Skip to main content
Log in

Some updates on isoperimetric problems

  • Article
  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. 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.

    Google Scholar 

  2. Maria Athanassenas, A variational problem for constant mean curvature surfaces with free boundary,J. Reine Angew. Math. 377 (1987), 97–107.

    MATH  MathSciNet  Google Scholar 

  3. F. Barthé and B. Maurey, Some remarks on isoperimetry of Gaussian type,Preprint ESI 721, 1999.

  4. 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.

  5. R. Courant and D. Hilbert,Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953.

    Google Scholar 

  6. C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante,J. Math. Pure et App. 16 (1841), 309–321.

    Google Scholar 

  7. Joel Foisy, Soap Bubble Clusters in R2 and in M3, Undergraduate thesis, Williams College, 1991.

  8. H. Hadwiger, Gitterperiodische Punktmengen und Isoperimetrie,Monatsh. Math. 76 (1972), 410–418.

    Article  MATH  MathSciNet  Google Scholar 

  9. Joel Hass and Roger Schlafly, Double bubbles minimize,Ann. of Math. (2)151 (2000), no. 2, 459–515.

    Article  MATH  MathSciNet  Google Scholar 

  10. Michael Hutchings, The structure of area-minimizing double bubbles,J. Geom. Anal. 7 (1997), no. 2, 285–304.

    Article  MATH  MathSciNet  Google Scholar 

  11. 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).

    Article  MATH  MathSciNet  Google Scholar 

  12. 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.

    Article  MATH  MathSciNet  Google Scholar 

  13. Wilbur R. Knorr,The ancient tradition of geometric problems, Dover Publications, Inc., New York, 1993.

    Google Scholar 

  14. Blaine Lawson and Keti Tenenblat (eds.),Differential geometry, A Symposium in Honor of Manfredo do Carmo. Longman Scientific & Technical, Harlow, 1991.

    Google Scholar 

  15. Frank Morgan,Geometric measure theory, A beginner’s guide. Third ed., Academic Press Inc., San Diego, CA, 2000.

    Google Scholar 

  16. 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.

    MATH  MathSciNet  Google Scholar 

  17. 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.

  18. Manuel Ritoré, Applications of compactness results for harmonic maps to stable constant mean curvature surfaces,Math. Z. 226 (1997), no. 3, 465–481.

    Article  MATH  MathSciNet  Google Scholar 

  19. —, Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere,Math. Z. 226 (1997), no. 1,127–146.

    Article  MATH  MathSciNet  Google Scholar 

  20. 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.

    Article  MATH  MathSciNet  Google Scholar 

  21. Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary,Geom. Dedicata 56 (1995), no. 1, 19–33.

    Article  MATH  MathSciNet  Google Scholar 

  22. Michael Spivak,A comprehensive introduction to differential geometry, vol. 4, Publish or Perish, Berkeley, 1979.

    Google Scholar 

  23. 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.

    Article  MATH  MathSciNet  Google Scholar 

  24. Thomas I. Vogel, Stability of a liquid drop trapped between two parallel planes,SIAM J. Appl. Math. 47 (1987), no. 3, 516–525.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Manuel Ritoré.

Rights and permissions

Reprints 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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03024725

Keywords

Navigation