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Comparison Geometry for Integral Radial Bakry-Émery Ricci Tensor Bounds

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Abstract

In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author’s previous results under integral Bakry-Émery Ricci tensor bounds, but also are generalizations of the Wei-Wylie’s pointwise results.

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Acknowledgements

The author would like to thank Homare Tadano for providing the manuscript [19] and pointing out a mini omission in the proof of Theorem 4.1. He also thanks the referee for a very careful reading of the paper and helpful suggestions. This work was partially supported by the Natural Science Foundation of Shanghai (17ZR1412800).

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Jia-Yong Wu

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  1. Jia-Yong Wu
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Correspondence to Jia-Yong Wu.

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Wu, JY. Comparison Geometry for Integral Radial Bakry-Émery Ricci Tensor Bounds. Potential Anal 58, 203–223 (2023). https://doi.org/10.1007/s11118-021-09937-w

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