The periodic isoperimetric problem

Given a discrete group $G$ of isometries of $\mathbb{R} ^3$, we study the $G$-isoperimetric problem, which consists of minimizing area (modulo $G$) among surfaces in $\mathbb{R} ^3$ which enclose a $G$-invariant region with a prescribed volume fraction. If $G$ is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where $G=Pm\overline{3}m$ (the group of symmetries of the integer rank three lattice $\mathbb{Z} ^3$) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than $1/6$, and we give an isoperimetric inequality for $G$-invariant regions that, for instance, implies that the area (modulo $\mathbb{Z} ^3$) of a surface dividing the three space in two $G$-invariant regions with equal volume fractions, is at least $2.19$ (the conjectured solution is the classical $P$ Schwarz triply periodic minimal surface whose area is $\sim 2.34$). Another consequence of this isoperimetric inequality is that $Pm\overline{3}m$-symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group $\mathbb{Z} ^3$.