{"paper":{"title":"Rigidity of area-minimizing hyperbolic surfaces in three-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Ivaldo Nunes","submitted_at":"2011-03-24T16:48:35Z","abstract_excerpt":"We prove that if $M$ is a three-manifold with scalar curvature greater than or equal to -2 and $\\Sigma\\subset M$ is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of $\\Sigma$ is greater than or equal to $4\\pi(g(\\Sigma)-1)$, where $g(\\Sigma)$ denotes the genus of $\\Sigma$. In the equality case, we prove that the induced metric on $\\Sigma$ has constant Gauss curvature equal to -1 and locally $M$ splits along $\\Sigma$. As a corollary, we obtain a rigidity result for cylinders $(I\\times\\Sigma,dt^2+g_{\\Sigma})$, where $I=[a,b]\\su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.4805","kind":"arxiv","version":1},"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"}