Comparison geometry for integral radial Bakry-\'Emery Ricci tensor bounds

TL;DR

This paper establishes mean curvature, volume, diameter, and eigenvalue comparison theorems on smooth metric measure spaces under integral bounds on the Bakry-Émery Ricci tensor, generalizing previous pointwise results.

Contribution

It introduces new comparison theorems under integral bounds for the Bakry-Émery Ricci tensor, extending classical pointwise bounds to integral conditions.

Findings

Mean curvature comparison under integral bounds

Volume comparison results

Diameter and eigenvalue estimates

Abstract

In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-\'Emery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author's previous results under integral Bakry-\'Emery Ricci tensor bounds, but also are generalizations of the Wei-Wylie's pointwise results.