For Riemannian manifolds with a measure (M, g, e−fdvolg) we prove mean curvature and volume comparison results when the∞- Bakry-Emery Ricci tensor is bounded from below and f or |∇f| is bounded, 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.
|Original language||English (US)|
|Number of pages||69|
|Journal||Journal of Differential Geometry|
|State||Published - 2009|
ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology