{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:52E3YLC5NFENTRZ4TOXWLFMGPN","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":"e63b837c3e69231efa0b8c1c049183fded871fd924e42272fc6681fc33d26041","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-02-03T15:56:35Z","title_canon_sha256":"004de024437892ff770b05fc8d1214e578f0f827a89d4c369312515f2f9106a7"},"schema_version":"1.0","source":{"id":"1102.0713","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0713","created_at":"2026-05-18T04:25:12Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0713v2","created_at":"2026-05-18T04:25:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0713","created_at":"2026-05-18T04:25:12Z"},{"alias_kind":"pith_short_12","alias_value":"52E3YLC5NFEN","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"52E3YLC5NFENTRZ4","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"52E3YLC5","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:123deb0e1dd3a3eaa6acddea4fc94ea91baaafabcb29448b04147f9932cb2618","target":"graph","created_at":"2026-05-18T04:25:12Z","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":"We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\\mathbb{R}^2$ introduced by Rivi\\`ere. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmonic and almost-harmonic maps from surfaces.","authors_text":"Ben Sharp, Peter Topping","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-02-03T15:56:35Z","title":"Decay estimates for Rivi\\`ere's equation, with applications to regularity and compactness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0713","kind":"arxiv","version":2},"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":"sha256:74cb5f1a278e9a0f388f9fd19fb8a05b3c859be37518cf53c524cb2aed0d43c4","target":"record","created_at":"2026-05-18T04:25:12Z","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":"e63b837c3e69231efa0b8c1c049183fded871fd924e42272fc6681fc33d26041","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-02-03T15:56:35Z","title_canon_sha256":"004de024437892ff770b05fc8d1214e578f0f827a89d4c369312515f2f9106a7"},"schema_version":"1.0","source":{"id":"1102.0713","kind":"arxiv","version":2}},"canonical_sha256":"ee89bc2c5d6948d9c73c9baf6595867b40253c29e5e35beb4be080ee68b28f30","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee89bc2c5d6948d9c73c9baf6595867b40253c29e5e35beb4be080ee68b28f30","first_computed_at":"2026-05-18T04:25:12.797563Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:12.797563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lAb9gGBj3Z6o87674MzUYKwNDwCYTfpHdLNOuMQzHlcB5Ct/zx/RAad1QUPel1u2dQQHLxRe2QJkruPhg8dxDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:12.798265Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.0713","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74cb5f1a278e9a0f388f9fd19fb8a05b3c859be37518cf53c524cb2aed0d43c4","sha256:123deb0e1dd3a3eaa6acddea4fc94ea91baaafabcb29448b04147f9932cb2618"],"state_sha256":"6b2020085d3402d428ac93d8ba9bf4eacecab187f05fead082b1b993cf510610"}