Comparison Geometry for the Bakry-Emery Ricci Tensor

For Riemannian manifolds with a measure $(M,g,e^{-f}dvo{l}_g)$ we prove mean curvature and volume comparison results when the $\mathrm{\infty }$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or ${\mathrm{\partial }}_{r}f$ 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.