Myers-type compactness theorem with the Bakry-Emery Ricci tensor

Abstract

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 $f$.