainly many valued and most likely quite complex Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 1925 Referential uses of definite descriptions Maureen Dowd in a recent opinion column in the New York Times uses President Bush Mr Bush W the President the Texas President he him to refer to the same person Claim In the context of the Dowd column these phrases are synonymous gt I would use them interchangeably in a report of the column gt I would use them interchangeably in a translation of the column to Greek gt In a translation of a Greek column to English I would translate nkowmo tpxng literally planet master as the US president or Busch Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 2025 Relativized meaning and synonymy gt A Iingustic convention is a tuple C C17 Ck W where 0 C17 c c c Ck are closed terms of the same type a o w 6 P0 gt The relativization of a term A to C is the term AC E AC1E 6 Qlt E c c a fresh constant7 c a gt The denotation ofA relative to C is denAC in the expanded language with Z W gt The referential intension of A relative to C is intAc gt Synonymy relative to a convention A we B ltgt Ac Z BC ltgt cfAc E A0 wherep1 A17 pn An cfBc E Bo wherep1 31 pn 8 and denA0 denBo7 7denAn denBn Yiannis N Moschovakis Relative meanings and other unexpected applications of the synonymy calculus 2125 Synonymy relative to a convention C1 7 C W gt IfC Bush the President he WaBush then Bush is ignorant we the President is ignorant zc gt IfC Hamlet7the Prince of Denmark7er then Hamlet was depressed we the Prince of Denmark was depressed In Greek immiuvo mi sdx y ltgt x and y are married to sisters gt If C pnut uvo txnkg brothers in law7 brothers in awthen O Nto tpxog mu 0 Qvo toong firm immiuvo m sg zc Niarchos and Onassis were brothers in law Yiannis N Moschovakis Reiative meanings and other unexpected appiications of the Synonymy caicuius 2225 Relative meaning and denotation gt If denC1 denCn W then denA denAC gt If denC1a denCna W and C17 7 C occur locally in A then denAa denAca gt In general denAa 7 denAca and it may be that Aa zc B but denAa 7 denBa Ywanms N Moschovakws Relamve meanmgs and other unexpected apphcamons of the Synonymy calculus 2325 Amendment to the basic setup The basic rendering operation becomes English expression l informal context m formal expression l linguistic conventions l state which determine global and local meaning relative to the conventions gt The relativization operation relative to a set of linguistic conventions is very similar to coordination but it uses a fresh constant rather than a variable Yiannis N Moschovakis Relative meanings and other unexpected applications of the Synonymy calculus 2425 Solving equations in algebra and in arithmetic Yiannis N Moschovakis UCLA and University of Athens Carnegie Mellon Summer School 26 June 2008 Outline We will consider equations px1xd 0 where px17 7Xd is a polynomial with integer coefficients in and of px X67X573X22X1 d1n6 px1X2X3 X15X2 7 X2X3 l 23X1X316 7 7 d 37 n 17 1 Algebra Are there real solutions of and which R the complete ordered field 07 73 r E R 2 Arithmetic Are there integer solutions of and which Z 7271012 gt l Logic Which is the more difficult problem7l Yiannis N Moschovakis Solving equations in algebra and m arithmetic 116 Alebraic equations in one unknown d 1 Equation It has solutions in R if The solutions are ax l b O a 7 O X 72 2X l 3 0 Yes X 7 aX2bxc0 72743620 X X23x10 3274520 Yes x3i px 0 Algorithm of approximation Sturm 1803 1855 algorithms X6 7 X5 4 solutions 17 z 17 38879 73X22X10 g 70 334734 71721465 Yiannis N Moschovakis Solving equations m algebra and m arithmetic 216 Polynomial long division Theorem For any two polynomials With rational coef cients fxgx if gx 3E O and degfx 2 deggx then there exists unique polys qx7 rx such that fx gxqx rx Where rx E O or degrx lt deggx With rx irx the division equation takes the form 00 gXqX x where again r X E O or degrx lt deggx Yiannis N Moschovakis Soiving equations m aigebra and m arithmetic 315 Sturm s algorithm for a real polynomial pX gt The Sturm sequence of p X p0x px7 p1x pX the derivative of px P0X P1Xq1X P200 P100 P2Xq2X P3X W p1xqr1x gt Wa the number of sign changes in the sequence p0017 p101 p201 p1oz for any real 04 prapb 7 0 then px has Wa 7 Wb roots in the interval 37 b Yiannis N Moschovakis Soivmg equations m aigebra and m arithmetic 415 An example thanks to Keith Matthews httpwwwnumbertheoryorgphpsturmhtml P000 p1X p200 p3X P400 psX P600 W72 W2 X67X573X22X1 6x5 7 5x4 7 6x 2 5x4 72x2 7 54x 7 38 12x3 719x2 7 2x 5 712053x2 8266X 5947 7107846X 63383 777249443861323 81 7258 438 7171 758797 279075 777249443861323 7 5 25 102 222 29 725733 7152309 777249443861323 7 1 number of roots in 722 5 71 4 gt The coefficients have been multiplied by some K gt These are all the real roots of this polynomial