Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group H¹

2015/01/29 by Jih-Hsin Cheng, Cheng, Jih-Hsin, Andrea Malchiodi +3
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1501.07357

openalex publication_date 2015/01/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to ℝ³ 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 H¹, characterizing the isoperimetric sets and constants for a family of Riemannian adapted metrics. Using Γ-convergence of the perimeter functionals, we also settle an isoperimetric conjecture in H¹ posed by P.Pansu.

Citations

• Embedded minimal tori in S3 and the Lawson conjecture

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