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.”

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 therefore do not move during mean curvature flow). They then defined a function to measure the distance from any given point on the surface to the nearest point on a neighboring region.

They found a way to analyze how this “separation function” changes over time, proving that it never goes to zero. That meant that the nightmare scenario could never happen.

The mathematicians could easily apply this method to closed surfaces that involve the same sorts of building blocks. But “a general [closed] surface may look really complicated in certain regions,” Bamler said — so complicated that it “could have precluded us from controlling the flow.” He and Kleiner then showed that those problematic regions have to be very small. It “only influences the flow, as a whole, in a very minimal way,” Bamler said. “So we can essentially ignore it.”

The separation function will never go to zero over time, no matter how complicated or strange the surface. In other words, neighboring regions can never converge, and complicated singularities cannot occur. Ilmanen’s conjecture is true.

In fact, Bamler and Kleiner showed that mean curvature flow almost always leads to one of two types of particularly simple singularities: spheres that shrink to a point, or cylinders that collapse to a line. “Any other type of singularity occurs only in rare, highly specific cases,” Bamler said, “where the singularities are so unstable that even the slightest perturbation would eliminate them.”

With the resolution of the multiplicity-one conjecture, “we now basically have a complete understanding of the mean curvature flow of surfaces in three-dimensional spaces,” said Otis Chodosh of Stanford. This knowledge, he added, could have significant applications in geometry and topology, particularly if mathematicians can prove the conjecture for three-dimensional surfaces living in 4D space. (Bamler and Kleiner are starting to look into this next case, though they say they’ll need to find a different approach than the one they used for two-dimensional surfaces.)

Already, Chodosh added, the proof might allow mathematicians to use mean curvature flow to re-prove an important problem about symmetries of spheres, called the Smale conjecture. Previous proofs of the conjecture were quite complicated, Bamler said. A proof that uses mean curvature flow could be easier to understand.

A related process known as Ricci flow has already been used to prove major conjectures, including the famous Poincaré conjecture (another statement about spheres). Mathematicians hope that Bamler and Kleiner’s work on mean curvature flow will help it become a similarly powerful method. “Bamler and Kleiner have given us a huge advance in our understanding of the singularities at the heart of mean curvature flow,” White said. “It definitely opens up the possibility of using it as a tool … to do all sorts of wonderful things.”