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

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Consider the scaling invariant, sharp log entropy (functional)

introduced by Weissler on noncompact manifolds with nonnegative Ricci

curvature. It can also be regarded as a sharpened version of

Perelman's W entropy 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 some well known flatness results in 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 C1 sense

at infinity. No second order condition on the metric is needed.