{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:FFUHBQHLO4OGVNVRMHMGT25UPB","short_pith_number":"pith:FFUHBQHL","canonical_record":{"source":{"id":"1306.5079","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2013-06-21T09:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"30e65b8c148a4304b1c45c7f9a6205bb9b28038cfafceb13aa4338dafd713e0b","abstract_canon_sha256":"1059c84699c3fa77ae10e2378b901e01c748a9ac627704bfd9a6b1c7cee76b9b"},"schema_version":"1.0"},"canonical_sha256":"296870c0eb771c6ab6b161d869ebb4785399b8e6ce09cce90e768abd9a7614e6","source":{"kind":"arxiv","id":"1306.5079","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.5079","created_at":"2026-05-18T02:38:17Z"},{"alias_kind":"arxiv_version","alias_value":"1306.5079v1","created_at":"2026-05-18T02:38:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.5079","created_at":"2026-05-18T02:38:17Z"},{"alias_kind":"pith_short_12","alias_value":"FFUHBQHLO4OG","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FFUHBQHLO4OGVNVR","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FFUHBQHL","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:FFUHBQHLO4OGVNVRMHMGT25UPB","target":"record","payload":{"canonical_record":{"source":{"id":"1306.5079","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2013-06-21T09:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"30e65b8c148a4304b1c45c7f9a6205bb9b28038cfafceb13aa4338dafd713e0b","abstract_canon_sha256":"1059c84699c3fa77ae10e2378b901e01c748a9ac627704bfd9a6b1c7cee76b9b"},"schema_version":"1.0"},"canonical_sha256":"296870c0eb771c6ab6b161d869ebb4785399b8e6ce09cce90e768abd9a7614e6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:17.172293Z","signature_b64":"UD6xXX6sRDSs49VzV1y1LO6IRVVH7dtmu69pVP1+/97RxCgfT+MomBLYk9/7GzwBBueD9iOuNLY6JDPdVkmxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"296870c0eb771c6ab6b161d869ebb4785399b8e6ce09cce90e768abd9a7614e6","last_reissued_at":"2026-05-18T02:38:17.171817Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:17.171817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.5079","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":"2026-05-18T02:38:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gWU9MnxpYmQqiSd78DrtNdZnTur8pqc5sq7fn1CG23WiKt0qz0aO2huHVS5j6uFD1Q1SYlZXlPMMJDA/mDREAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T23:34:17.561154Z"},"content_sha256":"658c3e2c6927633df4ad80b145b6c585fba8e6d66d60a04f529ce1058e570c07","schema_version":"1.0","event_id":"sha256:658c3e2c6927633df4ad80b145b6c585fba8e6d66d60a04f529ce1058e570c07"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:FFUHBQHLO4OGVNVRMHMGT25UPB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Comparison Theorems for Manifold with Mean Convex Boundary","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jian Ge","submitted_at":"2013-06-21T09:20:56Z","abstract_excerpt":"Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\\in \\RR$, we give a sharp estimate of the upper bound of $\\rho(x)=\\dis(x, \\partial M)$, in terms of the mean curvature bound of the boundary. When $\\partial M$ is compact, the upper bound is achieved if and only if $M$ is isometric to a disk in space form. A Kaehler version of estimation is also proved. Moreover we prove a Laplace comparison theorem for distance function to the boundary of Kaehler manifold and also estimate the first eigenvalue of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5079","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":"2026-05-18T02:38:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JccBGP3Lm19YSUKtRmm5VEwGz5jPvMr8K3QSTw7cjyMra1Kj9vaxpXvBnS6IqYgH+f3rBG8o9wVRhfmzTi8kBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T23:34:17.561823Z"},"content_sha256":"d16f0515055e2612652cea35fed86810b9d2fc9b216203cfde6f0999a0114b0f","schema_version":"1.0","event_id":"sha256:d16f0515055e2612652cea35fed86810b9d2fc9b216203cfde6f0999a0114b0f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FFUHBQHLO4OGVNVRMHMGT25UPB/bundle.json","state_url":"https://pith.science