Geometriae Dedicata

, Volume 116, Issue 1, pp 1–36 | Cite as

Closed Geodesics on Incomplete Surfaces



We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used


minimal surfaces geodesics toric 

Mathematics Subject Classifications (2000)

53A10 57N10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abreu M. Kahler geometry of toric manifolds in symplectic coordinates. In: Symplectic and Contact Topology: Interactions and Perspectives (Toronto/Montreal, 2001), Amer. Math. Soc. Providence, RI, Fields Inst. Comm. 35 2003, 1–24Google Scholar
  2. 2.
    F.J. Almgren Jr., The Structure of Limit Varifolds Associated with Minimizing Sequences of Mappings. London: Academic Press (1974).Google Scholar
  3. 3.
    K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem. Boca Raton FL: Chapman & Hall/CRC (2001).MATHGoogle Scholar
  4. 4.
    T. Delzant, Hamiltoniens périodiques et image convex de l'application moment. Bull. Soc. Math. France 116 (1988) 315-339MATHMathSciNetGoogle Scholar
  5. 5.
    M.E. Gage, Deforming curves on convex surfaces to simple closed geodesics. Indiana Univ. Math. J. 39 (1990) 1037-1059CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    V. Guillemin, Kahler structures on toric varieties. J. Differential Geom. 40 (1994) 285-309MATHMathSciNetGoogle Scholar
  7. 7.
    Guruprasad, K. and Haefliger, A.: Closed geodesics on orbifolds, arXiv:math.DG/ 0306238.Google Scholar
  8. 8.
    W.-T. Hsiang, W.-Y. Hsiang and P. Tomter, On the existence of minimal hyperspheres in compact symmetric spaces. Ann. Sci. Ecole Norm. Sup 21 (1988) 287-305MATHMathSciNetGoogle Scholar
  9. 9.
    W.-Y. Hsiang and H.B. Lawson, Minimal submanifolds of low cohomogeneity. J. Differential Geom. 5 (1971) 1-38MATHMathSciNetGoogle Scholar
  10. 10.
    J. Hass and P. Scott, Shortening curves on surfaces. Topology 33 (1994) 25-43CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    J. Milnor, Morse Theory. Princeton NJ: Princeton University Press (1963).MATHGoogle Scholar
  12. 12.
    J.T. Pitts, Regularity and Singularity of One Dimensional Stationary Integral Varifolds on Manifolds Arising from Variational Methods in the large. London: Academic Press (1974).Google Scholar
  13. 13.
    Pitts, J. T. and Rubinstein, J. H.: Applications of minimax to minimal surfaces and the topology of 3-manifolds In: Miniconference on Geometry and Partial Differential Equations, 2 (Canberra, 1986), Proc. Centre Math. Anal. ANU, 12, 1987, pp. 137–170Google Scholar
  14. 14.
    B. White, The space of minimal submanifolds for varying Riemannian metrics. Indiana Univ. Math. J. 40 (1991) 161-200CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MelbourneMelbourneAustralia

Personalised recommendations