{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:CQTRMKVCIKG4H7VXH5YCZH5IZL","short_pith_number":"pith:CQTRMKVC","canonical_record":{"source":{"id":"1204.0106","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-03-31T15:28:59Z","cross_cats_sorted":[],"title_canon_sha256":"4537687748a8b40f3cd536350ed50123f0c3dcaa27a3239684e4ec81e3b90938","abstract_canon_sha256":"9fc8c16030852a2d0ffdcae736fb2a4f5b8c509fa76a4b6ecdff8b92e361f54d"},"schema_version":"1.0"},"canonical_sha256":"1427162aa2428dc3feb73f702c9fa8caec7633c87670e2bdaaadd5450b6aa347","source":{"kind":"arxiv","id":"1204.0106","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.0106","created_at":"2026-05-18T03:58:50Z"},{"alias_kind":"arxiv_version","alias_value":"1204.0106v1","created_at":"2026-05-18T03:58:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.0106","created_at":"2026-05-18T03:58:50Z"},{"alias_kind":"pith_short_12","alias_value":"CQTRMKVCIKG4","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CQTRMKVCIKG4H7VX","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CQTRMKVC","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:CQTRMKVCIKG4H7VXH5YCZH5IZL","target":"record","payload":{"canonical_record":{"source":{"id":"1204.0106","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-03-31T15:28:59Z","cross_cats_sorted":[],"title_canon_sha256":"4537687748a8b40f3cd536350ed50123f0c3dcaa27a3239684e4ec81e3b90938","abstract_canon_sha256":"9fc8c16030852a2d0ffdcae736fb2a4f5b8c509fa76a4b6ecdff8b92e361f54d"},"schema_version":"1.0"},"canonical_sha256":"1427162aa2428dc3feb73f702c9fa8caec7633c87670e2bdaaadd5450b6aa347","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:50.373567Z","signature_b64":"hL4acDbxqVlA7a0WIRwW6tuvJG23jOe5n7TUs0ZG2o6BGMKmjBsRnJlQh/9huC0tSasvuPS9/1oJGCjYAshNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1427162aa2428dc3feb73f702c9fa8caec7633c87670e2bdaaadd5450b6aa347","last_reissued_at":"2026-05-18T03:58:50.372913Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:50.372913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.0106","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":"2026-05-18T03:58:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mZ6pYiKfdSN/Pe6F4uLhFPYNDfGvqkXNIoex6q8ppo9gPAgf8sB3SUebcOsJj4XDmDy/1D5BCGzYXCMeNfLuCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:53:09.577361Z"},"content_sha256":"757c056e5b7b416ed8aba0af58baf1e816f9efe541f144bbbdb8c61277efedbe","schema_version":"1.0","event_id":"sha256:757c056e5b7b416ed8aba0af58baf1e816f9efe541f144bbbdb8c61277efedbe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:CQTRMKVCIKG4H7VXH5YCZH5IZL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Deforming submanifolds of arbitrary codimension in a sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Entao Zhao, Hongwei Xu, Kefeng Liu","submitted_at":"2012-03-31T15:28:59Z","abstract_excerpt":"In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\\mathbb{S}^{n+d}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0106","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":"2026-05-18T03:58:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ypvXUYieKFlFoVxgKamp0B8XeXxuEXDbK9s6ZhZc54YwBhDL6SpFZXnvmQKJ9fxaoeA2J5BIpomaty/WmpoRBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:53:09.577706Z"},"content_sha256":"256dd921fb265592418cb4855b3a00c1d4ea78abc663f8700d38706c6832f1c0","schema_version":"1.0","event_id":"sha256:256dd921fb265592418cb4855b3a00c1d4ea78abc663f8700d38706c6832f1c0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CQTRMKVCIKG4H7VXH5YCZH5IZL/bundle.json","state_url":"https://pith.science/pith/CQTRMKVCIKG4H7VXH5YCZH5IZL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CQTRMKVCIKG4H7VXH5YCZH5IZL/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":"2026-05-28T12:53:09Z","links":{"resolver":"https://pith.science/pith/CQTRMKVCIKG4H7VXH5YCZH5IZL","bundle":"https://pith.science/pith/CQTRMKVCIKG4H7VXH5YCZH5IZL/bundle.json","state":"https://pith.science/pith/CQTRMKVCIKG4H7VXH5YCZH5IZL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CQTRMKVCIKG4H7VXH5YCZH5IZL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CQTRMKVCIKG4H7VXH5YCZH5IZL","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":"9fc8c16030852a2d0ffdcae736fb2a4f5b8c509fa76a4b6ecdff8b92e361f54d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-03-31T15:28:59Z","title_canon_sha256":"4537687748a8b40f3cd536350ed50123f0c3dcaa27a3239684e4ec81e3b90938"},"schema_version":"1.0","source":{"id":"1204.0106","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.0106","created_at":"2026-05-18T03:58:50Z"},{"alias_kind":"arxiv_version","alias_value":"1204.0106v1","created_at":"2026-05-18T03:58:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.0106","created_at":"2026-05-18T03:58:50Z"},{"alias_kind":"pith_short_12","alias_value":"CQTRMKVCIKG4","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CQTRMKVCIKG4H7VX","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CQTRMKVC","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:256dd921fb265592418cb4855b3a00c1d4ea78abc663f8700d38706c6832f1c0","target":"graph","created_at":"2026-05-18T03:58:50Z","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 prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\\mathbb{S}^{n+d}$.","authors_text":"Entao Zhao, Hongwei Xu, Kefeng Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-03-31T15:28:59Z","title":"Deforming submanifolds of arbitrary codimension in a sphere"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0106","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":"sha256:757c056e5b7b416ed8aba0af58baf1e816f9efe541f144bbbdb8c61277efedbe","target":"record","created_at":"2026-05-18T03:58:50Z","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":"9fc8c16030852a2d0ffdcae736fb2a4f5b8c509fa76a4b6ecdff8b92e361f54d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-03-31T15:28:59Z","title_canon_sha256":"4537687748a8b40f3cd536350ed50123f0c3dcaa27a3239684e4ec81e3b90938"},"schema_version":"1.0","source":{"id":"1204.0106","kind":"arxiv","version":1}},"canonical_sha256":"1427162aa2428dc3feb73f702c9fa8caec7633c87670e2bdaaadd5450b6aa347","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1427162aa2428dc3feb73f702c9fa8caec7633c87670e2bdaaadd5450b6aa347","first_computed_at":"2026-05-18T03:58:50.372913Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:50.372913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hL4acDbxqVlA7a0WIRwW6tuvJG23jOe5n7TUs0ZG2o6BGMKmjBsRnJlQh/9huC0tSasvuPS9/1oJGCjYAshNAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:50.373567Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.0106","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:757c056e5b7b416ed8aba0af58baf1e816f9efe541f144bbbdb8c61277efedbe","sha256:256dd921fb265592418cb4855b3a00c1d4ea78abc663f8700d38706c6832f1c0"],"state_sha256":"51b2628eabf2bd3279803b5b8b6d40b3c54501b4d9f87d9e0e9fc461df55ca79"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jw3F1dSRwrY9pqfWMiXeJ3iKfHk65aneGenXOPYvS+tVuOPQfZ5A7fz3p5lBv+ztYvJ/pWUNebv72Rsx3l96DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T12:53:09.579625Z","bundle_sha256":"47a2a6f5f0d41fbf7881ffde0cb20c579cc76222167572eaabcd6796a19b43d0"}}