Almost positive Ricci curvature in Kato sense -- an extension of Myers' theorem

TL;DR

This paper extends Myers' theorem by showing that small Kato constants of the negative Ricci curvature part imply volume bounds and manifold compactness, generalizing classical results.

Contribution

It introduces a new criterion involving Kato constants for Ricci curvature that generalizes the Bonnet-Myers theorem.

Findings

Small Kato constants lead to volume bounds

Generalization of Bonnet-Myers theorem

Connections to earlier curvature conditions

Abstract

It is shown that if the Kato constant of the negative part of the Ricci curvature below a positive level is small, then the volume of the corresponding manifold can be bounded above in terms of the Kato constant and the total Ricci curvature. Together with the results from [5] and [6], this yields a generalization of the famous Bonnet-Myers theorem. Connections to some earlier generalizations are discussed.