The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 828–867 | Cite as

Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

  • Jia-Yong Wu


We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry–Émery Ricci tensor bounds, extending Wei–Wylie’s comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates, and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry–Émery Ricci tensor. These give generalizations of some work of Petersen–Wei, Aubry, Petersen–Sprouse, Yau and more.


Bakry–Émery Ricci tensor Smooth metric measure space Integral curvature Comparison theorem Diameter estimate Eigenvalue estimate Volume growth estimate 

Mathematics Subject Classification

Primary 53C20 



The author sincerely thanks Professor Guofang Wei for her useful note, valuable advices, stimulating discussions and bringing my attention to the paper [13]. The author also thanks Peng Wu for many helpful discussions. This work was partially supported by the Natural Science Foundation of Shanghai (No. 17ZR1412800) and the National Natural Science Foundation of China (No. 11671141).


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Authors and Affiliations

  1. 1.Department of MathematicsShanghai Maritime UniversityShanghaiPeople’s Republic of China

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