{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:DFZCGC3ZP5PYXZ6UT7PANU7YFJ","short_pith_number":"pith:DFZCGC3Z","schema_version":"1.0","canonical_sha256":"1972230b797f5f8be7d49fde06d3f82a5ccdcdc93220bfd6538599014f1534be","source":{"kind":"arxiv","id":"1209.4150","version":3},"attestation_state":"computed","paper":{"title":"A stochastic target approach to Ricci flow on surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG"],"primary_cat":"math.PR","authors_text":"Ionel Popescu, Robert W. Neel","submitted_at":"2012-09-19T05:14:13Z","abstract_excerpt":"We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi's representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every $C^k$-norm. In the case of $C^0$ and $C^1$-convergence, we achieve this by coupling two p"},"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":"1209.4150","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-19T05:14:13Z","cross_cats_sorted":["math.AP","math.DG"],"title_canon_sha256":"fee212d0c7d6ec848461603cc948361318e6acf82238a985efa1ed2d11809f6b","abstract_canon_sha256":"19935c47c4d826c4dc64d54dec89e4cd7bc27b9e03c6b40a6764a0d2af8b610a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:04.862794Z","signature_b64":"BFSmINImnUyn49edu6Z+tuo7BOic4CL0WszsIvwWLzZ3hBInu1aXKHdpQC9SuaaoWTNjP2mowOnq8Rlf9gOpAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1972230b797f5f8be7d49fde06d3f82a5ccdcdc93220bfd6538599014f1534be","last_reissued_at":"2026-05-18T01:18:04.862190Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:04.862190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A stochastic target approach to Ricci flow on surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG"],"primary_cat":"math.PR","authors_text":"Ionel Popescu, Robert W. Neel","submitted_at":"2012-09-19T05:14:13Z","abstract_excerpt":"We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi's representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every $C^k$-norm. In the case of $C^0$ and $C^1$-convergence, we achieve this by coupling two p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4150","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.4150","created_at":"2026-05-18T01:18:04.862276+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.4150v3","created_at":"2026-05-18T01:18:04.862276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4150","created_at":"2026-05-18T01:18:04.862276+00:00"},{"alias_kind":"pith_short_12","alias_value":"DFZCGC3ZP5PY","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"DFZCGC3ZP5PYXZ6U","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"DFZCGC3Z","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ","json":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ.json","graph_json":"https://pith.science/api/pith-number/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/graph.json","events_json":"https://pith.science/api/pith-number/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/events.json","paper":"https://pith.science/paper/DFZCGC3Z"},"agent_actions":{"view_html":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ","download_json":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ.json","view_paper":"https://pith.science/paper/DFZCGC3Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.4150&json=true","fetch_graph":"https://pith.science/api/pith-number/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/graph.json","fetch_events":"https://pith.science/api/pith-number/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/action/storage_attestation","attest_author":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/action/author_attestation","sign_citation":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/action/citation_signature","submit_replication":"https://pith.science/pith/DFZCGC3ZP5PYXZ6UT7PANU7YFJ/action/replication_record"}},"created_at":"2026-05-18T01:18:04.862276+00:00","updated_at":"2026-05-18T01:18:04.862276+00:00"}