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

TL;DR

This paper proves that the minimizer of the sharp log entropy on noncompact manifolds with nonnegative Ricci curvature characterizes Euclidean space, leading to new flatness results based on metric convergence rather than curvature decay.

Contribution

It establishes that the sharp log entropy has a minimizer only on Euclidean space and derives new flatness criteria based on metric convergence in $C^1$ without second order conditions.

Findings

Minimizer of the sharp log entropy characterizes Euclidean space.

Manifolds with nonnegative Ricci curvature and $C^1$ convergence to Euclidean metric are flat.

New flatness results replace curvature decay conditions with metric convergence.

Abstract

Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler \cite{W:1} on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman's W entropy \cite{P:1} in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to $\R^n$. Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to $\R^n$. Comparing with the well known flatness results in \cite{An:1}, \cite{Ba:1} and \cite{BKN:1} 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.