New Proofs Probe Soap-Film Singularities

Summary

They started by re-proving Hardt and Simon’s decades-old result in eight dimensions, this time using a different method they hoped to test out. First, they assumed the opposite of what they wanted to show: that when you slightly perturb the wire frame that defines your surface, a singularity (a single point) always persists. Each time you make a perturbation, you get a new minimizing surface that still has a singularity. You can then stack all of these minimal surfaces on top of each other, so that the points where the singularities occur form a line. But that’s impossible. In 1970, the mathematician Herbert Federer found that any singularity on a minimizing surface in n-dimensional space can have a dimension of at most n − 8. That means that in eight dimensions, any singularity must be zero-dimensional: an isolated point. Lines aren’t allowed. Chodosh, Mantoulidis and Schulze extended Federer’s argument to apply to stacks of surfaces in eight dimensions as well. Yet in their proof, they’d produced a stack of surfaces with just such a line. The contradiction showed that their original assumption was false — meaning that you can perturb the wire frame to get rid of the singularity after all. Zhihan Wang and his colleagues proved that when singularities form on minimizing surfaces in 11-dimensional space, it’s possible to wiggle them away. They now felt ready to tackle the problem in nine dimensions. They started their proof in the same way: They assumed the worst, made a series of perturbations, and ended up with an infinite stack of minimizing surfaces that all had singularities. They then introduced a new tool called a separation function, which measures the distance between these singularities. If no perturbation can interfere with the singularity, then this separation function should always stay small. But the trio was able to show that sometimes the function could get large: Some perturbations could make the singularity disappear. The mathematicians had proved g...