{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:J2GH7U5D2ABZVM2WD2MW43ND27","short_pith_number":"pith:J2GH7U5D","canonical_record":{"source":{"id":"1811.11036","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2018-11-27T14:49:45Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"02d0595236d9ea5a26489faf2aeaa1c3bda04e6df014521790fe793a0c3ed3c3","abstract_canon_sha256":"b8ac3d05bde8cc98b289d014dcf6b18290b3ab476444549a96aaed485a523709"},"schema_version":"1.0"},"canonical_sha256":"4e8c7fd3a3d0039ab3561e996e6da3d7e8d61bb6ae2ce329a4d6f1997d9bc11c","source":{"kind":"arxiv","id":"1811.11036","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.11036","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"arxiv_version","alias_value":"1811.11036v1","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.11036","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"pith_short_12","alias_value":"J2GH7U5D2ABZ","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"J2GH7U5D2ABZVM2W","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"J2GH7U5D","created_at":"2026-05-18T12:32:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:J2GH7U5D2ABZVM2WD2MW43ND27","target":"record","payload":{"canonical_record":{"source":{"id":"1811.11036","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2018-11-27T14:49:45Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"02d0595236d9ea5a26489faf2aeaa1c3bda04e6df014521790fe793a0c3ed3c3","abstract_canon_sha256":"b8ac3d05bde8cc98b289d014dcf6b18290b3ab476444549a96aaed485a523709"},"schema_version":"1.0"},"canonical_sha256":"4e8c7fd3a3d0039ab3561e996e6da3d7e8d61bb6ae2ce329a4d6f1997d9bc11c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:45.415041Z","signature_b64":"vSrD2CWpq1N3gJhAHKw3lCAneXLySzcZGgww3Uyv+0cPYDf9od4TGlIKl4jZ8ji0B3d+6XOxuc8kvCX8yFgjDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e8c7fd3a3d0039ab3561e996e6da3d7e8d61bb6ae2ce329a4d6f1997d9bc11c","last_reissued_at":"2026-05-17T23:59:45.414615Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:45.414615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.11036","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6KNhsZeBTbymTs5l4EA8x8ko5x9Kpsqam0hKJbr2PuIo0snIOquXlf7pqGQxhKnF/1NTnupqeIJeb1e8GynTDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T10:15:21.741678Z"},"content_sha256":"4557f2ba1d2c12023431ec0f8cdfaeb38d7571b8f092283d3917777ee281ec42","schema_version":"1.0","event_id":"sha256:4557f2ba1d2c12023431ec0f8cdfaeb38d7571b8f092283d3917777ee281ec42"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:J2GH7U5D2ABZVM2WD2MW43ND27","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mean field equations on a closed Riemannian surface with the action of an isometric group","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Xiaobao Zhu, Yunyan Yang","submitted_at":"2018-11-27T14:49:45Z","abstract_excerpt":"Let $(\\Sigma,g)$ be a closed Riemannian surface, $\\textbf{G}=\\{\\sigma_1,\\cdots,\\sigma_N\\}$ be an isometric group acting on it. Denote a positive integer $\\ell=\\inf_{x\\in\\Sigma}I(x)$, where $I(x)$ is the number of all distinct points of the set $\\{\\sigma_1(x),\\cdots,\\sigma_N(x)\\}$. A sufficient condition for existence of solutions to the mean field equation $$\\Delta_g u=8\\pi\\ell\\left(\\frac{he^u}{\\int_\\Sigma he^udv_g}-\\frac{1}{{\\rm Vol}_g(\\Sigma)}\\right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $\\ell=1$ or equivalently $\\textbf{G}=\\{Id\\}$, where $Id$ is the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.11036","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"edY+MMNH9cWR1vJjbhMaY/hf1Sz+8bGeKBpON1bpMTAYnJKuE2uq7WRPJTMdGc0f7BDll5P7+FX1X52v4ew4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T10:15:21.742421Z"},"content_sha256":"d3f801a28ab0975285b0d026880f0a9009448fb2748221c61fc9cef5ca379399","schema_version":"1.0","event_id":"sha256:d3f801a28ab0975285b0d026880f0a9009448fb2748221c61fc9cef5ca379399"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J2GH7U5D2ABZVM2WD2MW43ND27/bundle.json","state_url":"https://pith.science/pith/J2GH7U5D2ABZVM2WD2MW43ND27/