Asian Journal of Mathematics

Volume 22 (2018)

Number 3

Special issue in honor of Ngaiming Mok (2 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

Pages: 577 – 598



Songzi Li (School of Mathematics, Renmin University of China, Beijing, China)

Xiang-Dong Li (Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)


In this paper, we prove the Li–Yau type Harnack inequality for the heat equation $\partial_t u = L u$ associated with the time-dependent Witten Laplacian on manifolds equipped with a variant of complete backward $(-K, m)$-super Perelman Ricci flows. Moreover, using a probabilistic approach we prove an improved Hamilton type Harnack inequality on manifolds equipped with complete $(-K)$-super Perelman Ricci flows.


Harnack inequality, super Perelman Ricci flows, Witten Laplacian

2010 Mathematics Subject Classification

Primary 58J35, 58J65. Secondary 60H30, 60J60.

Research supported by NSFC No. 11771430, Key Laboratory RCSDS, CAS, No. 2008DP173182, and by a Hua Luo-Keng Research Grant of the AMSS, CAS.

Received 31 October 2016

Accepted 7 June 2017

Published 8 August 2018