In this paper, we derive two subgradient estimates of the CR heat equation in a closed pseudohermitian 3-manifold which are served as the CR version of the Li-Yau gradient estimate. With its applications, we first get a subgradient estimate of the logarithm of the positive solution of the CR heat equation. Secondly, we have the Harnack inequality and upper bound estimate for the heat kernel. Finally, we obtain Perelman-type entropy formulae for the CR heat equation.
"Li-Yau gradient estimate and entropy formulae for the CR heat equation in a closed pseudohermitian 3-manifold." J. Differential Geom. 89 (2) 185 - 216, October 2011. https://doi.org/10.4310/jdg/1324477409