{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:NIQ6JX5BQCPP6IJ6M2UL5GEPTH","short_pith_number":"pith:NIQ6JX5B","schema_version":"1.0","canonical_sha256":"6a21e4dfa1809eff213e66a8be988f99e748ccfcd66e0ccc261b8c87c88124e8","source":{"kind":"arxiv","id":"1903.12317","version":1},"attestation_state":"computed","paper":{"title":"Proof of Bishop's volume comparison theorem using singular soap bubbles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Feng Gui, Hubert Bray, Yiyue Zhang, Zhenhua Liu","submitted_at":"2019-03-29T01:48:15Z","abstract_excerpt":"Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as \"soap bubbles,\" which minimize area for a given volume. Curiously, isoperimetric hypersurfaces can have codimension 7 singularities, an interesting challenge we are forced to overcome."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1903.12317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-03-29T01:48:15Z","cross_cats_sorted":[],"title_canon_sha256":"a8861d504754b4f359715942601fe9349af14911854eabfdd4e9721340c5d038","abstract_canon_sha256":"e343fa83470b9f5a1f87f9cc59e0509498e18d38dadf0820910dfd313d508d89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:54.341601Z","signature_b64":"H4ImVx8pHoq/uOpEWiOUdfmRmrhSwr82Iff/nhJRp2zncf4FmTYUFb48n7NVqUrriEwDa0Ug38LwFAulJwjFCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a21e4dfa1809eff213e66a8be988f99e748ccfcd66e0ccc261b8c87c88124e8","last_reissued_at":"2026-05-17T23:49:54.341188Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:54.341188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of Bishop's volume comparison theorem using singular soap bubbles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Feng Gui, Hubert Bray, Yiyue Zhang, Zhenhua Liu","submitted_at":"2019-03-29T01:48:15Z","abstract_excerpt":"Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as \"soap bubbles,\" which minimize area for a given volume. Curiously, isoperimetric hypersurfaces can have codimension 7 singularities, an interesting challenge we are forced to overcome."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.12317","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1903.12317","created_at":"2026-05-17T23:49:54.341246+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.12317v1","created_at":"2026-05-17T23:49:54.341246+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.12317","created_at":"2026-05-17T23:49:54.341246+00:00"},{"alias_kind":"pith_short_12","alias_value":"NIQ6JX5BQCPP","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_16","alias_value":"NIQ6JX5BQCPP6IJ6","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_8","alias_value":"NIQ6JX5B","created_at":"2026-05-18T12:33:24.271573+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2409.00583","citing_title":"Notes on scalar curvature lower bounds of steady gradient Ricci solitons","ref_index":3,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH","json":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH.json","graph_json":"https://pith.science/api/pith-number/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/graph.json","events_json":"https://pith.science/api/pith-number/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/events.json","paper":"https://pith.science/paper/NIQ6JX5B"},"agent_actions":{"view_html":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH","download_json":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH.json","view_paper":"https://pith.science/paper/NIQ6JX5B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.12317&json=true","fetch_graph":"https://pith.science/api/pith-number/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/graph.json","fetch_events":"https://pith.science/api/pith-number/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/action/storage_attestation","attest_author":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/action/author_attestation","sign_citation":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/action/citation_signature","submit_replication":"https://pith.science/pith/NIQ6JX5BQCPP6IJ6M2UL5GEPTH/action/replication_record"}},"created_at":"2026-05-17T23:49:54.341246+00:00","updated_at":"2026-05-17T23:49:54.341246+00:00"}