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The length of the shortest closed geodesic in a closed Riemannian $ 3$-manifold with nonnegative Ricci curvature


Authors: Ezequiel Barbosa and Yong Wei
Journal: Proc. Amer. Math. Soc. 144 (2016), 4001-4007
MSC (2010): Primary 53C42, 53C22
DOI: https://doi.org/10.1090/proc/13042
Published electronically: March 16, 2016
MathSciNet review: 3513555
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Abstract: In this note we discuss the problem of finding an upper bound on the length of the shortest closed geodesic in a closed Riemannian 3-manifold in terms of the volume. More precisely, we show that there exists a positive universal constant $ C$ such that, for every Riemannian 3-manifold $ (M^3,g)$ with $ Ric_g\geq 0$, at least one of the following assertions holds: (i). $ Sys_g(M)\leq C Vol_g(M)^{\frac 13}$, where $ Sys_g(M)$ denotes the length of the shortest closed geodesic in $ M^3$; (ii). $ M^3$ is diffeomorphic to $ \mathbb{S}^3$ and there exists a closed minimal surface $ \Sigma _0$ embedded in $ M^3$, with index 1, and $ A_g(\Sigma _0)\leq C Vol_g(M)^{\frac {2}{3}}$. This gives a partial answer to the problem proposed in Gromov's paper written in 1983.


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Ezequiel Barbosa
Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
Email: ezequiel@mat.ufmg.br

Yong Wei
Affiliation: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
Email: yong.wei@ucl.ac.uk

DOI: https://doi.org/10.1090/proc/13042
Keywords: Closed geodesic, systolic inequality, nonnegative Ricci curvature, minimal surfaces.
Received by editor(s): November 7, 2014
Received by editor(s) in revised form: June 9, 2015, and October 30, 2015
Published electronically: March 16, 2016
Communicated by: Michael Wolf
Article copyright: © Copyright 2016 American Mathematical Society