Soap bubbles and isoperimetric problems

Because of surface tension, soap bubbles or clusters thereof naturally try to minimize area for the volume(s) they enclose. This suggests the following mathematical questions. The isoperimetric problem, in an n-dimensional Riemannian manifold, is to enclose a region of a given (n-dimensional) volume v using a hypersurface of the smallest possible "area" (n-1 dimensional volume). For example, in Euclidean space, a classical theorem asserts that the unique area-minimizer is a sphere. The "soap bubble problem" is a generalization in which the problem is to enclose and separate m regions of prescribed volumes v_1, ..., v_m, using a (singular) hypersurface of minimal area. In many cases there is a natural conjecture for the area-minimizer, but proving it can be extremely difficult! For example, the Double Bubble Conjecture in R^3 was proved only recently. This conjecture, or now theorem, asserts that the least-area enclosure of two prescribed volumes in R^3 is the "standard double bubble", consisting of three pieces of spheres meeting along a circle at 120 degree angles. There is one such shape for each pair of volumes; if the two volumes are equal, the middle "sphere" is a flat disc. The standard double bubble looks like this:

The typical strategy for trying to prove something like this is to assume that you have an area-minimizing double bubble, and show that if it is nonstandard then you can modify it to decrease area. The first difficulty is proving that an area-minimizing surface exists! This is not at all trivial, and was accomplished by Almgren in the 1970's. Also, to get an existence theorem you generally have to enlarge the space of surfaces under consideration to include some "bad" surfaces that you don't like, and then struggle to rule these out. In particular, we cannot a priori guarantee that the enclosed regions, or the exterior region, will be connected. This difficulty is partially addressed below.

The triple bubble problem in R^3 currently seems hopeless without some brilliant new idea, although again there is a natural candidate surface. Indeed there are standard enclosures of m volumes in R^n for m &le n+1, given by stereographic projections of regular simplices in spheres.

Anyway, here are some papers on this subject.

Some further developments: Many additional results on the double bubble problem in R^n, S^n, and H^n, and other spaces have been proved by various groups of undergraduates working with Frank Morgan. For an excellent introduction to this whole subject I recommend his book, Geometric measure theory: a beginner's guide, third edition. In 2002, W. Wichiramala solved the triple bubble problem in R^2. (Paper [1] above only solved the easier version of this problem in which the enclosed and exterior regions are assumed to be connected.)

To demonstrate our ignorance, here are three embarassingly simple open problems.

The pictures above are copyright John Sullivan.

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