Subgradient estimate and Liouville-type theorem for the \(CR\) heat equation on Heisenberg groups
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.
scientific article; zbMATH DE number 5821930