Isoperimetric inequalities in crystallography

Ros, Antonio¹

¹ Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Journal of the American Mathematical Society, Tome 17 (2004) no. 2, pp. 373-388

Résumé

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 $3.09$).

Détail

DOI : 10.1090/S0894-0347-03-00447-8

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