{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7IRCQPTIDPOIYY4GMFJV4YZVKT","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":"e703bf812c2cc24015d125636790fdd8de9d165ed66a971bff994e90970ff2c6","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-21T07:27:03Z","title_canon_sha256":"c0248a9df27aaa9525396dba757369c8f0d1bb1b8f536738578e56e0d43db611"},"schema_version":"1.0","source":{"id":"1706.06766","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.06766","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"arxiv_version","alias_value":"1706.06766v1","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.06766","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"pith_short_12","alias_value":"7IRCQPTIDPOI","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7IRCQPTIDPOIYY4G","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7IRCQPTI","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:fd3bbae09339665d5aa6c0cf76fe9dc5b0b20cbf9bc6e548097e809ec02a15b1","target":"graph","created_at":"2026-05-18T00:41:56Z","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":"In this paper, we prove that the even solution of the mean field equation $\\Delta u=\\lambda(1-e^u) $ on $S^2$ must be axially symmetric when $4<\\lambda \\leq 8$. In particular, zero is the only even solution for $\\lambda=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.","authors_text":"Dongyi Wei, Gang Tian, Jiacheng Sun, Yuguang Shi","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-21T07:27:03Z","title":"Uniqueness of the Mean Field Equation and Rigidity of Hawking Mass"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.06766","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":"sha256:483e902d60079110c4072e4d4670c25336ff5674ef609b93eebbd23a3536c9a2","target":"record","created_at":"2026-05-18T00:41:56Z","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":"e703bf812c2cc24015d125636790fdd8de9d165ed66a971bff994e90970ff2c6","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-21T07:27:03Z","title_canon_sha256":"c0248a9df27aaa9525396dba757369c8f0d1bb1b8f536738578e56e0d43db611"},"schema_version":"1.0","source":{"id":"1706.06766","kind":"arxiv","version":1}},"canonical_sha256":"fa22283e681bdc8c638661535e633554c96b7f8f4e1fcc38813482f3f5fff96a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa22283e681bdc8c638661535e633554c96b7f8f4e1fcc38813482f3f5fff96a","first_computed_at":"2026-05-18T00:41:56.910308Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:56.910308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"20YfiX+yIU8KiAwN6qKX1hh6fS7ciRnTlEAXn0TWU8usgEyiTuwmY5J+7U3FzE4IcAjyHhJFUSlBBlV0ftNjDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:56.910812Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.06766","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:483e902d60079110c4072e4d4670c25336ff5674ef609b93eebbd23a3536c9a2","sha256:fd3bbae09339665d5aa6c0cf76fe9dc5b0b20cbf9bc6e548097e809ec02a15b1"],"state_sha256":"bc090775453a4828ac16d46c379e923c9bde999672104505515d067ee0648f27"}