The Periodic Isoperimetric Problem
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| The Gyroid is a triply periodic minimal surface with cubic symmetry discovered by A. Schoen. As a distintive symmetry it has 41-fold axes. It is conjectured that the Gyroid has least area among surfaces dividing the3-space into two I 4132 invariant regions with equal volume fractions. | P Schwarz triply periodic minimal surface is conjectured to have least area among surfaces dividing the 3-space in two Pm3m-invariant regions with equal volume fractions. |
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The periodic isoperimetric problem is an interesting open question in classical differential geometry: given a discrete group G of isometries of the Euclidean three space, it consists of describing, among surfaces dividing the space in two G-invariant regions with prescribed volume fraction, those which have least area (per unit cell). These minimizing surfaces always exist, are free of singularities and have constant mean curvature (some of them are minimal surfaces) . The periodic isoperimetric problem is also the simplest mathematical model to explain certain shapes appearing in a number of nanostructured interface phenomena in materials science, where spherical, cylindrical and lamellar configurations alternate with more sophisticated bicontinuous ones. In these last cases, interfaces are small perturbations of triply periodic constant mean curvature Gyroids (G=I4132), Primitive (G=Pm3m) or Diamond (G=Fd3m) Schwarz surfaces and other conjectural candidates to solving the G-periodic isoperimetric problem for various crystallographic groups G. |
Stable periodic constant mean curvature surfaces and mesoscopic phase separation. pdf, by Antonio Ros. Interfaces and Free Boundaries (to appear).
We give a comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice symmetry. This result is intended to elucidate the geometry of certain sophisticated interfaces appearing in mesoscale phase separation phenomena.
Isoperimetric Inequalities in Crystallography, pdf file (200 kb), by Antonio Ros, . Journal of the AMS 17 (2004), 373-388. We prove sharp isoperimetric inequalities for regions which are invariant under a given cubic space group.
The Periodic Isoperimetric Problem, pdf file (350 kb), by L. Hauswirth, J. Perez, P. Romon and A. Ros, Transactions of the AMS 356 (2004), no. 5, 2025--2047 Given a discrete group G of isometries in the 3-space, we study among G-invariant regions with prescribed volume fraction, those whose boundary has least area.
For an introduction to The Isoperimetric Problem, see the notes of the Lecture series at the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, summer 2001, Mathematical Sciences Research Institute, Berkeley, California. more... For a short presentation of this subject see the paper Some updates on isoperimetric problems, by M. Ritoré and A. Ros, The Mathematical Intelligencer, 24 (2002) 9-14.
