## The double bubble conjecture (1995)

Venue: | Electron. Res. Announc. Am. Math. Soc |

Citations: | 16 - 1 self |

### BibTeX

@ARTICLE{Hass95thedouble,

author = {Joel Hass and Michael Hutchings and Roger Schlafly and Communicated Richard Schoen},

title = {The double bubble conjecture},

journal = {Electron. Res. Announc. Am. Math. Soc},

year = {1995},

volume = {1},

pages = {98--102}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 2π/3. 1.

### Citations

168 | for Binary Floating-Point Arithmetic - Standard - 1985 |

63 |
Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints
- Almgren
- 1976
(Show Context)
Citation Context ...in R 3 which exhibits any of the singularities predicted by Plateau. 2. An outline of the proof Existence and regularity of a minimizer were established by F. Almgren and J. Taylor. Almgren showed in =-=[2]-=- that there exists an area minimizing surface in R 3 among the set of surfaces enclosing a given pair of volumes. Here surface refers to a generalized notion used in geometric measure theory, which in... |

62 | Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires. Paris: Gauthier-Villars, Trubner et. Cie - Plateau |

46 |
The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces
- Taylor
- 1976
(Show Context)
Citation Context ...ure theory, which includes piecewise-smooth surfaces. Almgren showed that the solution is a smooth surface almost everywhere. Taylor obtained additional information on the nature of the singularities =-=[14]-=-. She showed that a minimizer is a piecewise-smooth surface whose singularities consist of smooth triple curves along which three smooth surfaces come together at an angle of 120 o , and isolated poin... |

43 |
Sur la surface de révolution dont la courbure moyenne est constante
- Delaunay
(Show Context)
Citation Context ...ic theorem, and also tells us that a minimizing bubble enclosing two given regions in R 3 is a surface of revolution. Constant mean curvature surfaces of revolution in R 3 were classified by Delaunay =-=[6]-=-, and they form the pieces of our minimizing bubble. Almgren’s theorem provides no information about the topological complexity of the possible solutions. The existence theorem allows for the strange ... |

19 | Double bubbles minimize
- Hass, Schlafly
(Show Context)
Citation Context ...nterface, is the smallest possible surface which will contain the two volumes of air and keep them separate.” We have obtained a proof of this conjecture for the case of two equal volumes. Theorem 1. =-=[9]-=- The double bubble uniquely minimizes area among all surfaces in R 3 enclosing two equal volumes. We remark that a planar analogue has recently been solved in [1]. Our result can also be viewed as an ... |

16 | The surfaces of Delaunay - Eells - 1987 |

13 |
The standard double soap bubble in r 2 uniquely minimizes perimeter
- Foisy, Alfaro, et al.
- 1993
(Show Context)
Citation Context ...case of two equal volumes. Theorem 1. [9] The double bubble uniquely minimizes area among all surfaces in R 3 enclosing two equal volumes. We remark that a planar analogue has recently been solved in =-=[1]-=-. Our result can also be viewed as an isoperimetric inequality. Corollary 2. For any surface in R 3 enclosing two regions, each having volume V , the area A satisfies A 3 ≥ 243πV 2 with equality if an... |

13 |
The structure of area-minimizing double bubbles
- Hutchings
- 1997
(Show Context)
Citation Context ...n R n , for m < n, is rotationally symmetric about an (m − 1)-dimensional plane. The ideas behind this theorem are due to Brian White and Frank Morgan, and versions of it are written in [8], [11] and =-=[10]-=-. It implies the classical isoperimetric theorem, and also tells us that a minimizing bubble enclosing two given regions in R 3 is a surface of revolution. Constant mean curvature surfaces of revoluti... |

12 |
Clusters minimizing area plus length of singular curves
- Morgan
- 1994
(Show Context)
Citation Context ...volumes in R n , for m < n, is rotationally symmetric about an (m − 1)-dimensional plane. The ideas behind this theorem are due to Brian White and Frank Morgan, and versions of it are written in [8], =-=[11]-=- and [10]. It implies the classical isoperimetric theorem, and also tells us that a minimizing bubble enclosing two given regions in R 3 is a surface of revolution. Constant mean curvature surfaces of... |

5 |
Soap bubble clusters
- Foisy
- 1991
(Show Context)
Citation Context ...of m volumes in R n , for m < n, is rotationally symmetric about an (m − 1)-dimensional plane. The ideas behind this theorem are due to Brian White and Frank Morgan, and versions of it are written in =-=[8]-=-, [11] and [10]. It implies the classical isoperimetric theorem, and also tells us that a minimizing bubble enclosing two given regions in R 3 is a surface of revolution. Constant mean curvature surfa... |

4 | The geometry of soap films and soap - Almgren, Taylor - 1976 |

1 | first edition - Press - 1959 |