Frank Morgan

In a talk at Columbia University today (April 15, 2022), Gary Lawlor announced and described his proof that the standard triple soap bubble in R^3 is the least-perimeter way to enclose and separate three equal volumes.

Update 5/19/22: Milman and Neeman post proof of general triple and quadruple bubble conjectures in R^n and S^n https://arxiv.org/abs/2205.09102. See my review at https://amathr.org/milman-and-neeman and the beautiful Quanta article by Erica Klarreich.

Update 7/18/23: Milman and Neeman post proof of quintuple bubble conjecture as promised https://arxiv.org/abs/2307.08164.

Regularity theorems for compound bubbles or immiscible fluids require that the weights satisfy triangle inequalities. For example, if the interface between two regions of prescribed planar areas has high cost, two tangent disks minimizes weighted perimeter.

Here we note that even existence requires triangle inequalities, even for three fluids. Consider weight 3 for the 01, 02, and 12 interfaces (and weight 1 fo the rest), with say unit areas for 1 and 2 and small area for 3. Now the minimizer should be some kind of 12 double bubble inside a 3 bubble, with 13 and 23 interfaces coinciding with 30 interfaces. At a singularity, 13 and 30 interfaces meeting tangentially and then coincide, conjecturally the only new type of singularity.

Inspired by a question from Gary Lawlor.