Asian Journal of Mathematics

Volume 22 (2018)

Number 2

Special issue in honor of Ngaiming Mok (1 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

CR Li–Yau gradient estimate for Witten Laplacian via Bakry–Emery pseudohermitian Ricci curvature

Pages: 223 – 256



Der-Chen Chang (Department of Mathematics, Georgetown University, Washington, D.C., U.S.A.; and Department of Mathematics, Fu Jen Catholic University, Taipei, Taiwan)

Shu-Cheng Chang (Department of Mathematics, National Taiwan University, Taipei, Taiwan; and Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei, Taiwan)

Ting-Jung Kuo (Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan)

Sin-Hua Lai (Department of Mathematics, National Central University, Chungli, Taiwan)


In this paper, we derive the sub-gradient estimate of the CR heat equation associated with the Witten sub-Laplacian via the Bakry–Emery pseudohermitian Ricci curvature. With its applications, we first get a Harnack inequality for the positive solution of this CR heat equation in a closed pseudohermitian $(2n + 1)$-manifold. Secondly, we obtain Perelman-type linear entropy formulae for this CR heat equation.


CR heat equation, Li–Yau gradient estimate, Harnack inequality, Perelman entropy formulae, Bakry–Emery pseudohermitian Ricci curvature, Pseudohermitian manifold, Witten Laplacian

2010 Mathematics Subject Classification

Primary 32V05, 32V20. Secondary 53C56.

Received 15 October 2016

Accepted 15 June 2017

Published 15 June 2018