Yamabe flow and the Myers-type theorem on complete manifolds

TL;DR

This paper proves a Myers-type theorem for complete, locally conformally flat Riemannian manifolds with bounded Ricci curvature satisfying a Ricci pinching condition, showing such manifolds must be compact.

Contribution

It establishes a new compactness criterion for complete manifolds under Ricci pinching conditions, extending classical Myers theorems.

Findings

Complete locally conformally flat manifolds with Ricci pinching are compact.

The Ricci curvature bound implies manifold compactness.

The proof involves Yamabe flow techniques.

Abstract

In this paper,we prove the following Myers-type theorem: if $(M^n,g)$, $n\geq 3$, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition $Rc\geq \epsilon Rg>0$, where $\epsilon>0$ is an uniform constant, then $M^n$ must be compact.