For Riemannian manifolds with a measure (M,g,eβfdvolgβ) we prove mean
curvature and volume comparison results when the β-Bakry-Emery Ricci
tensor is bounded from below and f is bounded or βrβf 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.Comment: 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 adde