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arXiv · 0706.1120

Comparison Geometry for the Bakry-Emery Ricci Tensor

Guofang Wei·Will Wylie
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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.

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Guofang Wei, Will Wylie. 2007-09-14. Comparison Geometry for the Bakry-Emery Ricci Tensor. https://arxiv.org/abs/0706.1120

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