Isoperimetric domains of large volume in homogeneous three-manifolds.

Given a non-compact, simply connected homogeneous three-manifold X and a sequence { Ω n } n of isoperimetric domains in X with volumes tending to infinity, we prove that, as n → ∞ : (1) The radii of the Ω n tend to infinity. (2) The ratios Area ( ∂ Ω n ) Vol ( Ω n ) converge to the Cheeger constant Ch ( X ) , which we also prove to be equal to 2 H ( 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 ∂ Ω n converge to 1 2 Ch ( X ) . Furthermore, when Ch ( X ) is positive, we prove that for n large, ∂ Ω 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 ) . [ABSTRACT FROM AUTHOR]