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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$"},"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":"1303.4222","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-03-18T12:05:55Z","cross_cats_sorted":[],"title_canon_sha256":"209d3c3070b940dc3ca6b7de8ec1fd7762cfc7f657cd68bc3955aced5dd2e6a3","abstract_canon_sha256":"ffa30c5125acf984e9259287182ba8f22014f20a050427d2283c71897bb6ad45"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:05.022364Z","signature_b64":"L29OJFcdU2ejr0ARVI1aI2ZxjTGQEFEVVsd2aQLqzL9FMHdjEaP1H/rcMG+w1KpDEpKUKV5g9B6NYMD672oDDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39dab91449c3e3fd44d0657f2cd03780e1d3137f8d733cb4b67194219738cd79","last_reissued_at":"2026-05-18T03:29:05.021816Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:05.021816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\frac{1}{2}\\{Ch}(X)$. 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