{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:LCQOGQMTNQ2YGRVPI4IVOE2DR2","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":"e153d93f29aaef0148cdcb95e2ef48bed67cc118261054613f416c03507e89c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-23T09:28:55Z","title_canon_sha256":"cc0f557bff15c09e9d9aa666e0c2b75351544e47fa8abf343272e0e12f7e6997"},"schema_version":"1.0","source":{"id":"1204.5010","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.5010","created_at":"2026-05-18T03:57:14Z"},{"alias_kind":"arxiv_version","alias_value":"1204.5010v1","created_at":"2026-05-18T03:57:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.5010","created_at":"2026-05-18T03:57:14Z"},{"alias_kind":"pith_short_12","alias_value":"LCQOGQMTNQ2Y","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"LCQOGQMTNQ2YGRVP","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"LCQOGQMT","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:816acd8340d5411ee3ce1ecca21752a5d44ca4edf9b743b462855505234d7229","target":"graph","created_at":"2026-05-18T03:57:14Z","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":"In this paper, we formulate the notion of the $\\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\\mathcal{F}$-stable self-shrinkers in arbitrary codimension, in codimension one case, our results reduce to Colding-Minicozzi's results.","authors_text":"Ben Andrews, Haizhong Li, Yong Wei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-23T09:28:55Z","title":"$\\mathcal{F}$-stability for self-shrinking solutions to mean curvature flow"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.5010","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:b0107c4b753c09e91d91decb6cfa3f497824f3082e599cc3b35b8019012b691f","target":"record","created_at":"2026-05-18T03:57:14Z","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":"e153d93f29aaef0148cdcb95e2ef48bed67cc118261054613f416c03507e89c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-23T09:28:55Z","title_canon_sha256":"cc0f557bff15c09e9d9aa666e0c2b75351544e47fa8abf343272e0e12f7e6997"},"schema_version":"1.0","source":{"id":"1204.5010","kind":"arxiv","version":1}},"canonical_sha256":"58a0e341936c358346af47115713438ebbd4c2130994e7513decc993b540fe6b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58a0e341936c358346af47115713438ebbd4c2130994e7513decc993b540fe6b","first_computed_at":"2026-05-18T03:57:14.898840Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:14.898840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CqH3YEOxeVATgSYKZ+0Y3Sv0u4ElBTxqayjbcwKm87eezlgMopm5tWnaIWDDfgc9VzLJEAR1uUf8s0Ou2sXTDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:14.899288Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.5010","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0107c4b753c09e91d91decb6cfa3f497824f3082e599cc3b35b8019012b691f","sha256:816acd8340d5411ee3ce1ecca21752a5d44ca4edf9b743b462855505234d7229"],"state_sha256":"4877a39ae92ce7956d666e1eebeae8005c0211e74f3804c741c662279832572d"}