Optimal Integral Pinching Results

TL;DR

This paper extends classical theorems relating curvature conditions to topological properties of manifolds, providing new integral pinching results and characterizations in higher dimensions.

Contribution

It generalizes the Bochner-Weitzenb"ock theorem for manifolds with integral curvature pinching, extending Gursky's results to higher degrees and dimensions.

Findings

Vanishing of Betti numbers under integral pinching conditions

Characterization of equality cases in integral pinching

Extension of Gursky's results to higher dimensions and degrees

Abstract

In this article, we generalize the classical Bochner-Weitzenb\"ock theorem for manifolds satisfying an integral pinching on the curvature. We obtain the vanishing of Betti numbers under integral pinching assumptions on the curvature, and characterize the equality case. In particular, we reprove and extend to higher degrees and higher dimensions a number of integral pinching results obtained by M. Gursky for four-dimensional closed manifolds.