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Myers-type compactness theorem with the Bakry-Emery Ricci tensor. (English) Zbl 1486.53045

Summary: In this paper, we first prove the \(f\)-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded below and \(f\) is bounded by a linear function of distance. Based on this, we obtain Myers-type compactness theorems by generalizing the results of Cheeger, Gromov, and Taylor and Wan to the Bakry-Emery Ricci tensor. Moreover, we improve a result of Soylu by using a weaker condition on a derivative of \(f\).

MSC:

53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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References:

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