{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:RZOWR5U43A7THBCVWSLXZAFJ2K","short_pith_number":"pith:RZOWR5U4","canonical_record":{"source":{"id":"0906.2889","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-06-16T14:57:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"365c66821ff60ef6a650bdd60e48a7d5d42e5a0e840644d829d40d4b260da423","abstract_canon_sha256":"6d74ce40c71c54016ccd835c9b370851b0b1e535b7d464c5797ff3e07ebc59de"},"schema_version":"1.0"},"canonical_sha256":"8e5d68f69cd83f338455b4977c80a9d2b17d9e819ec0490e89961b129bad13f8","source":{"kind":"arxiv","id":"0906.2889","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.2889","created_at":"2026-05-18T04:23:22Z"},{"alias_kind":"arxiv_version","alias_value":"0906.2889v3","created_at":"2026-05-18T04:23:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.2889","created_at":"2026-05-18T04:23:22Z"},{"alias_kind":"pith_short_12","alias_value":"RZOWR5U43A7T","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"RZOWR5U43A7THBCV","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"RZOWR5U4","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:RZOWR5U43A7THBCVWSLXZAFJ2K","target":"record","payload":{"canonical_record":{"source":{"id":"0906.2889","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-06-16T14:57:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"365c66821ff60ef6a650bdd60e48a7d5d42e5a0e840644d829d40d4b260da423","abstract_canon_sha256":"6d74ce40c71c54016ccd835c9b370851b0b1e535b7d464c5797ff3e07ebc59de"},"schema_version":"1.0"},"canonical_sha256":"8e5d68f69cd83f338455b4977c80a9d2b17d9e819ec0490e89961b129bad13f8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:22.287041Z","signature_b64":"GtuU0F7nn+HL2qYrXFC9i39TabIj2RAGfMeLsJLpcrus6cXj5hq2YmEa88T+Sgr92sFR5zkJyV1FFJOqgUnfDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e5d68f69cd83f338455b4977c80a9d2b17d9e819ec0490e89961b129bad13f8","last_reissued_at":"2026-05-18T04:23:22.286393Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:22.286393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.2889","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZbOvu7DMGnI0KbJowwkuqlGBRLXemFvXaecnvjOzA6oFvEwQPViSs1yt/eLaZmhi8KNqsATpQGqywqiK1sACAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T06:27:04.892626Z"},"content_sha256":"410dc650e361babcb6befd390b0a9b62a7678b343ef637a613a2301a7918f803","schema_version":"1.0","event_id":"sha256:410dc650e361babcb6befd390b0a9b62a7678b343ef637a613a2301a7918f803"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:RZOWR5U43A7THBCVWSLXZAFJ2K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Normal curvature bounds along the mean curvature flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Hong Huang","submitted_at":"2009-06-16T14:57:27Z","abstract_excerpt":"Let $(M^n,g_0)$ and $(\\bar{M}^{n+1},\\bar{g})$ be complete Riemannian manifolds with $|\\bar{\\nabla}^k\\bar{Rm}|\\le \\bar{C}$ for $k \\le 2$, and suppose there is an isometric immersion $F_0: M^n \\rightarrow \\bar{M}^{n+1}$ with bounded second fundamental form. Let $F_t: M^n \\rightarrow \\bar{M}^{n+1}$ ($t\\in [0,T]$) be a family of immersions evolving by mean curvature flow with initial data $F_0$ and with uniformly bounded second fundamental forms.\n  We show that the supremum and infimum of the normal curvature of the immersions $F_t$ vary at a bounded rate. This is an analogue of a result of Rong a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.2889","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"doi++jy0wV1IHcelOUAhug23k6JoJ1OSj26BSy6K3KJVtexJnxzvY3mDG4jauI2BjOxqIU1ZODOgie1Gq2NYDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T06:27:04.892994Z"},"content_sha256":"5743d9cb77c7ccb182ac4107954d5f2e3ed3ee4aeb1a259276878110ba0dcdb4","schema_version":"1.0","event_id":"sha256:5743d9cb77c7ccb182ac4107954d5f2e3ed3ee4aeb1a259276878110ba0dcdb4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RZOWR5U43A7THBCVWSLXZAFJ2K/bundle.json","state_url":"https://pith.science/pith/RZOWR5U43A7THBCVWSLXZAFJ2K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RZOWR5U43A7THBCVWSLXZAFJ2K/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":"2026-05-25T06:27:04Z","links":{"resolver":"https://pith.science/pith/RZOWR5U43A7THBCVWSLXZAFJ2K","bundle":"https://pith.science/pith/RZOWR5U43A7THBCVWSLXZAFJ2K/bundle.json","state":"https://pith.science/pith/RZOWR5U43A7THBCVWSLXZAFJ2K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RZOWR5U43A7THBCVWSLXZAFJ2K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:RZOWR5U43A7THBCVWSLXZAFJ2K","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":"6d74ce40c71c54016ccd835c9b370851b0b1e535b7d464c5797ff3e07ebc59de","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-06-16T14:57:27Z","title_canon_sha256":"365c66821ff60ef6a650bdd60e48a7d5d42e5a0e840644d829d40d4b260da423"},"schema_version":"1.0","source":{"id":"0906.2889","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.2889","created_at":"2026-05-18T04:23:22Z"},{"alias_kind":"arxiv_version","alias_value":"0906.2889v3","created_at":"2026-05-18T04:23:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.2889","created_at":"2026-05-18T04:23:22Z"},{"alias_kind":"pith_short_12","alias_value":"RZOWR5U43A7T","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"RZOWR5U43A7THBCV","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"RZOWR5U4","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:5743d9cb77c7ccb182ac4107954d5f2e3ed3ee4aeb1a259276878110ba0dcdb4","target":"graph","created_at":"2026-05-18T04:23:22Z","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":"Let $(M^n,g_0)$ and $(\\bar{M}^{n+1},\\bar{g})$ be complete Riemannian manifolds with $|\\bar{\\nabla}^k\\bar{Rm}|\\le \\bar{C}$ for $k \\le 2$, and suppose there is an isometric immersion $F_0: M^n \\rightarrow \\bar{M}^{n+1}$ with bounded second fundamental form. Let $F_t: M^n \\rightarrow \\bar{M}^{n+1}$ ($t\\in [0,T]$) be a family of immersions evolving by mean curvature flow with initial data $F_0$ and with uniformly bounded second fundamental forms.\n  We show that the supremum and infimum of the normal curvature of the immersions $F_t$ vary at a bounded rate. This is an analogue of a result of Rong a","authors_text":"Hong Huang","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-06-16T14:57:27Z","title":"Normal curvature bounds along the mean curvature flow"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.2889","kind":"arxiv","version":3},"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":"sha256:410dc650e361babcb6befd390b0a9b62a7678b343ef637a613a2301a7918f803","target":"record","created_at":"2026-05-18T04:23:22Z","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":"6d74ce40c71c54016ccd835c9b370851b0b1e535b7d464c5797ff3e07ebc59de","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-06-16T14:57:27Z","title_canon_sha256":"365c66821ff60ef6a650bdd60e48a7d5d42e5a0e840644d829d40d4b260da423"},"schema_version":"1.0","source":{"id":"0906.2889","kind":"arxiv","version":3}},"canonical_sha256":"8e5d68f69cd83f338455b4977c80a9d2b17d9e819ec0490e89961b129bad13f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e5d68f69cd83f338455b4977c80a9d2b17d9e819ec0490e89961b129bad13f8","first_computed_at":"2026-05-18T04:23:22.286393Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:22.286393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GtuU0F7nn+HL2qYrXFC9i39TabIj2RAGfMeLsJLpcrus6cXj5hq2YmEa88T+Sgr92sFR5zkJyV1FFJOqgUnfDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:22.287041Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.2889","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:410dc650e361babcb6befd390b0a9b62a7678b343ef637a613a2301a7918f803","sha256:5743d9cb77c7ccb182ac4107954d5f2e3ed3ee4aeb1a259276878110ba0dcdb4"],"state_sha256":"46365cf2682a54b5887b5393d9ca0686566662cd4bb20070f30ac34fc283fd5b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ura2wl4+o/IU5lpsTSqEA/z4kP4nAEFS6+Cay3UHQ98JbP3ThEMgrES5yPsFEOTmHMkApwJkDgRKDJIuNspFDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T06:27:04.895363Z","bundle_sha256":"dbbd76c687f2ff827656b728ced6a8c0f4441b73b64b22ba5ec538864c2d4a11"}}