{"paper":{"title":"Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Qi S Zhang","submitted_at":"2017-08-03T08:37:53Z","abstract_excerpt":"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 $\\R^n$.\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 a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01049","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}