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# Topics in Topology MATH 697

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This 4 page Class Notes was uploaded by Mrs. Preston Lehner on Thursday October 29, 2015. The Class Notes belongs to MATH 697 at University of Michigan taught by Geoffrey Scott in Fall. Since its upload, it has received 16 views. For similar materials see /class/231480/math-697-university-of-michigan in Mathematics (M) at University of Michigan.

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Date Created: 10/29/15

Minimal surfaces in 3manifolds Math 697 Juan Souto 312008 Goal of this and the next couple of classes is to prove Theorem 1 Birkhoff Every closed Riemannian manifold contains a closed geodesic And the following more quantitative version for surfaces Theorem 2 Nabutovsky Rotman Every closed surface S has a closed geodesic with at most length 4diamS Let M be a closed manifold with Riemannian metric p and let V be the associated Levi Civita connection Given a smooth curve 77 01 a M de ne the length and the energy as follows M77 xpltnlttgtnlttgtgtdt Em pltnlttgtn lttgtgtdt It follows from the Cauchy Schwarz inequality that lM772 S EM77 with equality if 77 has constance velocity p77 t 77 t Observe that while the length of a curve is independent of the parametrization this is not true for the energy The distance dMxy between two points my 6 M the in mum of the lengths lM77 of curves 77 with 770 z and 771 y Question 1 Prove that M dM is a compact metric space Question 2 Given a continuous curve 77 01 a M we can de ne its dM length to be the supremum of 2f1dM77t77t71 over all partitions 0 to lt t1 lt lt tk 1 of the interval 01 Prove that if77 is smooth both de nitions coincide Question 3 Prove that for all my 6 M there is a Z Lipschitz curve 77 0dxy a M with 770 z and 77dxy y Prove that 77 is smooth Question 4 Prove that if a smooth curve yt minimizes the energy then it satis es the geodesic differential equation V 01 7 t 0 lt gt chm where V is the Levi Civita connection of M p Hint First variation formula Prove conversely that there is some 6 depending only on Mp such that if a curve 7 01 a M satis es 01 and has Ep y S 6 then 7 minimizes the energy 1 A curve 7 in M satisfying the geodesic differential equation is a geodesic If the domain of y is an interval in M then 7 is a geodesic arc If the domain is SI then 7 is a closed geodesic Using either the full answer to question 2 or the Picard Lindelo39f theorem and the last question we deduce Lemma 3 Existence of short arcs There is 6 such that if my 6 M are at most at distance 6 then there is a geodesic with unit speed 7 0dxy a M with 70 z and ydxy y Under a piece wise geodesic curve we understand a curve 7 17 b a M such that every point z 6 a7 b is contained perhaps in the bound ary of some interval a b such that the restriction of y to a b is geodesic In other words7 it is the concatenation of nitely many geodesics If 7 consists of at most h segment and each segment has at most energy 6 then we say that it is a 67 k piece wise geodesic curve Question 5 Prove that every two points can be joined by a piece wise geodesic curve Question 6 Prove that every smooth curve 7 can be approached by piece wise geodesic curves m with a E y l yl a l y Lemma 4 Straightening Let E be as in Lemma 3 Then for all e there is k such that every curve 7 with at most energy e is homotopic relative endpoints to a 67 k piece wise geodesic 77 with Ei7 S Question 7 Given 6 and h prove that the set of all eh piece wise quasi geodesics is compact Proposition 5 Every arc is homotopic relative to its end points to a geodesic arc Proof Take a minimizing sequence and7 using the lemma7 assume that they are e7 k piece wise quasi geodesics for some uniform choices of e and h Take a limit and prove that it is geodesic A similar argument applies now for closed curves and hence we ob tain Proposition 6 Every closed curve 7 is homotopic to a closed geodesic 77 which minimizes energy in the homotopy class The following observation shows that7 if 7T1 31 1 then it contains relatively small homotopically essential curves Lemma 7 Gromov Fip z E M Then 7T1M p is generated by loops of at most length 2diamM Here diamM max dMxy is the diameter of M 3 Combining the last proposition with Gromov7s observation we ob tain Corollary 8 Assume that M is not simply connected Then M con tains a closed geodesic of at most length 2 diamM Continuing with the same notation7 assume that 7T1M 1 and recall that Lemma 9 There is some h 2 2 with WkM 31 0 Assume for the sake of simplicity that h 2 and consider the set Z of maps fSlgtlt01aM which map fSl gtlt and fSl gtlt to points A map f E Z is homotopically trivial if it is by a homotopy within Z Let Zak be the set of those f E Z such that for all s the curve fSl gtlt is 6 k piece wise geodesic and de ne the energy of f E Zak to be 5f max EfSl X t te01 The straightening process above yields Proposition 10 There are 6 h and f E Z k which is a homotopically essential Question 8 Prove that there is 6 such that for all h and E and every f E Zak with 8f S 6 is homotopically trivial Fixing 6 k as in Proposition 10 take now a sequence of homo topically essential elements in Z k with limS inf S l f 1623 f Without loss of generality we may assume that for all i we have 50239 8fiSl gtlt The sequence with m j lSlL gtlt 05 is a so called minimacc sequence One shows that up to choice of a subsequence the curves 7 converge to a closed geodesic in M This shows the existence of a closed geodesic in the simply connected case concluding the proof of Birkhoff7s theorem Nabutovsky Rotman7s bound By Birckhoff7s theorem together with Gromov7s observation we can assume that out surface M2 is simply connected In other words7 M2 4 is the sphere We can also assume that M does not contained closed geodesics of length less than 4diam The proof is similar to the one of Birckhoff7s theorem Assume that there is no closed geodesic shorter than 4 diamM Additional remarks 1 It was conjectured that the constant in the Nabutovsky Rotman A D C40 4 V Cf theorem should be 2diamM but recently Balacheff Croke Katz have constructed counterexamples It is completely open if such bounds exist in higher dimensions Nabutovsky Rotman have quite a few results about this but they mainly treat rst cousins of geodesics like for example some types of graphs The case of geodesics is7 to my knowledge7 completely open The shortening process above is the so called curve shortening flow In order to see a computer animation of that you can go to Peter Scott7s web page and follow the links You can also look at Apart from the curve shortening ow7 there is a more or less equivalent approach to all this using Morse theory for the energy functional on the loop space The problem is that there are loops with in nite energy So7 one needs to decide on which subset of the loop space one is considering the energy functional Milnor 3 uses the same piece wise geodesic curves we work with here See 5 for a completely self contained account A different approach is much more functorial and much more complicated one can use the energy function to complete the space of smooth curves and get a so called Hilbert manifold on which the Energy is de ned and smooth and then one plays Morse theory This approach can be found in Klingenberg7s approach is much more complicated than the one we did here but it is quite pedagogical The point is that one is forced to use some tools of analysis like the Sobolev embedding lemma and so on I personally nd these things scary but in the case of dimension 1 they are quite easy REFERENCES 1 Jr Hass and Pi Scott7 Shortening curves on surfaces7 Topology 33 19947 not 17 2543 2 VW Klingenberg7 Riemannz39an geometry Second edition7 de Gruyteri 3 Jr Milnor7 Morse theory7 Annals of Mathematics Studies7 Not 51 Princeton University Press

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