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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

The length of a shortest closed geodesic and the area of a $ 2$-dimensional sphere


Author: R. Rotman
Journal: Proc. Amer. Math. Soc. 134 (2006), 3041-3047
MSC (2000): Primary 53C22; Secondary 58E10
Posted: April 10, 2006
MathSciNet review: 2231630
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a Riemannian manifold homeomorphic to $ S^2$. The purpose of this paper is to establish the new inequality for the length of a shortest closed geodesic, $ l(M)$, in terms of the area $ A$ of $ M$. This result improves previously known inequalities by C.B. Croke (1988), by A. Nabutovsky and the author (2002) and by S. Sabourau (2004).

References

  • 1. R. Bott, Lectures on Morse theory, old and new, Bull. Am. Math. Soc., 7 (1982), 331-358. MR 0663786 (84m:58026a)
  • 2. C.B. Croke, Area and the length of the shortest closed geodesic, J. Diff. Geom., 27 (1988), 1-21. MR 0918453 (89a:53050)
  • 3. C. B. Croke and M. Katz, Universal volume bounds in Riemannian manifolds, Surveys in differential geometry, VIII (2002), 109-137. MR 2039987 (2005d:53061)
  • 4. E. Calabi and J. Cao, Simple closed geodesics on convex surfaces, J. Diff Geom., 36 (1992), 517-549. MR 1189495 (93h:53039)
  • 5. A. Nabutovsky and R. Rotman, The length of a shortest closed geodesic on a $ 2$-dimensional sphere, IMRN, 2002 (2002), no. 39, 2121-2129.
  • 6. A. Nabutovsky and R. Rotman, Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem, J. Eur. Math. Soc., 5 (2003), 203-244. MR 2002213 (2004f:53043)
  • 7. A. Nabutovsky and R. Rotman, Volume, diameter and the minimal mass of a stationary $ 1$-cycle, GAFA, 14 (2004), 748-790. MR 2084979 (2005g:53069)
  • 8. J. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Ann. Math. Studies, 27 (1981), Princeton University Press. MR 0626027 (83e:49079)
  • 9. S. Sabourau, Filling radius and short closed geodesics of the $ 2$-sphere, Bull. Soc. Math. France, 132 (2004), 105-136. MR 2075918 (2005g:53065)

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Additional Information

R. Rotman
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 -- and -- Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: rotman@math.psu.edu, rina@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08297-9
PII: S 0002-9939(06)08297-9
Keywords: Geometric inequalities, closed geodesics
Received by editor(s): February 24, 2005
Received by editor(s) in revised form: April 14, 2005
Posted: April 10, 2006
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2006 American Mathematical Society




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