References
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.
Maria Athanassenas, A variational problem for constant mean curvature surfaces with free boundary,J. Reine Angew. Math. 377 (1987), 97–107.
F. Barthé and B. Maurey, Some remarks on isoperimetry of Gaussian type,Preprint ESI 721, 1999.
Yu. D. Burago and V. A. Zalgaller,Geometric inequalities, Springer-Verlag, Berlin, 1988, Translated from the Russian by A. B. Sosin-skil, Springer Series in Soviet Mathematics.
R. Courant and D. Hilbert,Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953.
C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante,J. Math. Pure et App. 16 (1841), 309–321.
Joel Foisy, Soap Bubble Clusters in R2 and in M3, Undergraduate thesis, Williams College, 1991.
H. Hadwiger, Gitterperiodische Punktmengen und Isoperimetrie,Monatsh. Math. 76 (1972), 410–418.
Joel Hass and Roger Schlafly, Double bubbles minimize,Ann. of Math. (2)151 (2000), no. 2, 459–515.
Michael Hutchings, The structure of area-minimizing double bubbles,J. Geom. Anal. 7 (1997), no. 2, 285–304.
Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture,Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 45–49 (electronic).
Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture,Annnals of Math. (2)155 (2002), no. 2, 459–489.
Wilbur R. Knorr,The ancient tradition of geometric problems, Dover Publications, Inc., New York, 1993.
Blaine Lawson and Keti Tenenblat (eds.),Differential geometry, A Symposium in Honor of Manfredo do Carmo. Longman Scientific & Technical, Harlow, 1991.
Frank Morgan,Geometric measure theory, A beginner’s guide. Third ed., Academic Press Inc., San Diego, CA, 2000.
Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems,Indiana Univ. Math. J. 48 (1999), no. 4, 1357–1394.
Ben Reichardt, Cory Heilmann, Yuan Y. Lai, and Anita Spielman, Proof of the double bubble conjecture in R4 and certain higher dimensions,Pacific J. Math. (to appear), 2000.
Manuel Ritoré, Applications of compactness results for harmonic maps to stable constant mean curvature surfaces,Math. Z. 226 (1997), no. 3, 465–481.
—, Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere,Math. Z. 226 (1997), no. 1,127–146.
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.
Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary,Geom. Dedicata 56 (1995), no. 1, 19–33.
Michael Spivak,A comprehensive introduction to differential geometry, vol. 4, Publish or Perish, Berkeley, 1979.
Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces,Ann. of Math. (2)103 (1976), no. 3, 489–539.
Thomas I. Vogel, Stability of a liquid drop trapped between two parallel planes,SIAM J. Appl. Math. 47 (1987), no. 3, 516–525.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ritoré, M., Ros, A. Some updates on isoperimetric problems. The Mathematical Intelligencer 24, 9–14 (2002). https://doi.org/10.1007/BF03024725
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03024725