Comparison Geometry for the Bakry-Emery Ricci Tensor

TL;DR

This paper extends classical geometric comparison theorems to manifolds with a weighted measure, under bounds on the Bakry-Emery Ricci tensor and the function f, generalizing results for Ricci curvature.

Contribution

It generalizes mean curvature and volume comparison theorems to the Bakry-Emery Ricci tensor with bounded f, extending classical comparison theorems to weighted manifolds.

Findings

Mean curvature comparison results under Bakry-Emery bounds

Volume comparison theorems for weighted manifolds

Necessity of bounds on f for these theorems to hold

Abstract

For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when $f$ is bounded. Simple examples show the bound on $f$ is necessary for these results.