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

DOI10.4310/MRL.2018.V25.N5.A14zbMATH Open1408.53092arXiv1708.01049OpenAlexW2963325082WikidataQ126101552 ScholiaQ126101552MaRDI QIDQ1727886

Qi S. Zhang

Publication date: 21 February 2019

Published in: Mathematical Research Letters (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1708.01049

zbMATH Keywords

decay

Mathematics Subject Classification ID