Isoperimetric inequalities in crystallography

Given a cubic space group $\mathcal G$ (viewed as a finite group of isometries of the torus $T=\mathbb{R} ^3/\mathbb{Z} ^3$), we obtain sharp isoperimetric inequalities for $\mathcal G$-invariant regions. These inequalities depend on the minimum number of points in an orbit of $\mathcal G$and on the largest Euler characteristic among nonspherical $\mathcal G$-symmetric surfaces minimizing the area under volume constraint (we also give explicit estimates of this second invariant for the various crystallographic cubic groups $\mathcal G$). As an example, we prove that any surface dividing $T$ into two equal volumes with the same (orientation-preserving) symmetries as the A. Schoen minimal Gyroid has area at least $3.00$ (the conjectured minimizing surface in this case is the Gyroid itself whose area is $3.09$).