Cambridge Journal of Mathematics

Volume 6 (2018)

Number 3

The local entropy along Ricci flow Part A: the no-local-collapsing theorems

Pages: 267 – 346



Bing Wang (Department of Mathematics, University of Wisconsin, Madison, Wi., U.S.A.)


We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li–Yau estimate along the Ricci flow. It can be used to show the Gromov–Hausdorff convergence of the Kähler Ricci flow on each minimal projective manifold of general type.


entropy, Ricci flow, Harnack inequality, projective manifold

2010 Mathematics Subject Classification

Primary 53C23, 53C25. Secondary 32J27.

This paper is partially supported by NSF grant DMS-1510401. The author would also like to acknowledge the invitation to MSRI Berkeley in spring 2016 supported by NSF grant DMS-1440140. Part of this work was done while the author was visiting AMSS(Academy of Mathematics and Systems Science) in Beijing and USTC (University of Science and Technology of China) in Hefei, during the summer of 2016. He wishes to thank AMSS and USTC for their hospitality. He would like to thank Mikhail Feldman, Jeff Viaclovsky, Lu Wang and Shaosai Huang for helpful discussions. He would also like to thank professor Q.S. Zhang to kindly point out the similarity between some of the results in this paper and the ones in his book(c.f. Remark 4.11).

Received 14 August 2017

Published 28 August 2018