{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:MSE4O6QOBAXDQKCM5NALUBE2IV","short_pith_number":"pith:MSE4O6QO","schema_version":"1.0","canonical_sha256":"6489c77a0e082e38284ceb40ba049a45552e45e4aa92dfb25a2cff0836297258","source":{"kind":"arxiv","id":"1707.04014","version":1},"attestation_state":"computed","paper":{"title":"Chord Shortening Flow and a Theorem of Lusternik and Schnirelmann","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Martin Li","submitted_at":"2017-07-13T07:49:27Z","abstract_excerpt":"We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord which is not orthogonal to the boundary would shrink to a point in finite time under the flow."},"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":"1707.04014","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-07-13T07:49:27Z","cross_cats_sorted":[],"title_canon_sha256":"ba18f8a583b898d2cad3ac3b3a09ba94b612614464da9e0fe24f6539a448410f","abstract_canon_sha256":"3d6da93fa2f91c40c02dfbf0fda1f93ab284914b39f9b63308cecafb7fcfd147"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:15.689910Z","signature_b64":"fdhOxtkIOT1kz5/1lJjQN051id+lwdNfLVnmCfYBQjt9bxMb18FbIipCFvN4f+qj5x87kuJzTXizo2Up6p1oCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6489c77a0e082e38284ceb40ba049a45552e45e4aa92dfb25a2cff0836297258","last_reissued_at":"2026-05-17T23:44:15.689398Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:15.689398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chord Shortening Flow and a Theorem of Lusternik and Schnirelmann","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Martin Li","submitted_at":"2017-07-13T07:49:27Z","abstract_excerpt":"We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord which is not orthogonal to the boundary would shrink to a point in finite time under the flow."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04014","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1707.04014","created_at":"2026-05-17T23:44:15.689485+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.04014v1","created_at":"2026-05-17T23:44:15.689485+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04014","created_at":"2026-05-17T23:44:15.689485+00:00"},{"alias_kind":"pith_short_12","alias_value":"MSE4O6QOBAXD","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_16","alias_value":"MSE4O6QOBAXDQKCM","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_8","alias_value":"MSE4O6QO","created_at":"2026-05-18T12:31:31.346846+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/MSE4O6QOBAXDQKCM5NALUBE2IV","json":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV.json","graph_json":"https://pith.science/api/pith-number/MSE4O6QOBAXDQKCM5NALUBE2IV/graph.json","events_json":"https://pith.science/api/pith-number/MSE4O6QOBAXDQKCM5NALUBE2IV/events.json","paper":"https://pith.science/paper/MSE4O6QO"},"agent_actions":{"view_html":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV","download_json":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV.json","view_paper":"https://pith.science/paper/MSE4O6QO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.04014&json=true","fetch_graph":"https://pith.science/api/pith-number/MSE4O6QOBAXDQKCM5NALUBE2IV/graph.json","fetch_events":"https://pith.science/api/pith-number/MSE4O6QOBAXDQKCM5NALUBE2IV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV/action/storage_attestation","attest_author":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV/action/author_attestation","sign_citation":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV/action/citation_signature","submit_replication":"https://pith.science/pith/MSE4O6QOBAXDQKCM5NALUBE2IV/action/replication_record"}},"created_at":"2026-05-17T23:44:15.689485+00:00","updated_at":"2026-05-17T23:44:15.689485+00:00"}