Length of a shortest closed geodesic in manifolds of dimension four

Nan Wu, Zhifei Zhu

Published 2017-02-22Version 1

In this paper, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric|>3$, 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 is based on a recent theorem of diffeomorphism finiteness of the manifolds satisfying the above conditions proven by J. Cheeger and A. Naber.