{"paper":{"title":"Isoperimetric domains of large volume in homogeneous three-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Antonio Ros, Joaquin Perez, Pablo Mira, William H. Meeks III","submitted_at":"2013-03-18T12:05:55Z","abstract_excerpt":"Given a non-compact, simply connected homogeneous three-manifold $X$ and a sequence $\\{\\Omega_n\\}_n$ of isoperimetric domains in $X$ with volumes tending to infinity, we prove that as $n\\to \\infty $: 1. The radii of the $\\Omega_n$ tend to infinity. 2. The ratios $\\{Area} (\\partial \\Omega_n)/\\{Vol}(\\Omega_n)$ converge to the Cheeger constant Ch$(X)$, which we also prove to be equal to $2H(X)$ where $H(X)$ is the critical mean curvature of $X$. 3. The values of the constant mean curvatures $H_n$ of the boundary surfaces $\\partial \\Omega_n$ converge to $\\frac{1}{2}\\{Ch}(X)$. Furthermore, when Ch$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4222","kind":"arxiv","version":3},"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"}