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

Abstract

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 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

The Length of the Shortest Closed Geodesic on a Surface of Finite Area

“…It was shown by Croke in [11] that l(M ) ≤ 31 √ A in the case that M is a sphere or a once-punctured sphere. For the spherical case, this estimate was further developed in [8], and has since been refined to l(M ) ≤ 4 √ 2A [14,16,19]. We prove that this sharper estimate also holds for the punctured sphere.…”

“…To that end, we prove the following lemma, a sharpening of the bound provided in [11] (cf. [16]). This lemma ensures the existence of short geodesic loops encircling the ends of M , as described in Lemma 3.…”

“…Secondly, we improve the bound on a complete, non-compact Riemannian manifold homeomorphic to a sphere with two punctures. Our improvements are due to the following modifications: an optimal use of the co-area formula as in [16]; utilizing min-max techniques on the space of 1-cycles instead of on the space of closed, piecewise differentiable curves on M , as in [8,14,16,19]; and using a length shortening technique on geodesic nets (which we realize as pairs of loops) as in [17,18]. Note that the main difficulty of proving the existence of (short) periodic geodesics on complete, non-compact manifolds manifests itself in the curve "running away to infinity" under the length shortening flow.…”

Untitled

“…More precisely, we are first able to recover the result of C. Croke [1988] on the existence of short closed geodesics for Riemannian two-spheres: we will deduce from Theorem 1.2 that any Riemannian two-sphere with unit area carries a closed geodesic of length at most 10.1; see Theorem 2.7. This is not as good as the current best constant, due to R. Rotman [2006] and equal to 4 √ 2 5.7, but it is not too far from it. Moreover, using Theorem 1.2, we can also recover Theorem VI of [Alvarez Paiva et al 2013] on the existence of a short closed geodesic for Finsler (eventually nonreversible) two-spheres.…”

An inequality for length and volume in the complex projective plane

“…where sys 2 (x) denotes the infimum of areas of surfaces representing the class x ∈ H 2 (M; Z). Gromov's stable systolic inequality [13] (see [5], [29]), for an arbitrary metric on the 2-sphere. The multiplicative constant 32 is not believed to be optimal.…”