Asymptotic estimates on the time derivative of $Φ$-entropy on Riemannian manifolds

Details Asymptotic estimates on the time derivative of $Φ$-entropy on Riemannian manifolds Asymptotic estimates on the time derivative of $Φ$-entropy on Riemannian manifolds

2011-04-16

arxiv.org/abs/1104.3204v2

Mathematics

Differential Geometry

This paper has been withdrawn by the author since we will modify it deeply

Scientific paper

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.

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