# Length of a shortest closed geodesic in manifolds of dimension four

## TL;DR

This paper establishes an upper bound on the length of the shortest closed geodesic in certain 4-dimensional manifolds with bounded Ricci curvature, volume, and diameter, using recent finiteness theorems.

## Contribution

It provides a new bound on shortest closed geodesic length for 4D manifolds based on Ricci curvature, volume, and diameter, extending geometric analysis results.

## Key findings

- Bound on shortest closed geodesic length in 4D manifolds
- Dependence of bound on volume and diameter
- Application of recent diffeomorphism finiteness theorem

## 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.