Length of a closed geodesic in 3-manifolds of positive scalar curvature

Abstract

Let $M$ be a closed $3$-dimensional Riemannian manifold with positive scalar curvature, $R_g\ge 6$. We show that $M$ contains a non-trivial closed geodesic of length less than $22500$. This confirms a conjecture of M. Gromov in dimension $3$.