{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:MF7GZAXFLGQ3U3RJLPBLAEMKUW","short_pith_number":"pith:MF7GZAXF","schema_version":"1.0","canonical_sha256":"617e6c82e559a1ba6e295bc2b0118aa5881560bc0430464c9245ab814845b3ef","source":{"kind":"arxiv","id":"1210.2112","version":2},"attestation_state":"computed","paper":{"title":"Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\\leq n\\leq 6$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DG","authors_text":"Xin Zhou","submitted_at":"2012-10-07T22:18:58Z","abstract_excerpt":"In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \\cite{A2}\\cite{P} corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature with $2\\leq n\\leq 6$. We characterize the Morse index, area and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces."},"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":"1210.2112","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-07T22:18:58Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"41183065387ea4853049382f1a1908cd6d3e488dcf8421ee58d9789ba835fe57","abstract_canon_sha256":"c5895583859ecae13620f2a5a0285c52d707a4bfbf224eb13761dc2e7354d219"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:45.813450Z","signature_b64":"Ba4MmAO7LorHwi4RaKgfh9J3ZFcHFuLMf49VRW1MStF0GDzmfwMEFM15DfdtZMKr+tKHEdlsv+BBwuZRbsN/CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"617e6c82e559a1ba6e295bc2b0118aa5881560bc0430464c9245ab814845b3ef","last_reissued_at":"2026-05-18T03:42:45.812910Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:45.812910Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\\leq n\\leq 6$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DG","authors_text":"Xin Zhou","submitted_at":"2012-10-07T22:18:58Z","abstract_excerpt":"In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \\cite{A2}\\cite{P} corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature with $2\\leq n\\leq 6$. We characterize the Morse index, area and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2112","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":"1210.2112","created_at":"2026-05-18T03:42:45.812985+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.2112v2","created_at":"2026-05-18T03:42:45.812985+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2112","created_at":"2026-05-18T03:42:45.812985+00:00"},{"alias_kind":"pith_short_12","alias_value":"MF7GZAXFLGQ3","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MF7GZAXFLGQ3U3RJ","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MF7GZAXF","created_at":"2026-05-18T12:27:14.488303+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/MF7GZAXFLGQ3U3RJLPBLAEMKUW","json":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW.json","graph_json":"https://pith.science/api/pith-number/MF7GZAXFLGQ3U3RJLPBLAEMKUW/graph.json","events_json":"https://pith.science/api/pith-number/MF7GZAXFLGQ3U3RJLPBLAEMKUW/events.json","paper":"https://pith.science/paper/MF7GZAXF"},"agent_actions":{"view_html":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW","download_json":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW.json","view_paper":"https://pith.science/paper/MF7GZAXF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.2112&json=true","fetch_graph":"https://pith.science/api/pith-number/MF7GZAXFLGQ3U3RJLPBLAEMKUW/graph.json","fetch_events":"https://pith.science/api/pith-number/MF7GZAXFLGQ3U3RJLPBLAEMKUW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW/action/storage_attestation","attest_author":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW/action/author_attestation","sign_citation":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW/action/citation_signature","submit_replication":"https://pith.science/pith/MF7GZAXFLGQ3U3RJLPBLAEMKUW/action/replication_record"}},"created_at":"2026-05-18T03:42:45.812985+00:00","updated_at":"2026-05-18T03:42:45.812985+00:00"}