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

## TL;DR

This paper extends classical Myers' compactness theorem to smooth metric measure spaces with Bakry-Emery Ricci tensor bounds, providing new comparison results and relaxing conditions on the weight function derivative.

## Contribution

It introduces a generalized Myers-type theorem for Bakry-Emery Ricci curvature, including a new $f$-mean curvature comparison and weaker derivative conditions.

## Key findings

- Established $f$-mean curvature comparison under Ricci bounds
- Proved a Myers-type compactness theorem for Bakry-Emery Ricci tensor
- Improved previous results with weaker derivative conditions on $f'$

## 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 from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing the results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Emery Ricci tensor. Moreover, we improve a result from Soylu by using a weaker condition on a derivative $f'(t)$.