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Isoperimetric inequalities in crystallography

Author: Antonio Ros
Journal: J. Amer. Math. Soc. 17 (2004), 373-388
MSC (2000): Primary 53A10, 53C42, 20H15
Published electronically: December 2, 2003
MathSciNet review: 2051615
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Abstract: 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 $3.09$).

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Additional Information

Antonio Ros
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Keywords: Isoperimetric problem, periodic minimal surfaces, cubic symmetry
Received by editor(s): March 17, 2003
Published electronically: December 2, 2003
Additional Notes: Partially supported by MCYT-FEDER research projects BFM2001-3318
Article copyright: © Copyright 2003 American Mathematical Society