{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:KE73FLUFW65EN5TI6BLMVJOW22","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":"ea6eac2c06b132d8553987a6ac3dcb802871f633065898e2043136da6e5108fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-30T12:30:41Z","title_canon_sha256":"4ed207a70b482628b8f7dcadadf58d92e498dfc3e6690e6e4a541b1afab857c1"},"schema_version":"1.0","source":{"id":"1208.6151","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.6151","created_at":"2026-05-18T03:07:41Z"},{"alias_kind":"arxiv_version","alias_value":"1208.6151v3","created_at":"2026-05-18T03:07:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.6151","created_at":"2026-05-18T03:07:41Z"},{"alias_kind":"pith_short_12","alias_value":"KE73FLUFW65E","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"KE73FLUFW65EN5TI","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"KE73FLUF","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:01863906e7a76338d459a9219bde4618a0b67057637a9d3bd1ddf87864f74d3f","target":"graph","created_at":"2026-05-18T03:07:41Z","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 $M$ be a complete Riemannian $3$-manifold with sectional curvatures between $0$ and $1$. A minimal $2$-sphere immersed in $M$ has area at least $4\\pi$. If an embedded minimal sphere has area $4\\pi$, then $M$ is isometric to the unit $3$-sphere or to a quotient of the product of the unit $2$-sphere with $\\mathbb{R}$, with the product metric. We also obtain a rigidity theorem for the existence of hyperbolic cusps. Let $M$ be a complete Riemannian $3$-manifold with sectional curvatures bounded above by $-1$. Suppose there is a $2$-torus $T$ embedded in $M$ with mean curvature one. Then the me","authors_text":"Harold Rosenberg, Laurent Mazet","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-30T12:30:41Z","title":"On minimal spheres of area $4\\pi$ and rigidity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.6151","kind":"arxiv","version":3},"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:c89f5d6bb61eff2fb32bc43e9ffb0806815a79e1fb6aea19e4cfeb58dc89544f","target":"record","created_at":"2026-05-18T03:07:41Z","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":"ea6eac2c06b132d8553987a6ac3dcb802871f633065898e2043136da6e5108fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-08-30T12:30:41Z","title_canon_sha256":"4ed207a70b482628b8f7dcadadf58d92e498dfc3e6690e6e4a541b1afab857c1"},"schema_version":"1.0","source":{"id":"1208.6151","kind":"arxiv","version":3}},"canonical_sha256":"513fb2ae85b7ba46f668f056caa5d6d6a0acdb24f3be1e25c662b6f20c0ea5ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"513fb2ae85b7ba46f668f056caa5d6d6a0acdb24f3be1e25c662b6f20c0ea5ac","first_computed_at":"2026-05-18T03:07:41.309967Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:41.309967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JvEEjUJ9zxah/CQON8pnVafNI+4sNWpr+naahAGfQ4H9a+GRNV3onVW9ipuHMrVtJ+Yyp/BNVJMqJ2VTzvYdCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:41.310586Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.6151","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c89f5d6bb61eff2fb32bc43e9ffb0806815a79e1fb6aea19e4cfeb58dc89544f","sha256:01863906e7a76338d459a9219bde4618a0b67057637a9d3bd1ddf87864f74d3f"],"state_sha256":"201a172cd82eded00784aba3567ccfb1b0361dc15662662c925040fad92b4974"}