Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The periodic isoperimetric problem
HTML articles powered by AMS MathViewer

by Laurent Hauswirth, Joaquín Pérez, Pascal Romon and Antonio Ros PDF
Trans. Amer. Math. Soc. 356 (2004), 2025-2047 Request permission

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 $\mathbb {Z}^3$.
References
  • D.M. Anderson, H.T. Davis, J.C.C. Nitsche, L.E. Scriven, Periodic Surfaces of Prescribed Mean Curvature, Advances in Chemical Physics, 77 (1990) 337-396.
  • Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
  • William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. MR 307015, DOI 10.2307/1970868
  • F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 420406, DOI 10.1090/memo/0165
  • Hiroshi Mori, On surfaces of right helicoid type in $H^{3}$, Bol. Soc. Brasil. Mat. 13 (1982), no. 2, 57–62. MR 735120, DOI 10.1007/BF02584676
  • F. Barthe, Extremal properties of central half-spaces for product measures, J. Funct. Anal. 182 (2001), no. 1, 81–107. MR 1829243, DOI 10.1006/jfan.2000.3708
  • Christophe Bavard and Pierre Pansu, Sur le volume minimal de $\textbf {R}^2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 479–490 (French). MR 875084
  • F. S. Bates & G. H. Fredrickson, Block copolymers-designer soft materials, Physics Today 52-2 (Feb. 1999) 32-38.
  • M. Carrion, J. Corneli, G. Walsh, S. Beheshti Double bubbles in the 3-torus, preprint.
  • J. Corneli, P. Holt, G. Lee, N. Leger, E. Schoenfeld & B. A. Steinhurst, The double bubble problem on the flat two-torus, preprint.
  • E. Gonzalez, U. Massari, and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), no. 1, 25–37. MR 684753, DOI 10.1512/iumj.1983.32.32003
  • Karsten Große-Brauckmann, New surfaces of constant mean curvature, Math. Z. 214 (1993), no. 4, 527–565. MR 1248112, DOI 10.1007/BF02572424
  • K. Grosse-Brauckmann, Cousins of constant mean curvature surfaces, preprint.
  • Michael Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rational Mech. Anal. 97 (1987), no. 3, 261–270. MR 862549, DOI 10.1007/BF00250810
  • H. Hadwiger, Gitterperiodische Punktmengen und Isoperimetrie, Monatsh. Math. 76 (1972), 410–418 (German). MR 324550, DOI 10.1007/BF01297304
  • T. Hahn, editor, International Tables for Crystallography, vol. A, fifth edition, Kluwer Academic Publishers, 2002.
  • Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
  • W. Y. Hsiang, A symmetry theorem on isoperimetric regions, PAM-409 (1988), UC Berkeley.
  • Wu-Yi Hsiang, Isoperimetric regions and soap bubbles, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 229–240. MR 1173044
  • S. T. Hyde, Identification of lyotropic liquid crystalline mesophases, Handbook of applied surface and colloid chemistry, Edited by K. Holmberg, John Wiley & Sons, Ltd. (2001).
  • Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. MR 1010168
  • E. Kroumova, J. M. Perez-Mato, M. I. Aroyo, S. Ivantchev, G. Madariaga, H. Wondratschek The Bilbao Crystallographic Server: a web site with crystallographic tools using the International Tables for Crystallography, 18th European Crystallographic Meeting, Praha, Czech Republic (1998), http://www.cryst.ehu.es.
  • H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
  • P. Lenz, C. Bechinger, C. Schäfle, P. Leiderer & R. Lipowsky, Perforated wetting layers from periodic patterns of lyophobic surface domains, Langmuir 17 (2001) 7814-7822.
  • Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407, DOI 10.1007/BF01399507
  • W. H. Meeks III, Lectures on Plateau’s problem, Scola de Geometria Diferencial, Universidade Federal do Ceará (Brazil), 1978.
  • William H. Meeks III, The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geom. 27 (1988), no. 3, 539–552. MR 940118
  • F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc. (to appear).
  • Frank Morgan and David L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J. 49 (2000), no. 3, 1017–1041. MR 1803220, DOI 10.1512/iumj.2000.49.1929
  • M. Ritoré, Superficies con curvatura media constante, Ph.D. Thesis, Granada, 1994, http://www.ugr.es/$\sim$ritore.
  • Manuel Ritoré, Index one minimal surfaces in flat three space forms, Indiana Univ. Math. J. 46 (1997), no. 4, 1137–1153. MR 1631568, DOI 10.1512/iumj.1997.46.1299
  • Manuel Ritoré and Antonio Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helv. 67 (1992), no. 2, 293–305. MR 1161286, DOI 10.1007/BF02566501
  • Manuel Ritoré and Antonio Ros, The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds, Trans. Amer. Math. Soc. 348 (1996), no. 1, 391–410. MR 1322955, DOI 10.1090/S0002-9947-96-01496-1
  • A. Ros, The isoperimetric problem, Proceedings of the Clay Mathematical Institute MSRI summer school on Minimal Surfaces (to appear).
  • Marty Ross, Schwarz’ $P$ and $D$ surfaces are stable, Differential Geom. Appl. 2 (1992), no. 2, 179–195. MR 1245555, DOI 10.1016/0926-2245(92)90032-I
  • H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer Verlag, Berlin (1890).
  • U. S. Schwarz & G. Gompper, Stability of bicontinuous cubic phases in ternary amphiphilic systems with spontaneous curvature, J. Chem. Phys. 112 (2000) 3792-3802.
  • J. Steiner, Sur le maximum et le minimum des figures dans le plan, sur la sphére et dans l’espace en général J. Reine Angew. Math. 24 (1842) 93-152.
  • E. L Thomas, D. M. Anderson, C.S. Henkee & D. Hoffman, Periodic area-minimizing surfaces in block copolymers, Nature 334 (1988) 598-602.
  • Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR 577325
  • Shing-Tung Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1-2, 109–158. MR 896385
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53A10, 53C42
  • Retrieve articles in all journals with MSC (2000): 53A10, 53C42
Additional Information
  • Laurent Hauswirth
  • Affiliation: Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France
  • MR Author ID: 649999
  • Email: hauswirth@univ-mlv.fr
  • Joaquín Pérez
  • Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain
  • Email: jperez@ugr.es
  • Pascal Romon
  • Affiliation: Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France
  • Email: romon@univ-mlv.fr
  • Antonio Ros
  • Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain
  • Email: aros@ugr.es
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 2025-2047
  • MSC (2000): Primary 53A10, 53C42
  • DOI: https://doi.org/10.1090/S0002-9947-03-03362-2
  • MathSciNet review: 2031051