The standard double bubble is the unique stable double bubble in

Authors:
Frank Morgan and Wacharin Wichiramala

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2745-2751

MSC (2000):
Primary 53A10, 49Q20, 53Cxx

Published electronically:
April 17, 2002

MathSciNet review:
1900881

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the only equilibrium double bubble in which is stable for fixed areas is the standard double bubble. This uniqueness result also holds for small stable double bubbles in surfaces, where it is new even for perimeter-minimizing double bubbles.

**[A]**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, viii+199. MR**0420406****[CF]**Andrew Cotton and David Freeman,*The double bubble problem in spherical and hyperbolic space*, preprint (2000).**[F]**Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba,*The standard double soap bubble in 𝑅² uniquely minimizes perimeter*, Pacific J. Math.**159**(1993), no. 1, 47–59. MR**1211384****[HMRR]**Michael Hutchings, Frank Morgan, Manuel Ritore, and Antonio Ros,*Proof of the double bubble conjecture*, Electron. Res. Announc. Amer. Math. Soc.**6**(2000), 45–49. MR**1777854**, 10.1090/S1079-6762-00-00079-2**[M1]**Frank Morgan,*Geometric measure theory*, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR**1775760****[M2]**Frank Morgan,*(𝑀,𝜀,𝛿)-minimal curve regularity*, Proc. Amer. Math. Soc.**120**(1994), no. 3, 677–686. MR**1169884**, 10.1090/S0002-9939-1994-1169884-3**[M3]**-,*Small perimeter-minimizing double bubbles in compact surfaces are standard*, Electronic Proceedings of the 78th annual meeting of the Louisiana/Mississippi Section of the MAA, Univ. of Miss., March 23-24, 2001, to appear.**[M4]**Frank Morgan,*Soap bubbles in 𝑅² and in surfaces*, Pacific J. Math.**165**(1994), no. 2, 347–361. MR**1300837****[RHLS]**Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, Anita Spielman,*Proof of the double bubble conjecture in and certain higher dimensional cases*, Pacific J. Math., to appear.**[SM]**John M. Sullivan and Frank Morgan,*Open problems in soap bubble geometry*, Internat. J. Math.**7**(1996), no. 6, 833–842. MR**1417788**, 10.1142/S0129167X9600044X**[T]**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. MR**0428181****[W]**Wacharin Wichiramala,*The planar triple bubble problem*, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2002.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
53A10,
49Q20,
53Cxx

Retrieve articles in all journals with MSC (2000): 53A10, 49Q20, 53Cxx

Additional Information

**Frank Morgan**

Affiliation:
Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267

Email:
Frank.Morgan@williams.edu

**Wacharin Wichiramala**

Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Email:
wichiram@math.uiuc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06640-6

Keywords:
Stable double bubble,
standard double bubble,
soap bubble

Received by editor(s):
April 18, 2001

Published electronically:
April 17, 2002

Communicated by:
Bennett Chow

Article copyright:
© Copyright 2002
by the authors