{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:5W6EPGJTTSX6WH4ILVTACMYWOP","short_pith_number":"pith:5W6EPGJT","canonical_record":{"source":{"id":"1204.6116","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-27T06:37:13Z","cross_cats_sorted":[],"title_canon_sha256":"a2a1768cf81d1a5c49302096fd67f079f54b5a0e134020f35a2183e76e5e97ad","abstract_canon_sha256":"0dc8b41a0d0f5f0022b45291062bbd46738a562b35f9775410907299dbf14b49"},"schema_version":"1.0"},"canonical_sha256":"edbc4799339cafeb1f885d6601331673fe2cf0151aeb3a6a7408c3ee8c17c7c8","source":{"kind":"arxiv","id":"1204.6116","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.6116","created_at":"2026-05-18T03:56:46Z"},{"alias_kind":"arxiv_version","alias_value":"1204.6116v1","created_at":"2026-05-18T03:56:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.6116","created_at":"2026-05-18T03:56:46Z"},{"alias_kind":"pith_short_12","alias_value":"5W6EPGJTTSX6","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"5W6EPGJTTSX6WH4I","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"5W6EPGJT","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:5W6EPGJTTSX6WH4ILVTACMYWOP","target":"record","payload":{"canonical_record":{"source":{"id":"1204.6116","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-27T06:37:13Z","cross_cats_sorted":[],"title_canon_sha256":"a2a1768cf81d1a5c49302096fd67f079f54b5a0e134020f35a2183e76e5e97ad","abstract_canon_sha256":"0dc8b41a0d0f5f0022b45291062bbd46738a562b35f9775410907299dbf14b49"},"schema_version":"1.0"},"canonical_sha256":"edbc4799339cafeb1f885d6601331673fe2cf0151aeb3a6a7408c3ee8c17c7c8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:46.972301Z","signature_b64":"lA1lMQtAPloCAmR5v013ujj9Ui0nUZ1WK7zK5rRzef9pqwGd4clLJkFyD9YDWDb2TlZu7McQWaveyo9zrYvMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"edbc4799339cafeb1f885d6601331673fe2cf0151aeb3a6a7408c3ee8c17c7c8","last_reissued_at":"2026-05-18T03:56:46.971662Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:46.971662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.6116","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:56:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4R8TmzPKdrwfLUt90rEBwnNevd4GgGS6A1t68xCCdr4OcJAfiHZtncDKvh6wmzkmkGRFTmfxTrZ+SFLCGFKFAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:45:43.055774Z"},"content_sha256":"227146416f4fbafb1f1c49bf50f25657e6fa70af7742145067fb33ee57e9cf61","schema_version":"1.0","event_id":"sha256:227146416f4fbafb1f1c49bf50f25657e6fa70af7742145067fb33ee57e9cf61"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:5W6EPGJTTSX6WH4ILVTACMYWOP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Stability of Self-Shrinkers of Mean Curvature Flow in Higher Codimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Yang-Kai Lue, Yng-Ing Lee","submitted_at":"2012-04-27T06:37:13Z","abstract_excerpt":"In this paper, we generalize Colding and Minicozzi's work \\cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the $F$-functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in \\cite{An} are unstable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6116","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:56:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Sm1rIiMrEgHoJIq7EZCoBGNRK1B5xL44blGPKMOIAZ32F2XtQWU9jCJ5O3xS7W8yxPxOEm7uAGHS4qVtMmqBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:45:43.056466Z"},"content_sha256":"67c2c93515912e3d59fbb474d86b741fdb2c252bf9906c3413b1abbad54b00a6","schema_version":"1.0","event_id":"sha256:67c2c93515912e3d59fbb474d86b741fdb2c252bf9906c3413b1abbad54b00a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5W6EPGJTTSX6WH4ILVTACMYWOP/bundle.json","state_url":"https://pith.science/pith/5W6EPGJTTSX6WH4ILVTACMYWOP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5W6EPGJTTSX6WH4ILVTACMYWOP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-12T09:45:43Z","links":{"resolver":"https://pith.science/pith/5W6EPGJTTSX6WH4ILVTACMYWOP","bundle":"https://pith.science/pith/5W6EPGJTTSX6WH4ILVTACMYWOP/bundle.json","state":"https://pith.science/pith/5W6EPGJTTSX6WH4ILVTACMYWOP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5W6EPGJTTSX6WH4ILVTACMYWOP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:5W6EPGJTTSX6WH4ILVTACMYWOP","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":"0dc8b41a0d0f5f0022b45291062bbd46738a562b35f9775410907299dbf14b49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-27T06:37:13Z","title_canon_sha256":"a2a1768cf81d1a5c49302096fd67f079f54b5a0e134020f35a2183e76e5e97ad"},"schema_version":"1.0","source":{"id":"1204.6116","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.6116","created_at":"2026-05-18T03:56:46Z"},{"alias_kind":"arxiv_version","alias_value":"1204.6116v1","created_at":"2026-05-18T03:56:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.6116","created_at":"2026-05-18T03:56:46Z"},{"alias_kind":"pith_short_12","alias_value":"5W6EPGJTTSX6","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"5W6EPGJTTSX6WH4I","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"5W6EPGJT","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:67c2c93515912e3d59fbb474d86b741fdb2c252bf9906c3413b1abbad54b00a6","target":"graph","created_at":"2026-05-18T03:56:46Z","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 generalize Colding and Minicozzi's work \\cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the $F$-functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in \\cite{An} are unstable.","authors_text":"Yang-Kai Lue, Yng-Ing Lee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-27T06:37:13Z","title":"The Stability of Self-Shrinkers of Mean Curvature Flow in Higher Codimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6116","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":"sha256:227146416f4fbafb1f1c49bf50f25657e6fa70af7742145067fb33ee57e9cf61","target":"record","created_at":"2026-05-18T03:56:46Z","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":"0dc8b41a0d0f5f0022b45291062bbd46738a562b35f9775410907299dbf14b49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-27T06:37:13Z","title_canon_sha256":"a2a1768cf81d1a5c49302096fd67f079f54b5a0e134020f35a2183e76e5e97ad"},"schema_version":"1.0","source":{"id":"1204.6116","kind":"arxiv","version":1}},"canonical_sha256":"edbc4799339cafeb1f885d6601331673fe2cf0151aeb3a6a7408c3ee8c17c7c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"edbc4799339cafeb1f885d6601331673fe2cf0151aeb3a6a7408c3ee8c17c7c8","first_computed_at":"2026-05-18T03:56:46.971662Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:56:46.971662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lA1lMQtAPloCAmR5v013ujj9Ui0nUZ1WK7zK5rRzef9pqwGd4clLJkFyD9YDWDb2TlZu7McQWaveyo9zrYvMBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:56:46.972301Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.6116","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:227146416f4fbafb1f1c49bf50f25657e6fa70af7742145067fb33ee57e9cf61","sha256:67c2c93515912e3d59fbb474d86b741fdb2c252bf9906c3413b1abbad54b00a6"],"state_sha256":"0ad20a35f0dcb11736decdaaa0094a5e754b6425f3ec2ff707789dd16dfd5c7e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"owMdQJ42Xl42VKzqLDPFghOhkgCeL4I/052oEp89GjaW94YqM4Ki7yQU2XUv6OmzoB2tpIK6elqG/4fEJvNsCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T09:45:43.059823Z","bundle_sha256":"88093936819fedd30ce1890b349827ef0eeb1fdc49d863a494dd99ce11ceaf55"}}