The length of the shortest closed geodesic in a closed Riemannian 3-manifold with nonnegative Ricci curvature (Q2814418)

For any compact Riemannian 3-manifold \(M\), with volume \(\text{Vol}_g M\), denote the length of its shortest closed geodesic as \(\text{Sys}_g M\). The authors prove that there is a constant \(C>0\) so that, for any compact Riemannian 3-manifold \(M\) with nonnegative Ricci curvature, either NEWLINE\[NEWLINE \text{Sys}_g M \leq C (\text{Vol}_g M)^{1/3} NEWLINE\]NEWLINE or \(M\) is diffeomorphic to the 3-sphere, and there is a closed embedded minimal surface \(\Sigma \subset M\) of index 1 with area NEWLINE\[NEWLINE \text{Area}_g \Sigma \leq C (\text{Vol}_g M)^{2/3}. NEWLINE\]NEWLINE This short paper uses older results of Gromov and more recent results of Rotman and Sabourau.