Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group $\mathsf{H}^1$

Paul C. Yang, Jih-Hsin Cheng, Andrea Malchiodi

Publication date: 29 January 2015

Abstract: 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.

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