Isoperimetric domains of large volume in homogeneous three-manifolds

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 $H(X)$.