The double bubble conjecture
Authors:
Joel Hass, Michael Hutchings and Roger Schlafly
Journal:
Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 98-102
MSC (1991):
Primary 53A10, 49Q10, 49Q25
DOI:
https://doi.org/10.1090/S1079-6762-95-03001-0
MathSciNet review:
1369639
Full-text PDF Free Access
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Abstract: The classical isoperimetric inequality states that the surface of smallest area enclosing a given volume in $R^3$ is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of $2 \pi / 3$.
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- M. Alfaro, J. Brock, J. Foisy, N. Hodges and J. Zimba The standard double soap bubble in $R^2$ uniquely minimizes perimeter, Pac. J. Math. 159, 47-59 (1993).
- F.J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs Amer. Math. Soc. 4, 165-199 (1976).
- F.J. Almgren and J. Taylor, The geometry of soap films and soap bubbles, Sci. Amer. 235,82-93 (1976).
- ANSI/IEEE Standard 754-1985 for Binary Floating-Point Arithmetic, The Institute of Electrical and Electronic Engineers, New York, 1985.
- C.V. Boys, Soap Bubbles, Dover Publ. Inc. NY 1959 (first edition 1911).
- C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pure et App. 16, 309-321 (1841).
- J. Eells, The surfaces of Delaunay, Math. Intelligencer 9, 53-57 (1987).
- J. Foisy, Soap bubble clusters in $R^2$ and $R^3$, undergraduate thesis, Williams College (1991).
- J. Hass and R. Schlafly, Double Bubbles Minimize, (preprint).
- M. Hutchings, The structure of area-minimizing double bubbles, to appear in J. Geom. Anal.
- F. Morgan, Clusters minimizing area plus length of singular curves, Math. Ann. 299, 697-714 (1994).
- R. E. Moore, Methods and Applications of Interval Analysis, SIAM, 1979.
- J. Plateau, Statique expérimentale et théorique des liquides soumis aux seules forces moleculaires, Gathier-Villars, Paris, 1873.
- J. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. 103, 489-539 (1976).
- D’arcy Thompson, On growth and form, Cambridge Univ. Press, NY, 1959, (first edition 1917).
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Additional Information
Joel Hass
Affiliation:
Department of Mathematics, University of California, Davis, CA 95616
Email:
hass@math.ucdavis.edu
Michael Hutchings
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138
Email:
hutching@math.harvard.edu
Roger Schlafly
Affiliation:
Real Software, PO Box 1680, Soquel, CA 95073
Email:
rschlafly@attmail.com
Keywords:
Double bubble; isoperimetric
Received by editor(s):
September 11, 1995
Additional Notes:
Hass was partially supported by the NSF
Hutchings was supported by an NSF Graduate Fellowship.
Communicated by:
Richard Schoen
Article copyright:
© Copyright 1996
Hass, Hutchings, Schlafly
