Stable Periodic Constant Mean Curvature Surfaces and Mesoscopic Phase Separation. Interfaces Free Bound

Abstract.- We give, for the first time, a comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice sym-metry. 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 dis-crete subgroup of translations with rank k, have genus ≤ k. Finally we extend the genus estimate to ambient spaces of the type M ×R, whereM is a nonnegatively curved surface. 1