{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:A4YVEMHWF6O66A2WORP3YCKNCY","short_pith_number":"pith:A4YVEMHW","canonical_record":{"source":{"id":"1603.02201","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-03-07T18:53:35Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c66a8d3f2f55425851015d9c5bb2f899f946cb37d99f1364989dbb0198bc0a5c","abstract_canon_sha256":"ee90170e2b1333ebc5054a1229eb32442413a4624874816244cd6d6a494cc04f"},"schema_version":"1.0"},"canonical_sha256":"07315230f62f9def0356745fbc094d163853221469b80673d9f26270a505b7be","source":{"kind":"arxiv","id":"1603.02201","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02201","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02201v1","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02201","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"pith_short_12","alias_value":"A4YVEMHWF6O6","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"A4YVEMHWF6O66A2W","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"A4YVEMHW","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:A4YVEMHWF6O66A2WORP3YCKNCY","target":"record","payload":{"canonical_record":{"source":{"id":"1603.02201","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-03-07T18:53:35Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c66a8d3f2f55425851015d9c5bb2f899f946cb37d99f1364989dbb0198bc0a5c","abstract_canon_sha256":"ee90170e2b1333ebc5054a1229eb32442413a4624874816244cd6d6a494cc04f"},"schema_version":"1.0"},"canonical_sha256":"07315230f62f9def0356745fbc094d163853221469b80673d9f26270a505b7be","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:31.379198Z","signature_b64":"X/z5QwB0svFmKAbsM/mDMnbKLsuBockK8hoCFiDkqylOJKRF8Ay6NzjL76FdDgN+AOFZu9S2L9/7lgdmbwNUCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07315230f62f9def0356745fbc094d163853221469b80673d9f26270a505b7be","last_reissued_at":"2026-05-18T01:19:31.378528Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:31.378528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.02201","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":"2026-05-18T01:19:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0MCY1NNEQ0N6NYQl92ugPXkaaKlSInv+INJtRHWNDKddXG0zCJL2ABfxLs7PlNLnjXe8M8ktHiGR2XyIQpUiDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:06:03.230391Z"},"content_sha256":"9643001e833df94a97d4360e65ee0512f3e3ec1040a29ad31fbd70692ccc4a85","schema_version":"1.0","event_id":"sha256:9643001e833df94a97d4360e65ee0512f3e3ec1040a29ad31fbd70692ccc4a85"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:A4YVEMHWF6O66A2WORP3YCKNCY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An integral formula and its applications on sub-static manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Chao Xia, Junfang Li","submitted_at":"2016-03-07T18:53:35Z","abstract_excerpt":"In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \\cite{Re2} and the recent result from \\cite{QX}. It provides a robust tool for sub-static manifolds regardless of the underlying topology.\n  Using this formula and suitable elliptic PDEs, we prove Heintze-Karcher type inequalities for bounded domains in general sub-static manifolds which recovers some of the results from Brendle \\cite{Br} as special cases.\n  On the other hand, we prove a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02201","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":"2026-05-18T01:19:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h87Lj5QVIyxoXo1+kx0QAzHOGfYxuJ0ki89pOwRIjJzjJ7rAhcbsFlR6KGOOU0Cy0kSetDP18B6xAuKkoOhxCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:06:03.230782Z"},"content_sha256":"b8563f417428084f92eaafca8ad2164537c110685e3f9c2d70b8db0a152d7e17","schema_version":"1.0","event_id":"sha256:b8563f417428084f92eaafca8ad2164537c110685e3f9c2d70b8db0a152d7e17"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A4YVEMHWF6O66A2WORP3YCKNCY/bundle.json","state_url":"https://pith.science/pith/A4YVEMHWF6O66A2WORP3YCKNCY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A4YVEMHWF6O66A2WORP3YCKNCY/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":"2026-06-22T22:06:03Z","links":{"resolver":"https://pith.science/pith/A4YVEMHWF6O66A2WORP3YCKNCY","bundle":"https://pith.science/pith/A4YVEMHWF6O66A2WORP3YCKNCY/bundle.json","state":"https://pith.science/pith/A4YVEMHWF6O66A2WORP3YCKNCY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A4YVEMHWF6O66A2WORP3YCKNCY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:A4YVEMHWF6O66A2WORP3YCKNCY","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":"ee90170e2b1333ebc5054a1229eb32442413a4624874816244cd6d6a494cc04f","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-03-07T18:53:35Z","title_canon_sha256":"c66a8d3f2f55425851015d9c5bb2f899f946cb37d99f1364989dbb0198bc0a5c"},"schema_version":"1.0","source":{"id":"1603.02201","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02201","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02201v1","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02201","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"pith_short_12","alias_value":"A4YVEMHWF6O6","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"A4YVEMHWF6O66A2W","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"A4YVEMHW","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:b8563f417428084f92eaafca8ad2164537c110685e3f9c2d70b8db0a152d7e17","target":"graph","created_at":"2026-05-18T01:19:31Z","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 article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \\cite{Re2} and the recent result from \\cite{QX}. It provides a robust tool for sub-static manifolds regardless of the underlying topology.\n  Using this formula and suitable elliptic PDEs, we prove Heintze-Karcher type inequalities for bounded domains in general sub-static manifolds which recovers some of the results from Brendle \\cite{Br} as special cases.\n  On the other hand, we prove a","authors_text":"Chao Xia, Junfang Li","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-03-07T18:53:35Z","title":"An integral formula and its applications on sub-static manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02201","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":"sha256:9643001e833df94a97d4360e65ee0512f3e3ec1040a29ad31fbd70692ccc4a85","target":"record","created_at":"2026-05-18T01:19:31Z","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":"ee90170e2b1333ebc5054a1229eb32442413a4624874816244cd6d6a494cc04f","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-03-07T18:53:35Z","title_canon_sha256":"c66a8d3f2f55425851015d9c5bb2f899f946cb37d99f1364989dbb0198bc0a5c"},"schema_version":"1.0","source":{"id":"1603.02201","kind":"arxiv","version":1}},"canonical_sha256":"07315230f62f9def0356745fbc094d163853221469b80673d9f26270a505b7be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"07315230f62f9def0356745fbc094d163853221469b80673d9f26270a505b7be","first_computed_at":"2026-05-18T01:19:31.378528Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:31.378528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X/z5QwB0svFmKAbsM/mDMnbKLsuBockK8hoCFiDkqylOJKRF8Ay6NzjL76FdDgN+AOFZu9S2L9/7lgdmbwNUCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:31.379198Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.02201","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9643001e833df94a97d4360e65ee0512f3e3ec1040a29ad31fbd70692ccc4a85","sha256:b8563f417428084f92eaafca8ad2164537c110685e3f9c2d70b8db0a152d7e17"],"state_sha256":"1ce389090e8162b88aa75db66d5c75073ee021a8be975b7f0bc2f9723e1fdeb4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aM2a/JR1lZS67up5hHgcXYJ9sivwx0/TycQN5vEp9nxm1d8pC6i9y58zDMri+iUYz3p+YOKQmGZWHcECuHooAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T22:06:03.233491Z","bundle_sha256":"e35705b28fec1011ce2f18c023afbe11e0bcba28ef5128df57fb8dd7d6769690"}}