{"paper":{"title":"Soap bubbles and isoperimetric regions in the product of a closed manifold with Euclidean space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jes\\'us Gonzalo P\\'erez","submitted_at":"2013-12-21T22:10:31Z","abstract_excerpt":"For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\\times{\\mathbb R}^n$ are of the form $X\\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\\times$(Euclidean sphere). We give an example of a surface $X$, with Gaussian curvature negative somewhere, such that the product $X\\times{\\mathbb R}$ contains stable soap bubbles of arbitrarily large enclosed volume which do not even project surjectively onto the $X$ factor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6311","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"}