In this paper, we first get a subgradient estimate of the $CR$ heat equation on a closed pseudohermitian $(2n + 1)$-manifold. Secondly, by deriving the $CR$ version of sub-Laplacian comparison theorem on an $(2n + 1)$-dimensional Heisenberg group $H^n$, we are able to establish a subgradient estimate and then the Liouville-type theorem for the $CR$ heat equation on $H^n$.
"Subgradient Estimate and Liouville-type Theorem for the $CR$ Heat Equation on Heisenberg Groups." Asian J. Math. 14 (1) 41 - 72, March 2010.