{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:3ZPTD6GGVUVFJU6K32QVXZZSIT","short_pith_number":"pith:3ZPTD6GG","canonical_record":{"source":{"id":"2412.04684","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-12-06T00:51:22Z","cross_cats_sorted":[],"title_canon_sha256":"648dd41eb93ff3665972ea6288edee0c5eec2b7be717e0a4a04394cc9581344a","abstract_canon_sha256":"9e00b9ee1ecbff295d58e4d49372c381f642199885b84eed45d4054a008b598a"},"schema_version":"1.0"},"canonical_sha256":"de5f31f8c6ad2a54d3cadea15be73244c17d32e0620bc70209fbea223f13e81e","source":{"kind":"arxiv","id":"2412.04684","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.04684","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"arxiv_version","alias_value":"2412.04684v1","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.04684","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"pith_short_12","alias_value":"3ZPTD6GGVUVF","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"pith_short_16","alias_value":"3ZPTD6GGVUVFJU6K","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"pith_short_8","alias_value":"3ZPTD6GG","created_at":"2026-07-05T09:45:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:3ZPTD6GGVUVFJU6K32QVXZZSIT","target":"record","payload":{"canonical_record":{"source":{"id":"2412.04684","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-12-06T00:51:22Z","cross_cats_sorted":[],"title_canon_sha256":"648dd41eb93ff3665972ea6288edee0c5eec2b7be717e0a4a04394cc9581344a","abstract_canon_sha256":"9e00b9ee1ecbff295d58e4d49372c381f642199885b84eed45d4054a008b598a"},"schema_version":"1.0"},"canonical_sha256":"de5f31f8c6ad2a54d3cadea15be73244c17d32e0620bc70209fbea223f13e81e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:45:21.996087Z","signature_b64":"57rTaQx/moMpkTSPPId8sT17ZeoVV/3B1pDmXRktQb7EiGxnhox0UHTGdr1y64GDaTEOoonsTX7IlPsoI5rqAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de5f31f8c6ad2a54d3cadea15be73244c17d32e0620bc70209fbea223f13e81e","last_reissued_at":"2026-07-05T09:45:21.995594Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:45:21.995594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.04684","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":"2026-07-05T09:45:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JBZJkmi9WAOTKlliRtr3iHJ0MHb88HnkcBu9kZvQdl7QqV4hcBvr+wxXQS+CtKOh20PPu5FzPORd3GPwASIcCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T07:26:12.386857Z"},"content_sha256":"c5df0c7046db4614655457d76a5315112386030370e19be80ceef212ed187b6b","schema_version":"1.0","event_id":"sha256:c5df0c7046db4614655457d76a5315112386030370e19be80ceef212ed187b6b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:3ZPTD6GGVUVFJU6K32QVXZZSIT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Liouville theorems for ancient solutions to the V-harmonic map heat flows II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongbing Qiu, Qun Chen","submitted_at":"2024-12-06T00:51:22Z","abstract_excerpt":"When the domain is a complete noncompact Riemannian manifold with nonnegative Bakry--Emery Ricci curvature and the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, by carrying out refined gradient estimates, we obtain a better Liouville theorem for ancient solutions to the V-harmonic map heat flows. Furthermore, we can also derive a Liouville theorem for quasi-harmonic maps under an exponential growth condition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.04684","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.04684/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:45:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PD0C1dEWgz0k9NI8wLOzX+OTGrff7TFGtxUcDBCOVSjqsRggHrhQ4oN54ZnEa3fDpSwJwH0dR5gQOyxgIjfPBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T07:26:12.387356Z"},"content_sha256":"1f16b85f06abf2f4ea57f567c6d65cd831ffb2aa72ae42c0929b2273ef01ad2a","schema_version":"1.0","event_id":"sha256:1f16b85f06abf2f4ea57f567c6d65cd831ffb2aa72ae42c0929b2273ef01ad2a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3ZPTD6GGVUVFJU6K32QVXZZSIT/bundle.json","state_url":"https://pith.science