{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:W6ABQXS4GP3XSHFUEZHWOPG5P5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9e32736eec6db0db0ee66591e1d113cf9ff0282c487a6458d0384a064815f5e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-08-03T08:37:53Z","title_canon_sha256":"6f7b4ff5aaf7dceb5e3cd133dde28ef23a1203d9d71de93637fe47308a25f2bf"},"schema_version":"1.0","source":{"id":"1708.01049","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01049","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01049v1","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01049","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"pith_short_12","alias_value":"W6ABQXS4GP3X","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"W6ABQXS4GP3XSHFU","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"W6ABQXS4","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:716a94ed239f5607dbb104e13c56f71967219ccfca2e48a29654aa031fd37541","target":"graph","created_at":"2026-05-18T00:38:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Qi S Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-08-03T08:37:53Z","title":"Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01049","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c5fe4bc312cb0f13b94bc64e2bfb54cf42342711bb5e34d8094e812b23fe219c","target":"record","created_at":"2026-05-18T00:38:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9e32736eec6db0db0ee66591e1d113cf9ff0282c487a6458d0384a064815f5e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-08-03T08:37:53Z","title_canon_sha256":"6f7b4ff5aaf7dceb5e3cd133dde28ef23a1203d9d71de93637fe47308a25f2bf"},"schema_version":"1.0","source":{"id":"1708.01049","kind":"arxiv","version":1}},"canonical_sha256":"b780185e5c33f7791cb4264f673cdd7f560f9d624b65b9fcad5636bfb82b727f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b780185e5c33f7791cb4264f673cdd7f560f9d624b65b9fcad5636bfb82b727f","first_computed_at":"2026-05-18T00:38:40.750794Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:40.750794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EgXM4OGqipW9XbtB12DUp/XvEBCKNJMMZDzgTRYW7cdMdLr0Q9uj+KEAeIP8kItSOS5BuzAVKKQdGcw0kVLIBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:40.751195Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.01049","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c5fe4bc312cb0f13b94bc64e2bfb54cf42342711bb5e34d8094e812b23fe219c","sha256:716a94ed239f5607dbb104e13c56f71967219ccfca2e48a29654aa031fd37541"],"state_sha256":"9576128ad8bb55909969ae72cd4b1fedfdddb25c83c34633536a0b405b7fb612"}