Space filling minimal surfaces and sphere packings

(Received 25 October 1993, accepted in final form 20 January 1994)

Abstract
A space filling minimal surface is defined to be any embedded minimal surface without boundary with the property that the area and genus enclosed by any large spherical region scales in proportion to the volume of the region. The triply periodic minimal surfaces are one realization, but not necessarily the only one. By using the genus per unit volume of the surface, a meaningful comparison of surface areas can be made even in cases where there is no unit cell. Of the known periodic minimal surfaces this measure of the surface area is smallest for Schoen's FRD surface. This surface is one of several that is closely related to packings of spheres. Its low area is largely due to the fact that the corresponding sphere packing (fcc) has the maximal kissing number.