Journal of Geometric Analysis

, Volume 21, Issue 2, pp 429–454 | Cite as

Closed Geodesics in Alexandrov Spaces of Curvature Bounded from Above



In this paper, we show a local energy convexity of W 1,2 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.


Width Sweepout Min-max Closed geodesic Alexandrov space of curvature bounded from above 

Mathematics Subject Classification (2000)

53C23 58E10 53C22 


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  1. 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
  2. 2.
    Alexandrov, A.D.: Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. 1, 33–84 (1957) MathSciNetGoogle Scholar
  3. 3.
    Almgren, F.J.: The Theory of Varifolds. Mimeographed Notes. Princeton University Press, Princeton (1965) Google Scholar
  4. 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
  5. 5.
    Birkhoff, G.D.: Dynamical systems with two degrees of freedom. Trans. Am. Math. Soc. 18(2), 199–300 (1917) MathSciNetMATHGoogle Scholar
  6. 6.
    Birkhoff, G.D.: Dynamical Systems. AMS Colloq. Publ., vol. 9. American Mathematical Society, Providence (1927) MATHGoogle Scholar
  7. 7.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence (2001) MATHGoogle Scholar
  8. 8.
    Busemann, H.: The Geometry of Geodesics. Academic Press, San Diego (1955) MATHGoogle Scholar
  9. 9.
    Cartan, E.: Leçons sur la Géométrie des Espaces de Riemann, 2nd edn. Gauthier-Villars, Paris (1928) 1951 MATHGoogle Scholar
  10. 10.
    Calabi, E., Cao, J.: Simple closed geodesics on convex surfaces. J. Differ. Geom. 36(3), 517–549 (1992) MathSciNetMATHGoogle Scholar
  11. 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
  12. 12.
    Colding, T.H., Minicozzi, W.P. II: Width and mean curvature flow. Geom. Topol. 12(5), 2517–2535 (2008) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Colding, T.H., Minicozzi, W.P. II: Width and finite extinction time of Ricci flow. Geom. Topol. 12(5), 2537–2586 (2008) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Colding, T.H., Minicozzi, W.P. II: Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman. JAMS 18(3), 561–569 (2005) MathSciNetMATHGoogle Scholar
  15. 15.
    Croke, C.B.: Area and the length of the shortest closed geodesic. J. Differ. Geom. 27(1), 1–21 (1988) MathSciNetMATHGoogle Scholar
  16. 16.
    Hadamard, J.: Les surfaces à courbures opposées et leurs lignes géodésique. J. Math. Pure Appl. 4, 27–75 (1898) Google Scholar
  17. 17.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). xii+544 pp ISBN: 0-521-79160-X; 0-521-79540-0 MATHGoogle Scholar
  18. 18.
    Korevaar, N., Schoen, R.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom. 1, 561–659 (1993) MathSciNetMATHGoogle Scholar
  19. 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. 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. 21.
    Lin, L., Wang, L.: Existence of good sweepouts on closed manifolds. Proc. Am. Math. Soc., to appear Google Scholar
  22. 22.
    Pitts, J.T.: Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Princeton University Press/University of Tokyo Press, Princeton/Tokyo (1981) MATHGoogle Scholar
  23. 23.
    Poincaré, H.: Sur les lignes g’eodésiques des surfaces convexes. Trans. Am. Math. Soc. 6, 237–274 (1904) CrossRefGoogle Scholar
  24. 24.
    Reshetnyak, Y.G.: Non-expanding mappings in a space of curvature not greater than K. Sib. Mat. Zh. 9, 918–927 (1968). (in Russian) English translation: Sib. Math. J. 9, 683–689 (1968) MATHGoogle Scholar
  25. 25.
    Serbinowski, T.: Harmonic maps into metric spaces with curvature bounded above. PhD thesis, University of Utah (1995) Google Scholar

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© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

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