Comparison geometry for the bakry-emery ricci tensor

For Riemannian manifolds with a measure (M, g, e^−fdvol_g) 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.