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

TL;DR

This paper characterizes isoperimetric sets in three-dimensional homogeneous manifolds, including the Heisenberg group, proving they are topological balls or specific surfaces, and confirms a conjecture about the isoperimetric constant in the Heisenberg group.

Contribution

It establishes the topological nature of isoperimetric sets in homogeneous 3-manifolds and confirms a conjecture about the isoperimetric constant in the Heisenberg group.

Findings

Isoperimetric sets in homogeneous 3-manifolds are topological balls.

In homogeneous spheres, isoperimetric sets are spheres or genus-one tori.

The isoperimetric constant in the Heisenberg group is characterized and the Pansu conjecture is settled.

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.