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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Additional Information

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 expĂ©rimentale et thĂ©orique 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).
- Dâ€™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