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

By R. Rotman and Communicated Jon G. Wolfson


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). Let l(M) denote the length of a shortest closed non-trivial geodesic on a closed Riemannian manifold M and let A be the area of M. In this paper we will prove the following theorem. Theorem 0.1. Let M be a manifold diffeomorphic to the 2-dimensional sphere. Then l(M) ≤ 4 √ 2 √ A. The first upper bounds for the length of a shortest closed geodesic on a 2-dimensional sphere were found by C.B. Croke (see [2]). In his paper Croke found estimates both in terms of the diameter and in terms of the area of a 2-dimensional sphere. Those results were later improved in [5] and in [9]. In particular, Croke proved that l(M) ≤ 31 √ A, Sabourau proved that l(M) ≤ 12 √ A, and A. Nabutovsk

Year: 2013
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