Isoperimetric domains of large volume in homogeneous three-manifolds

TL;DR

This paper investigates the geometric properties of large-volume isoperimetric domains in homogeneous three-manifolds, establishing their asymptotic behavior, relation to Cheeger constant, and approximation by constant mean curvature foliations.

Contribution

It proves that large isoperimetric domains' boundaries have mean curvatures approaching half the Cheeger constant and are approximated by CMC foliations when the Cheeger constant is positive.

Findings

Radii of isoperimetric domains tend to infinity.

Area-to-volume ratio converges to Cheeger constant.

Boundary mean curvatures approach half the Cheeger constant.

Abstract

Given a non-compact, simply connected homogeneous three-manifold $X$ and a sequence $\{\Omega_n\}_n$ of isoperimetric domains in $X$ with volumes tending to infinity, we prove that as $n\to \infty $: 1. The radii of the $\Omega_n$ tend to infinity. 2. The ratios $\{Area} (\partial \Omega_n)/\{Vol}(\Omega_n)$ converge to the Cheeger constant Ch$(X)$, which we also prove to be equal to $2H(X)$ where $H(X)$ is the critical mean curvature of $X$. 3. The values of the constant mean curvatures $H_n$ of the boundary surfaces $\partial \Omega_n$ converge to $\frac{1}{2}\{Ch}(X)$. Furthermore, when Ch$(X)$ is positive, we prove that for $n$ large, $\partial \Omega_n$ is well-approximated in a natural sense by the leaves of a certain foliation of $X$, where every leaf of the foliation is a surface of constant mean curvature $H(X)$.