Isoperimetric Inequalities in Crystallography

The study of the isoperimetric problem in the presence of crystallographic symmetries is an interesting unsolved question in classical differential geometry: Given a space group G, we want to describe, among surfaces dividing Euclidean 3-space into two G-invariant regions with prescribed volume fractions, those which have the least area per unit cell of the group. We know that this periodic isoperimetric problem always has solutions and any of these solutions is a smooth triply periodic surface with constant mean curvature ([3, 8, 12, 20, 21]). The explicit description of these surfaces, which we call G-isoperimetric surfaces, remains open in most of the cases. Let us consider, for instance, the following two simple situations. Assume the prefixed volume fractions are (both) equal to 1/2. If G is the cubic integer lattice Z, then it was proved by Hadwiger [10] that the G-isoperimetric surface is, up to symmetries, the family of parallel planes x = n/2, n ∈ Z. If G is the group of symmetries of Z (Pm3m in crystallographic notation), then it is conjectured that the isoperimetric surface is the classical P Schwarz triply periodic minimal surface. From the point of view of applications, it is also important to obtain explicit isoperimetric inequalities for periodic regions with prescribed symmetries. Triply periodic minimal surfaces and related surfaces are widely accepted in crystallography and materials science; see for instance the papers [6, 13, 17, 29, 30]. They appear as interface models in mesoscale self-assembled phenomena. Some of these are reminiscent of the familiar soap bubbles (such as lipid-water systems), but others are of a different nature (such as diblock copolymers). Roughly speaking, these interfaces are periodic and separate two immiscible liquid materials. The size of the periods varies between tens and hundreds of nanometers and the more interesting cases usually present cubic symmetry. Mesoscale interfaces can be treated as periodic surfaces minimizing (under volume fraction constraints) a suitable energy, whose dominant term is, in several cases, the area of the interface itself. So the periodic isoperimetric problem is the simplest geometric model to explain the shapes of these interfaces. Among the surfaces appearing most frequently in this context are periodic arrays of planes, spheres and cylinders, and small perturbations of constant mean curvature P (G = Pm3m) and D (G = Fd3m) Schwarz surfaces and the G Schoen Gyroid (G = I4132).