Subgradient estimate and Liouville-type theorem for the \(CR\) heat equation on Heisenberg groups
type: Item
instance of: scholarly article
title: Subgradient estimate and Liouville-type theorem for the \(CR\) heat equation on Heisenberg groups
zbMATH Open document ID: 1214.32011
author: Shu-Cheng Chang
author: Jingzhi Tie
author: Chin-Tung Wu
published in: The Asian Journal of Mathematics
publication date: 2010-12-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ajm/1286547518
review text: Recall that a pseudo-Hermitian manifold is a contact manifold with a given choice of contact form and a compatible complex structure on the contact subbundle. On such a manifold, there is a CR version of the heat equation, in which the Riemannian Laplacian is replaced by the sub-Laplacian. The first main result of this article is a subgradient estimate for positive solutions of the CR heat equation on a closed pseudo-Hermitian manifold. Second, the authors derive a comparison theorem for the sub-Laplacian of the Carnot-Carathéodory distance on the Heisenberg group. Combining this with the subgradient estimate, the authors obtain a Liouville type theorem for the CR heat equation, as well as a subgradient estimate for the CR heat kernel. The basis for these results is a CR version of the Bochner formula involving a third order operator which characterizes pluriharmonic functions and which can be related to the CR version of the Paeneitz operator.
Mathematics Subject Classification ID: 32V20
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 5821930
zbMATH Keywords: CR manifold
zbMATH Keywords: pseudo-Hermitian structure
zbMATH Keywords: CR heat equation
zbMATH Keywords: sub-Laplacian
zbMATH Keywords: heat kernel
zbMATH Keywords: subgradient estimate
zbMATH Keywords: Lie-Yau Harnack inequality
zbMATH Keywords: CR Paneitz operator
reviewed by: Andreas Cap
MaRDI profile type: Publication
OpenAlex ID: W1993853148
DOI: 10.4310/AJM.2010.V14.N1.A4
description: scientific article; zbMATH DE number 5821930