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Comparison geometry for the Bakry-Emery Ricci tensor. (English) Zbl 1189.53036

Mean curvature and volume comparison results are proved when the \(\infty\)–Bakry–Emery Ricci tensor of a Riemannian manifold with a measure \((M,g,e^{-f}d\:vol_g)\) is bounded from below and \(f\) or \(\left|\nabla f\right|\) is bounded. This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, extensions of all of the major comparison theorems when \(f\) is bounded are given. Some examples show that the bound of \(f\) is necessary for these results.

MSC:

53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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