In this paper we first discuss weighted mean curvature and volume comparisons on smooth metric measure space $$(M, g, e^{-f}dv)$$ ( M , g , e - f d v ) under the integral Bakry–Émery 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–Émery 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.

中文翻译：

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积分 Bakry-Émery Ricci 张量界的 Myers 类型定理

在本文中，我们首先讨论在积分 Bakry–Émery Ricci 下平滑度量空间 $$(M, g, e^{-f}d​​v)$$ ( M , g , e - fdv ) 上的加权平均曲率和体积比较张量界限。特别是，我们在势函数 f 上添加了一个附加条件，以确保先前结论对第二作者证明的某些情况的有效性。然后，我们应用比较结果在积分 Bakry-Émery Ricci 张量边界下获得新的直径估计和基本群有限性，这使 Wu (J Geom Anal 29:828-867, 2019) 中的定理 1.6 更加尖锐，并且可以查看作为迈尔斯和奥布里作品的延伸。