{"paper":{"title":"Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Keomkyo Seo, Sung-Hong Min","submitted_at":"2012-01-13T02:42:06Z","abstract_excerpt":"Let $\\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $\\Sigma$ and the ideal boundary $\\partial_\\infty \\Sigma$, say $\\rvol(\\Sigma)$ and $\\rvol(\\partial_\\infty \\Sigma)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $\\rvol(\\partial_\\infty \\Sigma) \\geq \\rvol(\\mathbb{S}^{k-1})$, then $\\Sigma$ satisfies the classi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2732","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"}