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Lengths of three simple periodic geodesics on a Riemannian 2-sphere

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Abstract

Let M be a Riemannian 2-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on M. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed 20d, where d is the diameter of M. We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem. Finally, we present better bounds for these three lengths for “thin” spheres, when the area A is much less than \(d^2\), where the bounds for the lengths of the first two simple periodic geodesics are asymptotically optimal, when \({A\over d^2}\longrightarrow 0\).

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Acknowledgments

This research was partially supported by NSERC Discovery Grant 155879-12 and NSERC Discovery Accelerator Supplement Grant of Alexander Nabutovsky and NSERC Discovery Grant 217655-13 of Regina Rotman.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S2E4, Canada

    Yevgeny Liokumovich, Alexander Nabutovsky & Regina Rotman

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  1. Yevgeny Liokumovich
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  2. Alexander Nabutovsky
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  3. Regina Rotman
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Correspondence to Alexander Nabutovsky.

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Liokumovich, Y., Nabutovsky, A. & Rotman, R. Lengths of three simple periodic geodesics on a Riemannian 2-sphere. Math. Ann. 367, 831–855 (2017). https://doi.org/10.1007/s00208-016-1402-5

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  • DOI: https://doi.org/10.1007/s00208-016-1402-5

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