Isoperimetric domains of large volume in homogeneous three-manifolds

2013/03/18 by William H. Meeks III, Meeks, William H., Pablo Mira +5
Mathematics · #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53A10 #Secondary 49Q05 #math.DG #msc:49Q05 #msc:53A10 #msc:53C42

paper · pdf · doi:10.48550/arxiv.1303.4222

47 pages, 2 figures; 1 conjecture removed from section 6 of last version

arxiv created 2013/04/03 · arxiv updated 2013/04/05

Abstract

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 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 ∂ 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).

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