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J2GH7U5D2ABZVM2WD2MW43ND27/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-23T10:15:21Z","links":{"resolver":"https://pith.science/pith/J2GH7U5D2ABZVM2WD2MW43ND27","bundle":"https://pith.science/pith/J2GH7U5D2ABZVM2WD2MW43ND27/bundle.json","state":"https://pith.science/pith/J2GH7U5D2ABZVM2WD2MW43ND27/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J2GH7U5D2ABZVM2WD2MW43ND27/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:J2GH7U5D2ABZVM2WD2MW43ND27","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":"b8ac3d05bde8cc98b289d014dcf6b18290b3ab476444549a96aaed485a523709","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2018-11-27T14:49:45Z","title_canon_sha256":"02d0595236d9ea5a26489faf2aeaa1c3bda04e6df014521790fe793a0c3ed3c3"},"schema_version":"1.0","source":{"id":"1811.11036","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.11036","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"arxiv_version","alias_value":"1811.11036v1","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.11036","created_at":"2026-05-17T23:59:45Z"},{"alias_kind":"pith_short_12","alias_value":"J2GH7U5D2ABZ","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"J2GH7U5D2ABZVM2W","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"J2GH7U5D","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:d3f801a28ab0975285b0d026880f0a9009448fb2748221c61fc9cef5ca379399","target":"graph","created_at":"2026-05-17T23:59:45Z","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":"Let $(\\Sigma,g)$ be a closed Riemannian surface, $\\textbf{G}=\\{\\sigma_1,\\cdots,\\sigma_N\\}$ be an isometric group acting on it. Denote a positive integer $\\ell=\\inf_{x\\in\\Sigma}I(x)$, where $I(x)$ is the number of all distinct points of the set $\\{\\sigma_1(x),\\cdots,\\sigma_N(x)\\}$. A sufficient condition for existence of solutions to the mean field equation $$\\Delta_g u=8\\pi\\ell\\left(\\frac{he^u}{\\int_\\Sigma he^udv_g}-\\frac{1}{{\\rm Vol}_g(\\Sigma)}\\right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $\\ell=1$ or equivalently $\\textbf{G}=\\{Id\\}$, where $Id$ is the","authors_text":"Xiaobao Zhu, Yunyan Yang","cross_cats":["math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2018-11-27T14:49:45Z","title":"Mean field equations on a closed Riemannian surface with the action of an isometric group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.11036","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":"sha256:4557f2ba1d2c12023431ec0f8cdfaeb38d7571b8f092283d3917777ee281ec42","target":"record","created_at":"2026-05-17T23:59:45Z","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":"b8ac3d05bde8cc98b289d014dcf6b18290b3ab476444549a96aaed485a523709","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2018-11-27T14:49:45Z","title_canon_sha256":"02d0595236d9ea5a26489faf2aeaa1c3bda04e6df014521790fe793a0c3ed3c3"},"schema_version":"1.0","source":{"id":"1811.11036","kind":"arxiv","version":1}},"canonical_sha256":"4e8c7fd3a3d0039ab3561e996e6da3d7e8d61bb6ae2ce329a4d6f1997d9bc11c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e8c7fd3a3d0039ab3561e996e6da3d7e8d61bb6ae2ce329a4d6f1997d9bc11c","first_computed_at":"2026-05-17T23:59:45.414615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:45.414615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vSrD2CWpq1N3gJhAHKw3lCAneXLySzcZGgww3Uyv+0cPYDf9od4TGlIKl4jZ8ji0B3d+6XOxuc8kvCX8yFgjDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:45.415041Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.11036","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4557f2ba1d2c12023431ec0f8cdfaeb38d7571b8f092283d3917777ee281ec42","sha256:d3f801a28ab0975285b0d026880f0a9009448fb2748221c61fc9cef5ca379399"],"state_sha256":"46be7392cc4d1d049c83f2551164df2237411937bf7aa77015268a10a48cc65d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mw4YaDHRSVoR/uPNwiptANK5JvAQfAqryHbpLwfz8NhvByBSqDTrFRixjaLJJ5bMuwAGQDgGM+r5qwViQmvsDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T10:15:21.746978Z","bundle_sha256":"77e6fe8349748bbd9b988db32bcd5a61b6a93875c968643973bb3b8a590ce280"}}