A sphere theorem for three dimensional manifolds with integral pinched curvature

TL;DR

This paper extends rigidity and sphere theorems to three-dimensional manifolds with integral pinched curvature using an adapted Bochner technique and topological classification, providing optimal geometric conditions.

Contribution

It introduces a new application of integral Bochner techniques to three-dimensional manifolds, establishing optimal sphere theorems based on integral curvature conditions.

Findings

Proved rigidity results for 3D manifolds with integral pinched curvature.

Established optimal sphere theorems for three-dimensional manifolds.

Utilized classification of nonnegative scalar curvature manifolds in proofs.

Abstract

In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend the results in dimension three. Then, by using the classification of closed three-manifolds with nonnegative scalar curvature and a few topological considerations, we deduce optimal sphere theorems for three-dimensional manifolds with integral pinched curvature.