/pith/3ZPTD6GGVUVFJU6K32QVXZZSIT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3ZPTD6GGVUVFJU6K32QVXZZSIT/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":"2026-08-21T07:26:12Z","links":{"resolver":"https://pith.science/pith/3ZPTD6GGVUVFJU6K32QVXZZSIT","bundle":"https://pith.science/pith/3ZPTD6GGVUVFJU6K32QVXZZSIT/bundle.json","state":"https://pith.science/pith/3ZPTD6GGVUVFJU6K32QVXZZSIT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3ZPTD6GGVUVFJU6K32QVXZZSIT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:3ZPTD6GGVUVFJU6K32QVXZZSIT","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":"9e00b9ee1ecbff295d58e4d49372c381f642199885b84eed45d4054a008b598a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-12-06T00:51:22Z","title_canon_sha256":"648dd41eb93ff3665972ea6288edee0c5eec2b7be717e0a4a04394cc9581344a"},"schema_version":"1.0","source":{"id":"2412.04684","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.04684","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"arxiv_version","alias_value":"2412.04684v1","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.04684","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"pith_short_12","alias_value":"3ZPTD6GGVUVF","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"pith_short_16","alias_value":"3ZPTD6GGVUVFJU6K","created_at":"2026-07-05T09:45:21Z"},{"alias_kind":"pith_short_8","alias_value":"3ZPTD6GG","created_at":"2026-07-05T09:45:21Z"}],"graph_snapshots":[{"event_id":"sha256:1f16b85f06abf2f4ea57f567c6d65cd831ffb2aa72ae42c0929b2273ef01ad2a","target":"graph","created_at":"2026-07-05T09:45:21Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2412.04684/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"When the domain is a complete noncompact Riemannian manifold with nonnegative Bakry--Emery Ricci curvature and the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, by carrying out refined gradient estimates, we obtain a better Liouville theorem for ancient solutions to the V-harmonic map heat flows. Furthermore, we can also derive a Liouville theorem for quasi-harmonic maps under an exponential growth condition.","authors_text":"Hongbing Qiu, Qun Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-12-06T00:51:22Z","title":"Liouville theorems for ancient solutions to the V-harmonic map heat flows II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.04684","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:c5df0c7046db4614655457d76a5315112386030370e19be80ceef212ed187b6b","target":"record","created_at":"2026-07-05T09:45:21Z","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":"9e00b9ee1ecbff295d58e4d49372c381f642199885b84eed45d4054a008b598a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-12-06T00:51:22Z","title_canon_sha256":"648dd41eb93ff3665972ea6288edee0c5eec2b7be717e0a4a04394cc9581344a"},"schema_version":"1.0","source":{"id":"2412.04684","kind":"arxiv","version":1}},"canonical_sha256":"de5f31f8c6ad2a54d3cadea15be73244c17d32e0620bc70209fbea223f13e81e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de5f31f8c6ad2a54d3cadea15be73244c17d32e0620bc70209fbea223f13e81e","first_computed_at":"2026-07-05T09:45:21.995594Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:45:21.995594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"57rTaQx/moMpkTSPPId8sT17ZeoVV/3B1pDmXRktQb7EiGxnhox0UHTGdr1y64GDaTEOoonsTX7IlPsoI5rqAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:45:21.996087Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.04684","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c5df0c7046db4614655457d76a5315112386030370e19be80ceef212ed187b6b","sha256:1f16b85f06abf2f4ea57f567c6d65cd831ffb2aa72ae42c0929b2273ef01ad2a"],"state_sha256":"7ed8563fbda21d274db7a2f47cdedba66361608dd62aca84d8ab8e15e7e03a56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WimRNEs4atsb+rKlvcmVGuMO53BzPx64uWsrmgfmjrGgFpohC6TOD9lXLRR9p73RhEYXC172B0Mtc+sIxxzkAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T07:26:12.391752Z","bundle_sha256":"671af1370c3286293ba29266baded5d0617a98927f11a4d46874fc67cdbee4e5"}}