Tohoku Mathematical Journal
2021
September
SECOND SERIES VOL. 73, NO. 3

Tohoku Math. J.
73 (2021), 421-432

Title MYERS-TYPE COMPACTNESS THEOREM WITH THE BAKRY-EMERY RICCI TENSOR

Author Seungsu Hwang and Sanghun Lee

(Received August 27, 2019, revised May 12, 2020)
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 $f$.

Mathematics Subject Classification. Primary 53C20; Secondary 53C21.
Key words and phrases. Bakry--Emery Ricci curvature, myers theorem, mean curvature comparison theorem, Riccati inequality.

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