{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2002:ECPNTYPX7C3YIQBYQOGOSECG4E","short_pith_number":"pith:ECPNTYPX","canonical_record":{"source":{"id":"math/0211191","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2002-11-12T17:54:48Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"6fa8f4f8a1509d6a8f1fd8ee194f3faa03ed05f9050340ca121eae3636b1479e","abstract_canon_sha256":"75a4bc718ef4e28faa3eb02d816d916f55126d8079d9a74b4de6e433923158f8"},"schema_version":"1.0"},"canonical_sha256":"209ed9e1f7f8b7844038838ce91046e13bbc2675396ec2681c8732e594ea389a","source":{"kind":"arxiv","id":"math/0211191","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0211191","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0211191v3","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0211191","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"pith_short_12","alias_value":"ECPNTYPX7C3Y","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"ECPNTYPX7C3YIQBY","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"ECPNTYPX","created_at":"2026-05-18T12:25:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2002:ECPNTYPX7C3YIQBYQOGOSECG4E","target":"record","payload":{"canonical_record":{"source":{"id":"math/0211191","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2002-11-12T17:54:48Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"6fa8f4f8a1509d6a8f1fd8ee194f3faa03ed05f9050340ca121eae3636b1479e","abstract_canon_sha256":"75a4bc718ef4e28faa3eb02d816d916f55126d8079d9a74b4de6e433923158f8"},"schema_version":"1.0"},"canonical_sha256":"209ed9e1f7f8b7844038838ce91046e13bbc2675396ec2681c8732e594ea389a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:00.530128Z","signature_b64":"q0zjiIjW/HkACZluYAwCP5Ns4Qy2GvwuthWo6RyAvTs9mm5VAGB7Xbr7i/nMdR0EkzurP09g7izuB7gMS7BsBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"209ed9e1f7f8b7844038838ce91046e13bbc2675396ec2681c8732e594ea389a","last_reissued_at":"2026-05-18T02:38:00.529523Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:00.529523Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0211191","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":"2026-05-18T02:38:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1gTJ8O2JeElm2hNte+sKkiRXv2r8dW6uTUvjmGV59ZAfsT9mugjO4rK2Q6BKR+UPCN9CVhC5jb/739TYzZ9ZBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T01:20:24.438744Z"},"content_sha256":"dd442a8e2f7c39c515fa92f11875b841fdcf6c67d9ac3b97b41e16f8ddbd9d54","schema_version":"1.0","event_id":"sha256:dd442a8e2f7c39c515fa92f11875b841fdcf6c67d9ac3b97b41e16f8ddbd9d54"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2002:ECPNTYPX7C3YIQBYQOGOSECG4E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates","license":"","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"David Glickenstein","submitted_at":"2002-11-12T17:54:48Z","abstract_excerpt":"Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n-dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdorff sense to a space which i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0211191","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":"2026-05-18T02:38:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GN8m2Stc3hIv/KdzNAd+VCY76cJIfIppNJraCceFmSv2KAqbdH6R844KDjNJoz6JjsUK0kt6wal+hhcX34P3DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T01:20:24.439114Z"},"content_sha256":"e9983ac78061101be69a3558193fde9125248acfc65398603b3a6dc32560b714","schema_version":"1.0","event_id":"sha256:e9983ac78061101be69a3558193fde9125248acfc65398603b3a6dc32560b714"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ECPNTYPX7C3YIQBYQOGOSECG4E/bundle.json","state_url":"https://pith.science/pith/ECPNTYPX7C3YIQBYQOGOSECG4E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ECPNTYPX7C3YIQBYQOGOSECG4E/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":"2026-05-23T01:20:24Z","links":{"resolver":"https://pith.science/pith/ECPNTYPX7C3YIQBYQOGOSECG4E","bundle":"https://pith.science/pith/ECPNTYPX7C3YIQBYQOGOSECG4E/bundle.json","state":"https://pith.science/pith/ECPNTYPX7C3YIQBYQOGOSECG4E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ECPNTYPX7C3YIQBYQOGOSECG4E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2002:ECPNTYPX7C3YIQBYQOGOSECG4E","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":"75a4bc718ef4e28faa3eb02d816d916f55126d8079d9a74b4de6e433923158f8","cross_cats_sorted":["math.MG"],"license":"","primary_cat":"math.DG","submitted_at":"2002-11-12T17:54:48Z","title_canon_sha256":"6fa8f4f8a1509d6a8f1fd8ee194f3faa03ed05f9050340ca121eae3636b1479e"},"schema_version":"1.0","source":{"id":"math/0211191","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0211191","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0211191v3","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0211191","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"pith_short_12","alias_value":"ECPNTYPX7C3Y","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"ECPNTYPX7C3YIQBY","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"ECPNTYPX","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:e9983ac78061101be69a3558193fde9125248acfc65398603b3a6dc32560b714","target":"graph","created_at":"2026-05-18T02:38:00Z","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":"Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n-dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdorff sense to a space which i","authors_text":"David Glickenstein","cross_cats":["math.MG"],"headline":"","license":"","primary_cat":"math.DG","submitted_at":"2002-11-12T17:54:48Z","title":"Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0211191","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:dd442a8e2f7c39c515fa92f11875b841fdcf6c67d9ac3b97b41e16f8ddbd9d54","target":"record","created_at":"2026-05-18T02:38:00Z","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":"75a4bc718ef4e28faa3eb02d816d916f55126d8079d9a74b4de6e433923158f8","cross_cats_sorted":["math.MG"],"license":"","primary_cat":"math.DG","submitted_at":"2002-11-12T17:54:48Z","title_canon_sha256":"6fa8f4f8a1509d6a8f1fd8ee194f3faa03ed05f9050340ca121eae3636b1479e"},"schema_version":"1.0","source":{"id":"math/0211191","kind":"arxiv","version":3}},"canonical_sha256":"209ed9e1f7f8b7844038838ce91046e13bbc2675396ec2681c8732e594ea389a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"209ed9e1f7f8b7844038838ce91046e13bbc2675396ec2681c8732e594ea389a","first_computed_at":"2026-05-18T02:38:00.529523Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:00.529523Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q0zjiIjW/HkACZluYAwCP5Ns4Qy2GvwuthWo6RyAvTs9mm5VAGB7Xbr7i/nMdR0EkzurP09g7izuB7gMS7BsBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:00.530128Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0211191","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd442a8e2f7c39c515fa92f11875b841fdcf6c67d9ac3b97b41e16f8ddbd9d54","sha256:e9983ac78061101be69a3558193fde9125248acfc65398603b3a6dc32560b714"],"state_sha256":"07d30b6b91e021372a2cebab7e2b6a707c5c6fe354b6be445a6b7cce291e4214"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jqgOOaTaEqEY3TceDU9/JgyELt4wjNfH8X4iwhWD8dAQeWokkPLZUtR1ljoMdjAj6ecXlOgtLEQkI0qMWGU/Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T01:20:24.441420Z","bundle_sha256":"a8305b00511150ff9c74930aa95b5615f8e87d895666f8d7dc99ee7c30f3f70c"}}