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

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 -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.