A New Proof Smooths Out the Math of Melting

Summary

For any closed, compact surface — that is, a surface that is finite in diameter and has a distinct inside and outside — mean curvature flow is destined to lead to a singularity. (For a simple sphere, this singularity is the final point the surface shrinks to.) “We have this flow that is supposed to make surfaces simpler, but we know that the flow always becomes singular,” Bamler said. “So if we want to understand what the flow does, we need to understand its singularity formation.” That’s where the multiplicity-one conjecture comes in. Separation Is the Key to Success Simple singularities, like pinch points, can be removed in a straightforward manner, enabling mean curvature flow to proceed unimpeded. But if a singularity is more complicated — if, say, two sheets within a surface come together, overlapping over an entire region rather than affecting just one point — this won’t be possible. In such cases, Bamler said, “we don’t know how the flow behaves.” Richard Bamler studies processes called geometric flows, which transform general geometric objects into something simpler and more symmetric. Ilmanen formulated his conjecture to rule out these troublesome situations. Decades later, Bamler and Kleiner set out to prove him right. To do so, they imagined an unusual shape — what Kleiner called “an evil catenoid.” It consists of two spheres, one inside the other, connected by a small cylinder, or neck, to form a single surface. If the neck were to shrink so fast that it pulled the two spherical regions together, Kleiner noted, that would be the “nightmare scenario.” In order to rule out this scenario, he and Bamler wanted to understand how the two regions would interact with each other, and how the separation between them would change over time. So the two mathematicians broke the shape into different building blocks — regions that looked like parallel sheets when you zoomed in to them, and special regions called minimal surfaces (which have zero mean curvature, and the...