{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MF7GZAXFLGQ3U3RJLPBLAEMKUW","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":"c5895583859ecae13620f2a5a0285c52d707a4bfbf224eb13761dc2e7354d219","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-07T22:18:58Z","title_canon_sha256":"41183065387ea4853049382f1a1908cd6d3e488dcf8421ee58d9789ba835fe57"},"schema_version":"1.0","source":{"id":"1210.2112","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.2112","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"arxiv_version","alias_value":"1210.2112v2","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2112","created_at":"2026-05-18T03:42:45Z"},{"alias_kind":"pith_short_12","alias_value":"MF7GZAXFLGQ3","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MF7GZAXFLGQ3U3RJ","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MF7GZAXF","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:d4719c7af03acfd4ad49d434b9af8bd674a19f4b4036cd4e9ea992013fa24fbf","target":"graph","created_at":"2026-05-18T03:42:45Z","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 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.","authors_text":"Xin Zhou","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-07T22:18:58Z","title":"Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\\leq n\\leq 6$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2112","kind":"arxiv","version":2},"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":"sha256:887f776862cb140032e922979ec3331072014c790bfb29e3f6397708abeb0453","target":"record","created_at":"2026-05-18T03:42:45Z","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":"c5895583859ecae13620f2a5a0285c52d707a4bfbf224eb13761dc2e7354d219","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-07T22:18:58Z","title_canon_sha256":"41183065387ea4853049382f1a1908cd6d3e488dcf8421ee58d9789ba835fe57"},"schema_version":"1.0","source":{"id":"1210.2112","kind":"arxiv","version":2}},"canonical_sha256":"617e6c82e559a1ba6e295bc2b0118aa5881560bc0430464c9245ab814845b3ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"617e6c82e559a1ba6e295bc2b0118aa5881560bc0430464c9245ab814845b3ef","first_computed_at":"2026-05-18T03:42:45.812910Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:45.812910Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ba4MmAO7LorHwi4RaKgfh9J3ZFcHFuLMf49VRW1MStF0GDzmfwMEFM15DfdtZMKr+tKHEdlsv+BBwuZRbsN/CA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:45.813450Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.2112","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:887f776862cb140032e922979ec3331072014c790bfb29e3f6397708abeb0453","sha256:d4719c7af03acfd4ad49d434b9af8bd674a19f4b4036cd4e9ea992013fa24fbf"],"state_sha256":"710ee23709bd64774fbf60798faaff9f6a8b79cb0484f4750037d10280801375"}