Double Bubble Conjecture

In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a ''standard double bubble'': three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002.

The proof combines multiple ingredients. Compactness of rectifiable currents (a generalized definition of surfaces) shows that a solution exists. A symmetry argument proves that the solution must be a surface of revolution, and it can be further restricted to having a bounded number of smooth pieces. Jean Taylor proof of Plateau's laws describes how these pieces must be shaped and connected to each other, and a final case analysis shows that, among surfaces of revolution connected in this way, only the standard double bubble has locally-minimal area.

The double bubble theorem extends the isoperimetric inequality, according to which the minimum-perimeter enclosure of any area is a circle, and the minimum-surface-area enclosure of any single volume is a sphere. Analogous results on the optimal enclosure of two volumes generalize to weighted forms of surface energy, to Gaussian measure of surfaces, and to Euclidean spaces of any dimension.

Statement

According to the isoperimetric inequality, the minimum-perimeter enclosure of any area is a circle, and the minimum-surface-area enclosure of any single volume is a sphere. The existence of a shape with bounded surface area that encloses two volumes is obvious: just enclose them with two separate spheres. It is less obvious that there must exist some shape that encloses two volumes and has the minimum possible surface area: it might instead be the case that a sequence of shapes converges to a minimum (or to zero) without reaching it. This problem also raises tricky definitional issues: what is meant by a shape, the surface area of a shape, and the volume that it encloses, when such things may be non-smooth or even fractal? Nevertheless, it is possible to formulate the problem of optimal enclosures rigorously using the theory of rectifiable currents, and to prove using compactness in the space of rectifiable currents that every two volumes have a minimum-area enclosure.

Plateau's laws state that any minimum area piecewise-smooth shape that encloses any volume or set of volumes must take a form commonly seen in soap bubbles in which surfaces of constant mean curvature meet in threes, forming dihedral angles of 120° ($2\pi/3$ radians). In a ''standard double bubble'', three patches of spheres meet at this angle along a shared circle. Two of these spherical surfaces form the outside boundary of the double bubble and a third one in the interior separates the two volumes from each other. In physical bubbles, the radii of the spheres are inversely proportional to the pressure differences between the volumes they separate, according to the Young–Laplace equation. This connection between pressure and radius is reflected mathematically in the fact that, for any standard double bubble, the three radii $r_1$, $r_2$, and $r_3$ of the three spherical surfaces obey the equation $\frac=\frac+\frac,$ where $r_1$ is the smaller radius of the two outer bubbles. In the special case when the two volumes and two outer radii are equal, calculating the middle radius using this formula leads to a division by zero. In this case, the middle surface is instead a flat disk, which can be interpreted as a patch of an infinite-radius sphere. The double bubble theorem states that, for any two volumes, the standard double bubble is the minimum area shape that encloses them; no other set of surfaces encloses the same amount of space with less total area.

In the Euclidean plane, analogously, the minimum perimeter of a system of curves that enclose two given areas is formed by three circular arcs, with the same relation between their radii, meeting at the same angle of 120°. For two equal areas, the middle arc degenerates to a straight line segment. The three-dimensional standard double bubble can be seen as a surface of revolution of this two-dimensional double bubble. In any higher dimension, the optimal enclosure for two volumes is again formed by three patches of hyperspheres, meeting at the same 120° angle.

History

The three-dimensional isoperimetric inequality, according to which a sphere has the minimum surface area for its volume, was formulated by Archimedes but not proven rigorously until the 19th century, by Hermann Schwarz. In the 19th century, Joseph Plateau studied the double bubble, and the truth of the double bubble theorem was assumed without proof by C. V. Boys in the 1912 edition of his book on soap bubbles. Plateau formulated Plateau's laws, describing the shape and connections between smooth pieces of surfaces in compound soap bubbles; these were proven mathematically for minimum-volume enclosures by Jean Taylor in 1976.

By 1989, the double bubble problem had become a "serious focus of research". In 1991, Joel Foisy, an undergraduate student at Williams College, was the leader of a team of undergraduates that proved the two-dimensional analogue of the double bubble conjecture. In his undergraduate thesis, Foisy was the first to provide a precise statement of the three-dimensional double bubble conjecture, but he was unable to prove it.

A proof for the restricted case of the double bubble conjecture, for two equal volumes, was announced by Joel Hass