Asymptotic estimates on the time derivative of $\Phi$-entropy on   Riemannian manifolds
  In this note, we obtain the asymptotic estimate for the time derivative of the $\Phi$-entropy in terms of the lower bound on the Bakry-Emery $\Gamma_2$ curvature. In the cases of Hyperbolic space and Heisenberg group, we show that the time derivative of the $\Phi$-entropy is non-increasing, and we also get sharp asymptotic bound for the time derivative of the entropy in the Heisenberg group. 
Bin Qian
math.DG