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 $|Ric|\leq 3$, volume $vol(M)>v>0$, and diameter $diam(M)<D$, 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.