Frank Morgan

Steve Zottoli and I, in attempting to model biological cells, came up with the following theorem on the shape of planar double cells in which the tension of an interface is a strictly convex function of the linear density. One could conjecture a similar result in 3D.

 Theorem. For a double planar cell of prescribed areas A1, A2, consisting of three interfaces of prescribed mass meeting at two points, the configuration of least energy, computed for each interface as the integral of the tension with respect to arclength, is given by three circular arcs of constant density, as in Figure 1. The angles at which they meet are determined by tension equilibrium and their curvatures k_i and tensions t_i satisfy

                    k₁t₁ = k₂t₂ + k₃t₃.

This is the same as the energy minimizing cluster for two immiscible fluids inside a third.

Figure 1. An energy-minimizing double cell consists of three circular arcs in equilibrium. Figure from Slobozhanin and Alexander.

Remark. Even if the enclosed masses instead of areas are prescribed, the minimizer will still be minimizing for whatever areas it has and will hence have the stated form. If total mass rather than the mass of each interface is prescribed, then each interface will have the same density and tension and the double cell will be the same as the energy minimizing soap bubble cluster, with angles of 120 degrees.

Proof. A minimizer has constant density on each interface (because on any fixed curve of prescribed mass, constant density uniquely minimizes energy, because tension is a strictly convex function of density). Circular arcs are best. A fortiori, the minimizer minimizes weighted length, with a different weighting constant for each interface, which is the immiscible fluids problem.

Remark. The immiscible fluids problem can be posed in great generality in geometric measure theory [M1, Chapter 16], allowing very general, disconnected regions. That the minimizer still takes the above form can be proved by the same simple argument that Hutchings [M2] provided for the case when all weightings are equal, the “double soap bubble problem,” as was pointed out by Cotton and Freeman [CF, §2.1].

[CF]  Andrew Cotton and David Freeman, The double bubble problem in spherical and hyperbolic space, Intern. J. Math. Math. Sci. 32 (2002) 641-699.

[M1]  Frank Morgan, Geometric Measure Theory: a Beginner’s Guide, Academic Press, third edition, 2000.

[M2]  Frank Morgan, Proof of the double bubble conjecture, Amer. Math. Monthly 108 (March 2001) 193-205.

Seeing my old friend A. R. Gurney, the illustrious playright, at his 60th Reunion here at Williams College and addressing the 25th Reunion dinner at MIT, all last Saturday, brought back many happy memories.

Gurney and I taught a freshman seminar at MIT on “Math and Literature,” in which he had to do all the homework I assigned on the Euler characteristic and I had to do his assignments in literature, including a parody of Dante’s Inferno which I call Dante’s MIT.

At the MIT 25th Reunion Dinner, in the dazzling new Media Lab with its breathtaking view of the Boston skyline and the Charles River, I began with an old segment from the nationally syndicated PM Magazine from May 30, 1984, referred to in some old exams. My old students followed my talk on “Soap Bubbles and Mathematics” with their old enthusiasm and acuity. I’m very proud of them and all that they have accomplished.