The periodic isoperimetric problem

Transactions of the American Mathematical Society

Volumn 356, Issue 5, 2004, Pages 2025-2047

  Hauswirth, Laurent ^a   Pérez, Joaquín ^b   Romon, Pascal ^a   Ros, Antonio ^b  

^a UNIVERSITÉ GUSTAVE EIFFEL   (France)

^b UNIVERSITY OF GRANADA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2142834706     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-03-03362-2     Document Type: Article

References (41)

1. D.M. Andersen, H.T. Davis, J.C.C. Nitsche, L.E. Scriven, Periodic Surfaces of Prescribed Mean Curvature, Advances in Chemical Physics, 77 (1990) 337-396.
2. A.D. Alexandrov, Uniqueness theorems for surfaces in the large I, Vestnik Leningrad Univ. Math. 11 (1956) 5-17. MR 19:167c
3. W. K. Allard, On the first variation of a varifold, Ann. of Math. 95 (1972) 417-491. MR 46:6136
4. F. J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. AMS 165 (1976). MR 54:8420
5. J. L. Barbosa & M. do Carao, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984) 339-353. MR 85k:58021c
6. F. Barthe, F. Extremal properties of central half-spaces for product measures, J. Funct. Anal. 182 (2001) 81-107. MR 2003a:60024
7. 2, Ann. Sci. École Norm. Sup. (4) 10 (1986) 479-490. MR 88b:53048
8. F. S. Bates & G. H. Fredrickson, Block copolymers-designer soft materials, Physics Today 52-2 (Feb. 1999) 32-38.
9. M. Carrion, J. Corneli, G. Walsh, S. Beheshti Double bubbles in the 3-torus, preprint.
10. J. Corneli, P. Holt, G. Lee, N. Leger, E. Schoenfeld & B. A. Steinhurst, The double bubble problem on the flat two-torus, preprint.
11. E. Gonzalez, U. Massari & I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983) 25-37. MR 84d:49043
12. K. Grosse-Brauckmann, New surfaces of constant mean curvature Math. Z. 214 (1993) 527565. MR 94k:53012
13. K. Grosse-Brauckmann, Cousins of constant mean curvature surfaces, preprint.
14. M. Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rat. Mech. Anal. 97 (1987) 261-270. MR 87k:49050
15. H. Hadwiger, Gitterperiodische Punktmengen und Isoperimetrie, Monatsh. Math. 76 (1972) 410-418. MR 48:2902
16. T. Hahn, editor, International Tables for Crystallography, vol. A, fifth edition Kluwer Academic Publishers, 2002.
17. J. Hersch, Quatre propriétés isopérimétrigues des membranes sphériques homogènes, C. R. Acad. Sci. Paris Ser. A-B 270 (1970) A1645-A1648. MR 45:1444
18. W. Y. Hsiang, A symmetry theorem on isoperimetric regions, PAM-409 (1988), UC Berkeley.
19. W. Y. Hsiang, Isoperimetric regions and soap bubbles, Proceedings conference in honor of Manfrede do Carmo, Pitman surveys in pure and appl. math. 82 (1991) 229-240. MR 93h:49060
20. 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).
21. N. Korevaar, R. Kusner & B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Diff. Geom. 30 (1989) 465-503. MR 90g:53011
22. E. Kroumova, J. M. Perez-Mato, M. I. Aroyo, S. Ivantchev, G. Madariaga, H. Wondratschek The Bilbao Crystallographic Server: a web site with cristallographic tools using the International Tables for Crystallography, 18th European Crystallographic Meeting, Praha, Czech Republic (1998), http://www.cryst.ehu.es.
23. 3, Ann. Math. 92 (1970) 335-374. MR 42:5170
24. 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.
25. P. Li & S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and first eigenvalue of compact surfaces, Invent. Math. 69 (1982) 269-291. MR 84f:53049
26. W. H. Meeks III, Lectures on Plateau's problem, Scola de Geometria Diferencial, Universidade Federal do Ceará (Brazil), 1978.
27. _, The topology and geometry of embedded surfaces of constant mean curvature, J. Diff. Geom. 27 (1988) 539-552, MR 89h:53025
28. F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc. (to appear).
29. F. Morgan & D. L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J. 49 (2000) 1017-1041. MR 2002e:53043
30. M. Ritoré, Superficies con curvatura media constante, Ph.D. Thesis, Granada, 1994, http://www.ugr.es/~ritore.
31. _, Index one minimal surfaces in flat three-space forms, Indiana Univ. Math. J. 46 (1997) 1137-1153. MR 99h:58041
32. M. Ritoré & A. Ros, Stable Constant Mean Curvature Tori and the Isoperimetric Problem in Three Space Forms, Comment. Math. Helvet. 67 (1992) 293-305. MR 93a:53055
33. _, The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds, Trans. Amer. Math. Soc. 348 (1996) 391-410. MR 96f: 58038
34. A. Ros, The isoperimetric problem, Proceedings of the Clay Mathematical Institute MSRI summer school on Minimal Surfaces (to appear).
35. M. Ross, Schwarz' P and D surfaces are stable, Differential Geom. Appl. 2 (1992) 179-195. MR 94j:53010
36. H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer Verlag, Berlin (1890).
37. U. S. Schwarz & G. Gompper, Stability of bicontinuous cubic phases in ternary amphiphilic systems with spontaneous curvature, J. Chem. Phys. 112 (2000) 3792-3802.
38. 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.
39. E. L Thomas, D. M. Anderson, C.S. Henkee & D. Hoffman, Periodic area-minimizing surfaces in block copolymers, Nature 334 (1988) 598-602.
40. P. Yang & S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980) 55-63. MR 81m:58084
41. S. T. Yau, Nonlinear analysis in geometry, L'enseignement Math. 33 (1987) 109-158. MR 88g:58003