{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:MYAOQRVYNZX23HD6FNHZLX3RVZ","short_pith_number":"pith:MYAOQRVY","schema_version":"1.0","canonical_sha256":"6600e846b86e6fad9c7e2b4f95df71ae7a93a836925f5fb3417a0393d8d41801","source":{"kind":"arxiv","id":"1504.00966","version":2},"attestation_state":"computed","paper":{"title":"Min-max hypersurface in manifold of positive Ricci curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Xin Zhou","submitted_at":"2015-04-04T02:09:54Z","abstract_excerpt":"In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \\cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface.\n  As an essential technical tool, we "},"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":"1504.00966","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-04T02:09:54Z","cross_cats_sorted":[],"title_canon_sha256":"f38cbc2f19ab10bc8c6c9665b9b5359a57c8d24327001b9f6efab7a09931f5f5","abstract_canon_sha256":"dda5c591ec89900fd6001e1994d2c01330d99d9ee18177743603e4385a4f461a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:12.567747Z","signature_b64":"Wkmo7DRFraBMrfbscZhH2DBFhHh7cC17mP1+kCaN1uJ+JsAy+DF4KN2pUr7385mnmsGy7GrP1B3mVUjSIdD4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6600e846b86e6fad9c7e2b4f95df71ae7a93a836925f5fb3417a0393d8d41801","last_reissued_at":"2026-05-18T00:56:12.567236Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:12.567236Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Min-max hypersurface in manifold of positive Ricci curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Xin Zhou","submitted_at":"2015-04-04T02:09:54Z","abstract_excerpt":"In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \\cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface.\n  As an essential technical tool, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00966","kind":"arxiv","version":2},"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":"1504.00966","created_at":"2026-05-18T00:56:12.567329+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.00966v2","created_at":"2026-05-18T00:56:12.567329+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00966","created_at":"2026-05-18T00:56:12.567329+00:00"},{"alias_kind":"pith_short_12","alias_value":"MYAOQRVYNZX2","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_16","alias_value":"MYAOQRVYNZX23HD6","created_at":"2026-05-18T12:29:32.376354+00:00"},{"alias_kind":"pith_short_8","alias_value":"MYAOQRVY","created_at":"2026-05-18T12:29:32.376354+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ","json":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ.json","graph_json":"https://pith.science/api/pith-number/MYAOQRVYNZX23HD6FNHZLX3RVZ/graph.json","events_json":"https://pith.science/api/pith-number/MYAOQRVYNZX23HD6FNHZLX3RVZ/events.json","paper":"https://pith.science/paper/MYAOQRVY"},"agent_actions":{"view_html":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ","download_json":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ.json","view_paper":"https://pith.science/paper/MYAOQRVY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.00966&json=true","fetch_graph":"https://pith.science/api/pith-number/MYAOQRVYNZX23HD6FNHZLX3RVZ/graph.json","fetch_events":"https://pith.science/api/pith-number/MYAOQRVYNZX23HD6FNHZLX3RVZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ/action/storage_attestation","attest_author":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ/action/author_attestation","sign_citation":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ/action/citation_signature","submit_replication":"https://pith.science/pith/MYAOQRVYNZX23HD6FNHZLX3RVZ/action/replication_record"}},"created_at":"2026-05-18T00:56:12.567329+00:00","updated_at":"2026-05-18T00:56:12.567329+00:00"}