The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature

TL;DR

This paper establishes bounds on the first cohomology group of compact manifolds using integral Ricci curvature conditions and the Kato constant, providing quantitative insights beyond previous qualitative results.

Contribution

It introduces a method to estimate the dimension of the first cohomology group based on integral Ricci curvature bounds and the Kato constant, offering new quantitative results.

Findings

Bound on the first cohomology dimension in terms of Kato constant

Quantitative estimates contrasting previous qualitative results

Extension of Ricci curvature analysis to integral bounds

Abstract

We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, contrary to results by Elworthy and Rosenberg.