Length of a shortest closed geodesic in manifolds of dimension four

Abstract

In this paper, we show that for any closed $4$-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $\lvert \operatorname{Ric} \rvert \leq 3$, volume $\textrm{vol} (M) \gt v \gt 0$, and diameter $\textrm{diam} (M) \leq D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$.

The proofs of our result are based on a recent theorem of diffeomorphism finiteness of the manifolds satisfying the above conditions proven by J. Cheeger and A. Naber.