The entropy formulas for the CR heat equation and their applications on pseudohermitian \((2n+1)\)-manifolds
For a CR-manifold \(M\) of hypersurface type, the choice of a contact form is usually referred to as a pseudo-Hermitian structure. Having chosen such a structure, one has the CR-heat equation, which involves a sub-Laplacian instead of a Riemannian Laplace operator. This paper is devoted to the analysis of the CR-heat equation on strictly-pseudoconvex pseudo-Hermitian manifolds. The main results of the article are entropy formulae of Perelman type and of Nash-type for the CR heat equation, which apply under certain conditions on the positivity of the Ricci curvature, respectively, a combination of Ricci curvature and torsion. The proofs that these entropies are monotone non-increasing on solutions of the CR heat equation are based on a CR Bochner formula, which involves a third order operator closely related to the (fourth order) CR analog of the Paeneitz-operator, and on a CR Harnack-type estimate. As applications of their results, the authors derive an integral version of the subgradient estimate for solutions of the CR heat equation as well as upper and lower bounds on the first positive eigenvalue of the sub-Laplacian.
scientific article; zbMATH DE number 5704129