Asymptotic estimates on the time derivative of entropy on a Riemannian manifold

TL;DR

This paper investigates the asymptotic behavior of the entropy's time derivative for heat equation solutions on Riemannian manifolds, providing bounds based on curvature and spectral gap, with explicit results for hyperbolic space.

Contribution

It offers new estimates on the entropy's rate of change on compact Riemannian manifolds, linking geometric properties to entropy dynamics.

Findings

Bounds on entropy derivative in terms of Ricci curvature

Bounds in terms of spectral gap

Explicit asymptotic bounds for hyperbolic space

Abstract

We consider the entropy of the solution to the heat equation on a Riemannian manifold. When the manifold is compact, we provide two estimates on the rate of change of the entropy in terms of the lower bound on the Ricci curvature and the spectral gap respectively. Our explicit computation for the three dimensional hyperbolic space shows that the time derivative of the entropy is asymptotically bounded by two positive constants.