Comparison Geometry for the Bakry-Emery Ricci Tensor

Abstract

For Riemannian manifolds with a measure $(M,g,e^{-f}dvo{l}_g)$ 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.

21 pages, Some of the estimates have been improved. In light of some new references, and to improve the exposition, the paper has been reorganized. An appendix is also added

References in corpus (2)

• Some Geometry and Analysis on Ricci Solitons
• Complete Shrinking Ricci Solitons have Finite Fundamental Group

Cited by in corpus (8)

• Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor
• Geometry of Complete Gradient Shrinking Ricci Solitons
• Perelman's reduced volume and a gap theorem for the Ricci flow
• Rigidity of gradient Ricci Solitons
• Remarks on non-compact gradient Ricci solitons
• On gradient Ricci solitons with Symmetry
• Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Émery Ricci curvature
• Properties of Complete Noncompact Warped Product Gradient Yamabe Solitons