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Transactions of the American Mathematical Society

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The periodic isoperimetric problem

Authors: Laurent Hauswirth, Joaquín Pérez, Pascal Romon and Antonio Ros
Journal: Trans. Amer. Math. Soc. 356 (2004), 2025-2047
MSC (2000): Primary 53A10, 53C42
Published electronically: October 28, 2003
MathSciNet review: 2031051
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Abstract: 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 $\mathbb{Z} ^3$.

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Additional Information

Laurent Hauswirth
Affiliation: Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France

Joaquín Pérez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

Pascal Romon
Affiliation: Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France

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

Received by editor(s): February 6, 2003
Received by editor(s) in revised form: April 11, 2003
Published electronically: October 28, 2003
Additional Notes: The first and third authors were partially supported by Picasso program 02669WB and J. Pérez and A. Ros by MCYT-FEDER research projects BFM2001-3318 and HF2000-0088
Article copyright: © Copyright 2003 American Mathematical Society