Comparison Geometry for the Bakry-Emery Ricci Tensor

Details Comparison Geometry for the Bakry-Emery Ricci Tensor Comparison Geometry for the Bakry-Emery Ricci Tensor

2007-06-08

arxiv.org/abs/0706.1120v2

Mathematics

Differential Geometry

21 pages, Some of the estimates have been improved. In light of some new references, and to improve the exposition, the paper

Scientific paper

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.

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