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

TL;DR

This paper extends comparison geometry results to smooth metric measure spaces with integral bounds on the Bakry-Émery Ricci tensor, leading to new diameter, eigenvalue, and volume growth estimates.

Contribution

It generalizes existing comparison theorems to the integral case for Bakry-Émery Ricci bounds, broadening their applicability.

Findings

Established mean curvature and volume comparison estimates under integral bounds.

Derived diameter, eigenvalue, and volume growth estimates from these comparison results.

Generalized previous results by Petersen-Wei, Aubry, and others.

Abstract

We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-\'{E}mery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry-\'{E}mery Ricci tensor. These give generalizations of some work of Petersen-Wei, Aubry, Petersen-Sprouse, Yau and more.