Existence results for closed Finsler geodesics via spherical complexities
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Calculus of Variations
Existence results for closed Finsler geodesics via spherical complexities Stephan Mescher1 Received: 17 March 2020 / Accepted: 30 June 2020 © The Author(s) 2020
Abstract We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric yield the existence of multiple closed geodesics as well as upper bounds on their lengths. Mathematics Subject Classification Primary 58E10 · Secondary 53C60; 37C27
1 Introduction In [10], the author has introduced integer-valued homotopy-invariants based on spaces of continuous maps from spheres to topological spaces. Given a closed manifold M, these invariants can be used to estimate numbers of orbits of critical points of G-invariant differentiable functions on Hilbert submanifolds of C 0 (S n , M), where G is a subgroup of O(n + 1) and where we consider the O(n + 1)-action by reparametrization, i.e. the one induced by the standard O(n + 1)-action on S n . For n = 1, this leads to estimates of critical orbits of O(2)-invariant or S O(2)-invariant functions on Hilbert manifolds of free loops in M. A typical situation in which this method applies is the study of energy functionals of Riemannian or Finsler metrics on H 1 (S 1 , M), the Hilbert manifold of free loops in M that is locally modelled on the Sobolev space H 1 (S 1 , Rdim M ) = W 1,2 (S 1 , Rdim M ). The critical points of such functionals are precisely the closed geodesics of the metric under consideration and methods from Morse theory and Lusternik-Schnirelmann theory have been very successfully implemented to derive existence results for closed geodesics in the previous decades, we refer e.g. to [13] for an overview. Here, we only want to present certain recent results that are close to or intersect with the results shown in this note. In the following, we will always let λ denote the reversibility of the Finsler metric under consideration, see [15]
Communicated by J.Jost.
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Stephan Mescher [email protected] Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany 0123456789().: V,-vol
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or [14] for the definition of reversibility. For x ∈ R we further let x denote the smallest integer that is bigger than or equal to x. • In [16, Theorem 3], H.-B. Rademacher has shown that any Finsler metric on S n whose n λ2 flag curvature satisfies (1+λ) 2 < K ≤ 1 admits 2 − 1 distinct prime closed geodesics whose lengths are less than 2nπ. • Rademacher has further shown that any Finsler metric on S 2n , n ≥ 3, whose flag curvature 1 satisfies (n−1) 2 < K ≤ 1 admits two distinct prime closed geodesics of length less than 2nπ. • In [7] and [3], Y. Long and H. Duan have proven that every Finsler
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