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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1708.01049","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-08-03T08:37:53Z","cross_cats_sorted":[],"title_canon_sha256":"6f7b4ff5aaf7dceb5e3cd133dde28ef23a1203d9d71de93637fe47308a25f2bf","abstract_canon_sha256":"9e32736eec6db0db0ee66591e1d113cf9ff0282c487a6458d0384a064815f5e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:40.751195Z","signature_b64":"EgXM4OGqipW9XbtB12DUp/XvEBCKNJMMZDzgTRYW7cdMdLr0Q9uj+KEAeIP8kItSOS5BuzAVKKQdGcw0kVLIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b780185e5c33f7791cb4264f673cdd7f560f9d624b65b9fcad5636bfb82b727f","last_reissued_at":"2026-05-18T00:38:40.750794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:40.750794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\R^n$. 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