{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:HA54DJH46CDDZXQNNQJJDM5BKZ","short_pith_number":"pith:HA54DJH4","canonical_record":{"source":{"id":"1211.2792","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-12T20:54:49Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"f29e49694c711a95403a14ae9a57f0b9f3ec603318211fa2323486640fa78e3d","abstract_canon_sha256":"0e3a42e3bab0d81a15ea90ec4c4999952db4d9728247f31a9cc5007705706303"},"schema_version":"1.0"},"canonical_sha256":"383bc1a4fcf0863cde0d6c1291b3a1565045441fcea6b511758057e5ca86c7a8","source":{"kind":"arxiv","id":"1211.2792","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2792","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2792v1","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2792","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"HA54DJH46CDD","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HA54DJH46CDDZXQN","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HA54DJH4","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:HA54DJH46CDDZXQNNQJJDM5BKZ","target":"record","payload":{"canonical_record":{"source":{"id":"1211.2792","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-12T20:54:49Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"f29e49694c711a95403a14ae9a57f0b9f3ec603318211fa2323486640fa78e3d","abstract_canon_sha256":"0e3a42e3bab0d81a15ea90ec4c4999952db4d9728247f31a9cc5007705706303"},"schema_version":"1.0"},"canonical_sha256":"383bc1a4fcf0863cde0d6c1291b3a1565045441fcea6b511758057e5ca86c7a8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:14.815775Z","signature_b64":"siG5UBtxqn5Dskfvtyt6aBeqjKBu5AuDhsdS5CgWyqvFtB4TykFqi8FGvEYzJIdx9s0JhUtIaTPu0fUq+MlsDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"383bc1a4fcf0863cde0d6c1291b3a1565045441fcea6b511758057e5ca86c7a8","last_reissued_at":"2026-05-18T00:10:14.815165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:14.815165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.2792","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":"2026-05-18T00:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DNN4AvtF6sMeKx+afsC+w/OwGaBJvEJXIu/dsolRfJATp/DsvqdbbjZ0jKuKdt3HgVpKwl+jGlg69V/2hrvCDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:12:46.864215Z"},"content_sha256":"82362f2e41a2b620397b610a8bef03aed70604e5cda5dcbb1129aa0099b951f6","schema_version":"1.0","event_id":"sha256:82362f2e41a2b620397b610a8bef03aed70604e5cda5dcbb1129aa0099b951f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:HA54DJH46CDDZXQNNQJJDM5BKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Super Ricci flow for disjoint unions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Michael Munn, Sajjad Lakzian","submitted_at":"2012-11-12T20:54:49Z","abstract_excerpt":"In this paper we consider compact, Riemannian manifolds $M_1, M_2$ each equipped with a one-parameter family of metrics $g_1(t), g_2(t)$ satisfying the Ricci flow equation. Motivated by a characterization of the super Ricci flow developed by McCann-Topping, we introduce the notion of a super Ricci flow for a family of distance metrics defined on the disjoint union $\\MM$. In particular, we show such a super Ricci flow property holds provided the distance function between points in $M_1$ and $M_2$ evolves by the heat equation. We also discuss possible applications and examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2792","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":"2026-05-18T00:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"prkm3V0vM+nV389aa/HF7Dun8cvZOG8k1/2eaqnCASoWFlyQgBWPEe/h9hpZ9MOFWKHk+agU3Oyu4NLX4TNvAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T22:12:46.864840Z"},"content_sha256":"1f4c4095f5f2cb4c42b1197009f29af4f40d3330c785a0862e382af9f0c1e0d8","schema_version":"1.0","event_id":"sha256:1f4c4095f5f2cb4c42b1197009f29af4f40d3330c785a0862e382af9f0c1e0d8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HA54DJH46CDDZXQNNQJJDM5BKZ/bundle.json","state_url":"https://pith.science/pith/HA54DJH46CDDZXQNNQJJDM5BKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HA54DJH46CDDZXQNNQJJDM5BKZ/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":"2026-05-31T22:12:46Z","links":{"resolver":"https://pith.science/pith/HA54DJH46CDDZXQNNQJJDM5BKZ","bundle":"https://pith.science/pith/HA54DJH46CDDZXQNNQJJDM5BKZ/bundle.json","state":"https://pith.science/pith/HA54DJH46CDDZXQNNQJJDM5BKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HA54DJH46CDDZXQNNQJJDM5BKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HA54DJH46CDDZXQNNQJJDM5BKZ","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":"0e3a42e3bab0d81a15ea90ec4c4999952db4d9728247f31a9cc5007705706303","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-12T20:54:49Z","title_canon_sha256":"f29e49694c711a95403a14ae9a57f0b9f3ec603318211fa2323486640fa78e3d"},"schema_version":"1.0","source":{"id":"1211.2792","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2792","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2792v1","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2792","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"HA54DJH46CDD","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HA54DJH46CDDZXQN","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HA54DJH4","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:1f4c4095f5f2cb4c42b1197009f29af4f40d3330c785a0862e382af9f0c1e0d8","target":"graph","created_at":"2026-05-18T00:10:14Z","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 consider compact, Riemannian manifolds $M_1, M_2$ each equipped with a one-parameter family of metrics $g_1(t), g_2(t)$ satisfying the Ricci flow equation. Motivated by a characterization of the super Ricci flow developed by McCann-Topping, we introduce the notion of a super Ricci flow for a family of distance metrics defined on the disjoint union $\\MM$. In particular, we show such a super Ricci flow property holds provided the distance function between points in $M_1$ and $M_2$ evolves by the heat equation. We also discuss possible applications and examples.","authors_text":"Michael Munn, Sajjad Lakzian","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-12T20:54:49Z","title":"Super Ricci flow for disjoint unions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2792","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":"sha256:82362f2e41a2b620397b610a8bef03aed70604e5cda5dcbb1129aa0099b951f6","target":"record","created_at":"2026-05-18T00:10:14Z","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":"0e3a42e3bab0d81a15ea90ec4c4999952db4d9728247f31a9cc5007705706303","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-12T20:54:49Z","title_canon_sha256":"f29e49694c711a95403a14ae9a57f0b9f3ec603318211fa2323486640fa78e3d"},"schema_version":"1.0","source":{"id":"1211.2792","kind":"arxiv","version":1}},"canonical_sha256":"383bc1a4fcf0863cde0d6c1291b3a1565045441fcea6b511758057e5ca86c7a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"383bc1a4fcf0863cde0d6c1291b3a1565045441fcea6b511758057e5ca86c7a8","first_computed_at":"2026-05-18T00:10:14.815165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:14.815165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"siG5UBtxqn5Dskfvtyt6aBeqjKBu5AuDhsdS5CgWyqvFtB4TykFqi8FGvEYzJIdx9s0JhUtIaTPu0fUq+MlsDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:14.815775Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.2792","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82362f2e41a2b620397b610a8bef03aed70604e5cda5dcbb1129aa0099b951f6","sha256:1f4c4095f5f2cb4c42b1197009f29af4f40d3330c785a0862e382af9f0c1e0d8"],"state_sha256":"898d4f111d2c26a86f08aacee0ebe7d23e94dd787483533dbe1a6ff9d2f4bcd3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9VVuKkM/ZNOD2ETjhDjGwqqC6pt+959C1C5TuZ+xZNIq+IBFcf8Y3hS4GH1j96M+udezBZKkH9R6vowT+qMXDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T22:12:46.868437Z","bundle_sha256":"7befa80ae995265ab2fac5263e17509698182505ba420d7dff6480338eac3a26"}}