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Abstract

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estimate, the no-local-collapsing theorem, the pseudo-locality theorem and the strong maximum principle for curvature tensors, are essentially improved for Ricci flows induced by Ricci shrinkers. Our results provide many necessary tools to analyze short time singularities of the Ricci flows of general dimension.

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Acknowledgements

Yu Li would like to thank Jiyuan Han and Shaosai Huang for helpful comments. Bing Wang would like to thank Haozhao Li and Lu Wang for their interests in this work. Part of this work was done while both authors were visiting IMS (Institute of Mathematical Sciences) at ShanghaiTech University during the summer of 2018. They wish to thank IMS for its hospitality. Last but not least, the authors would like to thank the anonymous referees for several valuable comments that help improve the exposition of the paper.

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Yu Li is partially supported by research fund from SUNY Stony Brook and Bing Wang is partially supported by NSF Grant DMS-1510401 and research funds from USTC and UW-Madison.

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Li, Y., Wang, B. Heat kernel on Ricci shrinkers. Calc. Var. 59, 194 (2020). https://doi.org/10.1007/s00526-020-01861-y

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