The entropy formulas for the CR heat equation and their applications on pseudohermitian (2n+ 1)-manifolds

Abstract

We derive Perelman's and Nash-type entropy formulas for the CR heat equation on closed pseudohermitian (2n + 1)-manifolds and show that it is monotone nonincreasing if its pseudohermitian Ricci curvature minus (n + 1)/2 times the pseudohermitian torsion is nonnegative. As results, we are able to obtain an integral version of the subgradient estimate for the CR heat equation and an upper bound estimate for the first positive eigenvalue of the sublaplacian by using the CR Bochner formulas and CR Harnack-type inequality. As a byproduct, we obtain a sharp lower bound estimate for the first positive eigenvalue of the sublaplacian on a closed pseudohermitian (2n + 1)-manifold.