Mathematical Research Letters

Volume 25 (2018)

Number 5

Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness

Pages: 1673 – 1693



Qi S. Zhang (School of Mathematical Sciences, Fudan University, Shanghai, China; and Department of Mathematics, University of California at Riverside)


Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler [W] on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman’s W entropy [P] in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to $\mathbf{R}^n$.

Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to $\mathbf{R}^n$. Comparing with the well known flatness results in [An], [Ba] and [BKN] on asymptotically flat manifolds and asymptotically locally Euclidean (ALE) manifolds, their decay or integral condition on the curvature tensor is replaced by the condition that the metric converges to the Euclidean one in $C^1$ sense at infinity. No second order condition on the metric is needed.

Received 7 August 2017

Accepted 2 April 2018

Published 1 February 2019