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Sharp upper diameter bounds for compact shrinking Ricci solitons

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Abstract

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman’s entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

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Acknowledgements

The author thanks Peng Wu for helpful discussions. The author also thanks the referee for valuable comments and suggestions, which helped to improve the paper. This work is supported by the NSFC (11671141) and 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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Wu, JY. Sharp upper diameter bounds for compact shrinking Ricci solitons. Ann Glob Anal Geom 60, 19–32 (2021). https://doi.org/10.1007/s10455-021-09764-7

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