Closed Geodesics in Alexandrov Spaces of Curvature Bounded from Above
In this paper, we show a local energy convexity of W1,2 maps into CAT(K) spaces. This energy convexity allows us to extend Colding and Minicozzi’s width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.
KeywordsWidth Sweepout Min-max Closed geodesic Alexandrov space of curvature bounded from above
Mathematics Subject Classification (2000)53C23 58E10 53C22
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- 1.Alexandrov, A.D.: A theorem on triangles in a metric space and some of its applications. Trudy Mat. Inst. Steklov. 38, 5–23 (1951) Google Scholar
- 3.Almgren, F.J.: The Theory of Varifolds. Mimeographed Notes. Princeton University Press, Princeton (1965) Google Scholar
- 4.Berg, I.D., Nikolaev, I.G.: On an Extremal Property of Quadrilaterals in an Alexandrov Space of Curvature ≤K. Contemporary Mathematics, vol. 424. American Mathematical Society, Providence (2007) Google Scholar
- 11.Colding, T.H., De Lellis, C.: The min-max construction of minimal surfaces. Surv. Differ. Geom. 8, 75–107 (2003). Lectures on Geometry and Topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, May 3–5, 2002, Sponsored by JDG Google Scholar
- 16.Hadamard, J.: Les surfaces à courbures opposées et leurs lignes géodésique. J. Math. Pure Appl. 4, 27–75 (1898) Google Scholar
- 19.Lyusternik, L.A.: Topology of Functional Spaces and Calculus of Variations in the Large. Trudy Inst. Steklov., vol. 19, Izdat. Akad. Nauk SSSR, Moscow (1947) (in Russian); translated from the Russian by J.M. Danskin. Translations of Mathematical Monographs, vol. 16. American Mathematical Society, Providence (1966). vii+96 pp Google Scholar
- 20.Lyusternik, L.A., Šnirel’man, L.: Topological methods in variational problems and their application to the differential geometry of surfaces. Usp. Mat. Nauk (N.S.) 1(17), 166–217 (1947) (in Russian) Google Scholar
- 21.Lin, L., Wang, L.: Existence of good sweepouts on closed manifolds. Proc. Am. Math. Soc., to appear Google Scholar
- 25.Serbinowski, T.: Harmonic maps into metric spaces with curvature bounded above. PhD thesis, University of Utah (1995) Google Scholar