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After assuming strict stability of $\\Sigma$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(- \\epsilon,\\epsilon)\\times \\Sigma$. We also show that if $\\Sigma$ is a critical point for the Hawking mass on the deSitter-Schwarzschild manifold $\\mathbb{R}\\times\\Sph^2$ and can be written as a graph over a slice $\\Sigma_r=\\{r\\}\\times\\mathbb{S}^2$, then $\\Sigma$ itself must b"},"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":"1206.5511","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-06-24T15:47:48Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"bd71e2d48525c6c434955717676c290818654bfa2ac125217092b9e7a71cc239","abstract_canon_sha256":"d690a9b7d9406b5cdeecaed1459a7821fd356cf13dfb33efd90fcf0e5d35ea11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:39.885428Z","signature_b64":"Og3aIizaw0DgN+SpZQ8nZ7z72Nj2XxW40+c+W+Hc14IVaUCWbiVLNnV4eTDxMwqL0XtnwXtESBI+lv8s2qySBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ba9ff7dbe4a02a1bb9c8f78a8dda8aff55a9a951bee516fd71338e8a2593828","last_reissued_at":"2026-05-18T03:52:39.884699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:39.884699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hawking mass and local rigidity of minimal two-spheres in three-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.DG","authors_text":"Davi M\\'aximo, Ivaldo Nunes","submitted_at":"2012-06-24T15:47:48Z","abstract_excerpt":"We study rigidity of minimal two-spheres $\\Sigma$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. 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