Myers’ Type Theorem for Integral Bakry–Émery Ricci Tensor Bounds

In this paper we first discuss weighted mean curvature and volume comparisons on smooth metric measure space $$(M, g, e^{-f}dv)$$ under the integral Bakry–Emery Ricci tensor bounds. In particular, we add an additional condition on the potential function f to ensure the validity of previous conclusions for some cases proved by the second author. Then, we apply the comparison results to get a new diameter estimate and a fundamental group finiteness under the integral Bakry–Emery Ricci tensor bounds, which sharpens Theorem 1.6 in Wu (J Geom Anal 29:828–867, 2019) and can be viewed as the extension of the works of Myers and Aubry.