{"paper":{"title":"Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group $\\mathsf{H}^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Andrea Malchiodi, Jih-Hsin Cheng, Paul Yang","submitted_at":"2015-01-29T07:18:34Z","abstract_excerpt":"In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to $\\mathbb{R}^3$ are topological balls. We also prove that in three-dimensional homogeneous spheres isopermetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dimensional Heisenberg group $\\mathsf{H}^1$, characterizing the isoperimetric sets and constants for a family of Riemannian adapted metrics. Using $\\Gamma$-convergence of the perimeter functionals, we also settle an isoperimetric conjecture in $\\mathsf{H}^1$ posed by P.Pansu."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07357","kind":"arxiv","version":2},"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"}