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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Isoperimetric inequalities in crystallography
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by Antonio Ros
J. Amer. Math. Soc. 17 (2004), 373-388
Published electronically: December 2, 2003


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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Bibliographic Information
  • Antonio Ros
  • Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
  • Email:
  • Received by editor(s): March 17, 2003
  • Published electronically: December 2, 2003
  • Additional Notes: Partially supported by MCYT-FEDER research projects BFM2001-3318
  • © Copyright 2003 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 373-388
  • MSC (2000): Primary 53A10, 53C42, 20H15
  • DOI:
  • MathSciNet review: 2051615