Comparison geometry for the Bakry-Emery Ricci tensor

For Riemannian manifolds with a measure (M, g, e -f dvol 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.