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

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.