Double bubble theorem
The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989,[1] but was not proven until 2002.
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.
[2] In a standard double bubble, three patches of spheres meet at this angle along a shared circle.
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.
[3] This connection between pressure and radius is reflected mathematically in the fact that, for any standard double bubble, the three radii
[4] 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.
[1] 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°.
[6] In any higher dimension, the optimal enclosure for two volumes is again formed by three patches of hyperspheres, meeting at the same 120° angle.
[1][7] 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.
[5][8] 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.
[10] A proof for the restricted case of the double bubble conjecture, for two equal volumes, was announced by Joel Hass and Roger Schlafly in 1995, and published in 2000.
[6][10][13] After earlier work on the four-dimensional case,[14] the full generalization to higher dimensions was published by Reichardt in 2008,[7] and in 2014, Lawlor published an alternative proof of the double bubble theorem generalizing both to higher dimensions and to weighted forms of surface energy.
[15] A lemma of Brian White shows that the minimum area double bubble must be a surface of revolution.
For, if not, one could use a similar argument to the ham sandwich theorem to find two orthogonal planes that bisect both volumes, replace surfaces in two of the four quadrants by the reflections of the surfaces in the other quadrants, and then smooth the singularities at the reflection planes, reducing the total area.
[8] Based on this lemma, Michael Hutchings was able to restrict the possible shapes of non-standard optimal double bubbles, to consist of layers of toroidal tubes.
[16] Additionally, Hutchings showed that the number of toroids in a non-standard but minimizing double bubble could be bounded by a function of the two volumes.
[8][12] The eventual proof of the full double bubble conjecture also uses Hutchings' method to reduce the problem to a finite case analysis, but it avoids the use of computer calculations, and instead works by showing that all possible nonstandard double bubbles are unstable: they can be perturbed by arbitrarily small amounts to produce another surface with lower area.
[22] For the same problem in three dimensions, the optimal solution is not known; Lord Kelvin conjectured that it was given by a structure combinatorially equivalent to the bitruncated cubic honeycomb, but this conjecture was disproved by the discovery of the Weaire–Phelan structure, a partition of space into equal volume cells of two different shapes using a smaller average amount of surface area per cell.
For instance, the curve-shortening flow is a process in which curves in the plane move at a speed proportionally to their curvature.
