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.