Periodic Surfaces of Prescribed Mean Curvature

Abstract

While there are eighteen triply periodic minimal surfaces that reportedly are free of self-intersections, to date there is no known example of a triply periodic surface of constant, nonzero mean curvature that is embedded in R ³ (three dimensional Euclidean space). We have computed and displayed [1,2] five families of such surfaces, where every surface in a given family has the same space group, the same Euler characteristic per lattice-fundamental region, and the same dual pair of triply periodic graphs that define the connectivity of the two labyrinthine subvolumes created by the infinitely connected surface. Each family comprises two branches, corresponding to the two possible signs of the mean curvature, and a minimal surface. The branches have been tracked in mean curvature, and the surface areas and volume fractions recorded, with the region dA = 2HdV carefully checked to hold. The three families that contain the minimal surfaces P and D of Schwarz and the I-WP minimal surface of Schoen terminate in configurations that are close-packed spheres. However, one branch of the family that includes the Neovius surface C(P) contains self-intersecting solutions and terminates at self-intersecting spheres. On approaching the sphere limit, whether self-intersecting or close-packed, the gradual disappearance of small ‘neck’ or ‘connector’ regions between neighboring ’sphere-like’ regions is in close analogy with the rotationally symmetric unduloids of Delauney. We give what we suspect are analytical values for the areas of the I-WP and F-RD minimal surfaces, and a possible limit on the magnitude of the mean curvature in such families is proposed and discussed. We also report that the I-WP and F-RD minimal surfaces each divide R ³ into two subspaces of unequal volume fractions.