Matrix Li–Yau–Hamilton inequality for the CR heat equation in pseudohermitian $$(2n+1)$$ ( 2 n + 1 ) -manifolds

Abstract

In this paper, we first derive the CR analogue of matrix Li–Yau–Hamilton inequality for the positive solution to the CR heat equation in a closed pseudohermitian $$(2n+1)$$ -manifold with nonnegative bisectional curvature and bitorsional tensor. We then obtain the CR Li–Yau gradient estimate in the Heisenberg group. We apply this CR gradient estimate and extend the CR matrix Li–Yau–Hamilton inequality to the case of the Heisenberg group. As a consequence, we derive the Hessian comparison property for the Heisenberg group.