#### Vol. 298, No. 2, 2019

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Sharp logarithmic Sobolev inequalities along an extended Ricci flow and applications

### Guoqiang Wu and Yu Zheng

Vol. 298 (2019), No. 2, 483–509
##### Abstract

We prove a sharp Logarithmic Sobolev inequality along an extended Ricci flow. As applications, we derive an integral bound for the conjugate heat kernel and also obtain Lipschitz continuity of the pointed Nash entropy. Finally, based on these results, we prove an $\epsilon$-regularity theorem for this extended Ricci flow.

##### Keywords
conjugate heat kernel, logarithmic Sobolev inequalities, $\varepsilon$-regularity.
Primary: 53C44
Secondary: 53C21
##### Milestones
Revised: 15 May 2018
Accepted: 24 June 2018
Published: 8 March 2019
##### Authors
 Guoqiang Wu School of Science Zhejiang Sci-Tech University Hangzhou China Yu Zheng School of Mathematical Sciences East China Normal University Shanghai China