Stable periodic constant mean curvature surfaces and mesoscopic phase separation

Abstract

We give a first comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice symmetry. This result is intended to elucidate the geometry of certain sophisticated interfaces appearing in mesoscale phase separation phenomena. We prove that closed, stable, constant mean curvature surfaces in ℝ3/Γ, Γ ∈ ℝ3 being a discrete subgroup of translations with rank k, have genus ≤ k. Finally, we extend the genus estimate to ambient spaces of the type M × ℝ, where M is a nonnegatively curved surface. © European Mathematical Society 2007.