/pith/FFUHBQHLO4OGVNVRMHMGT25UPB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FFUHBQHLO4OGVNVRMHMGT25UPB/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":"2026-05-30T23:34:17Z","links":{"resolver":"https://pith.science/pith/FFUHBQHLO4OGVNVRMHMGT25UPB","bundle":"https://pith.science/pith/FFUHBQHLO4OGVNVRMHMGT25UPB/bundle.json","state":"https://pith.science/pith/FFUHBQHLO4OGVNVRMHMGT25UPB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FFUHBQHLO4OGVNVRMHMGT25UPB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:FFUHBQHLO4OGVNVRMHMGT25UPB","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":"1059c84699c3fa77ae10e2378b901e01c748a9ac627704bfd9a6b1c7cee76b9b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2013-06-21T09:20:56Z","title_canon_sha256":"30e65b8c148a4304b1c45c7f9a6205bb9b28038cfafceb13aa4338dafd713e0b"},"schema_version":"1.0","source":{"id":"1306.5079","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.5079","created_at":"2026-05-18T02:38:17Z"},{"alias_kind":"arxiv_version","alias_value":"1306.5079v1","created_at":"2026-05-18T02:38:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.5079","created_at":"2026-05-18T02:38:17Z"},{"alias_kind":"pith_short_12","alias_value":"FFUHBQHLO4OG","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FFUHBQHLO4OGVNVR","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FFUHBQHL","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:d16f0515055e2612652cea35fed86810b9d2fc9b216203cfde6f0999a0114b0f","target":"graph","created_at":"2026-05-18T02:38:17Z","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":"Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\\in \\RR$, we give a sharp estimate of the upper bound of $\\rho(x)=\\dis(x, \\partial M)$, in terms of the mean curvature bound of the boundary. When $\\partial M$ is compact, the upper bound is achieved if and only if $M$ is isometric to a disk in space form. A Kaehler version of estimation is also proved. Moreover we prove a Laplace comparison theorem for distance function to the boundary of Kaehler manifold and also estimate the first eigenvalue of","authors_text":"Jian Ge","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2013-06-21T09:20:56Z","title":"Comparison Theorems for Manifold with Mean Convex Boundary"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5079","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":"sha256:658c3e2c6927633df4ad80b145b6c585fba8e6d66d60a04f529ce1058e570c07","target":"record","created_at":"2026-05-18T02:38:17Z","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":"1059c84699c3fa77ae10e2378b901e01c748a9ac627704bfd9a6b1c7cee76b9b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2013-06-21T09:20:56Z","title_canon_sha256":"30e65b8c148a4304b1c45c7f9a6205bb9b28038cfafceb13aa4338dafd713e0b"},"schema_version":"1.0","source":{"id":"1306.5079","kind":"arxiv","version":1}},"canonical_sha256":"296870c0eb771c6ab6b161d869ebb4785399b8e6ce09cce90e768abd9a7614e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"296870c0eb771c6ab6b161d869ebb4785399b8e6ce09cce90e768abd9a7614e6","first_computed_at":"2026-05-18T02:38:17.171817Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:17.171817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UD6xXX6sRDSs49VzV1y1LO6IRVVH7dtmu69pVP1+/97RxCgfT+MomBLYk9/7GzwBBueD9iOuNLY6JDPdVkmxDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:17.172293Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.5079","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:658c3e2c6927633df4ad80b145b6c585fba8e6d66d60a04f529ce1058e570c07","sha256:d16f0515055e2612652cea35fed86810b9d2fc9b216203cfde6f0999a0114b0f"],"state_sha256":"73089993d16ef8c5f1b4f4f9606015701bb4c3a78be30e1639b2a69f0243d419"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9MvPVfaD5RMAyJSTd1Ka3QOMHV3e8IBGxSan2QydjM7RGbABCKqZc/+0G+7Gf2pKtR8C8NvVE2oKcXwG5atDDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T23:34:17.565459Z","bundle_sha256":"7f03fae9c711b1b13ae59bdf15c168e5206c88bbc1acb932bf7202656795465b"}}