Yiannis N Moschovakis Solving equations m algebra and m arithmetic 516 Tarski39s algorithm Theorem Tarski 1930 There is an algorithm Which decides Whether an arbitrary elementary first order sentence of algebra is true or false Examples of elementary sentences of algebra gt The equation px O has 5 real solutions gt For all Y X17X2 7Xnp O or q gt 0 gt There exist real numbers 2 X17X2 7Xn such that p0and 0620 and qZO where p px17 7X 7 712 q are polynomials Yiannis N Moschovakis Solving equations in algebra and m arithmetic 616 The elementary first order sentences of algebra are the syntactically correct words finite sequences from the alphabet of 16 symbols 0 1 i 7 lt field operations not amp and V or sentential operators 3 there exists V for every quantifiers punctuation X l variablesxl xll For every number there is a bigger one English VX3yX lt y math English Vxl xllxl lt formal elementary sentence gt the variables are interpreted by real numbers in R Yiannis N Moschovakis Solving equations in algebra and m arithmetic 715 Analytic geometry X72y0 16x 71 16y 71 9 Ywanms N Moschovakxs So vmg equauons m a gebra and m amhmeuc 816 Euclidean geometry By using Cartesian coordinates the problems of Euclidean geometry are translated into algebra problems many of which can be further translated into elementary sentences hence Corollary Tarski 1930 Elementary Euclidean geometry is decidable ie there is an algorithm Which decides Whether an arbitrary elementary proposition of Euclidean geometry is true or false gt The circle of Apollonius gt The 3 point line and the 9 point circle of Euler gt gt There are also very substantial applications to computer graphics Yiannis N Moschovakis Solving equations in algebra and m arithmetic 915 Geometry intuitively simple sentences are not always elementary gt Elementary Every angle can be trisected Not elementary Every angle can be trisected using ruler and compass gt Elementary Every cube can be doubled Not elementary Every cube can be doubled using ruler and compass gt Not elementary The circle of radius 1 can be squared Because 7139 is not an algebraic number The elementary first order sentences of geometry are by definition those Which can be expressed in the first order language of algebra by the use of coordinates Yiannis N Moschovakis Solving equations m algebra and m arithmetic 1016 The elementary first order sentences of arithmetic are exactly the same as for algebra ie the syntactically correct words finite sequences from the alphabet of 16 symbols 0 1 l 7 lt field operations not amp and V or sentential operators 3 there exists V for every quantifiers punctuation X l variablesxl xll For every number there is a bigger one English VX3yX lt y math English Vxl xllxl lt formal elementary sentence gt But the variables are interpreted by integers in Z Yiannis N Moschovakis Solving equations m algebra and m arithmetic 1116 Algebra and arithmetic gt 2X l 3 O has a solution True in algebra X 7 False in arithmetic gt X l 2X3 l X2 l 5X l 6 O has a solution 2 solutions in algebra by Sturm or more simply The integer solutions must divide 6 so we try 07 i1 i2 i3 i6 and verify that the only integer solution is X 72 1216 Yiannis N Moschovakis Solving equations in algebra and m arithmetic Arithmetic is more difficult than algebra Theorem Andrew Wiles 1994 The equation X l y 2 has no integer solutions when n gt 2 This was conjectured in 1640 by Fermat who believed he had proved it only the proof did not fit in the margin of his notebook and so it is known as Fermat39s Last Theorem but no correct proof was known before Wiles39 in 1994 Yiannis N Moschovakis Solving equations m algebra and m arithmetic 1316 Prime numbers A number X gt1 is prime if it is divisible only by 1 and X Primes 2 3 5 7 11 13 17 192329 gt There are 1229 prime numbers lt 10000 gt There are infinitely many prime numbers Euclid A number X gt1 is a twin prime if both X and x 2 are primes Twin primes 35111729415971101107 gt There are 205 twin primes lt 10000 gt Are there infinitely many twin primes Open problem famously and apparently hopelessly for now Yiannis N Moschovakis Solving equations m algebra and m arithmetic 1416 Arithmetical truths Theorem Turing Church 1936 There is no algorithm Which decides for an arbitrary sentence of arithmetic Whether it is true or false in other words The problem of arithmetical truth is undecidable Theorem Matiyasevich 1970 Davis Putnam Robinson There is no algorithm Which decides for an arbitrary polynomial px17 Xn With integer coef cients Whether the equation pix1xn 0 has integer roots in other words Hilbert39s 10th problem is unsolvable Hilbert 1900 23 problems Which Will occupy the mathematicians of the 20th century Yiannis N Moschovakis Solving equations in algebra and m arithmetic 1516
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