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The other one is about the convergence. It says that in a closed {\\Rm} $\\tilde{M}$, assume the curve shortening flow $\\ct$ exists for all $t\\in[0,\\infty)$ and its length converges to a positive limit, then $ \\lim\\limits_{t\\rightarrow\\infty}max_{\\ct}|\\nabla^{m}A|^{2}=0$ for all $m=0,1"},"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":"1212.5515","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2012-12-21T16:39:34Z","cross_cats_sorted":[],"title_canon_sha256":"8d718044f016ea29006b9d984574632298405459cb7964011c33323169867edf","abstract_canon_sha256":"e13612dbe759243a1f49db95e200f46379039b8130a93afac50aafa102ae1a79"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:52.127650Z","signature_b64":"IHA8CX36RcnijC+pt2k+ie4pjzoui4amSy8NSolA7Q/p5T2VCs1OtI90n7vkJ1WZYstbrJ3Ls8737lacpMk7CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7090127c1ddf66136ef2887ec97738a63444bb51148fccf5cbced57245c8dc62","last_reissued_at":"2026-05-18T03:37:52.127023Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:52.127023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The curve shortening flow with parallel 1-form","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hengyu Zhou","submitted_at":"2012-12-21T16:39:34Z","abstract_excerpt":"Let $M$ be a closed Riemannian manifold with a parallel 1-form $\\Omega$. 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It says that in a closed {\\Rm} $\\tilde{M}$, assume the curve shortening flow $\\ct$ exists for all $t\\in[0,\\infty)$ and its length converges to a positive limit, then $ \\lim\\limits_{t\\rightarrow\\infty}max_{\\ct}|\\nabla^{m}A|^{2}=0$ for all $m=0,1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5515","kind":"arxiv","version":2},"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":"1212.5515","created_at":"2026-05-18T03:37:52.127135+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.5515v2","created_at":"2026-05-18T03:37:52.127135+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5515","created_at":"2026-05-18T03:37:52.127135+00:00"},{"alias_kind":"pith_short_12","alias_value":"OCIBE7A535TB","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"OCIBE7A535TBG3XS","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"OCIBE7A5","created_at":"2026-05-18T12:27:16.716162+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/OCIBE7A535TBG3XSRB7MS5ZYUY","json":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY.json","graph_json":"https://pith.science/api/pith-number/OCIBE7A535TBG3XSRB7MS5ZYUY/graph.json","events_json":"https://pith.science/api/pith-number/OCIBE7A535TBG3XSRB7MS5ZYUY/events.json","paper":"https://pith.science/paper/OCIBE7A5"},"agent_actions":{"view_html":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY","download_json":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY.json","view_paper":"https://pith.science/paper/OCIBE7A5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.5515&json=true","fetch_graph":"https://pith.science/api/pith-number/OCIBE7A535TBG3XSRB7MS5ZYUY/graph.json","fetch_events":"https://pith.science/api/pith-number/OCIBE7A535TBG3XSRB7MS5ZYUY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY/action/storage_attestation","attest_author":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY/action/author_attestation","sign_citation":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY/action/citation_signature","submit_replication":"https://pith.science/pith/OCIBE7A535TBG3XSRB7MS5ZYUY/action/replication_record"}},"created_at":"2026-05-18T03:37:52.127135+00:00","updated_at":"2026-05-18T03:37:52.127135+00:00"}