A New Proof Smooths Out the Math of Melting
Introduction
Imagine an ice cube floating in a glass of water. Eventually, it will melt down to a tiny frozen speck before disappearing. As it shrinks, its surface gets smoother, and any irregularities or sharp edges gradually vanish. Mathematicians want to understand this process in greater detail, to be able to say exactly how the surface of the ice — or, say, the shape of a gradually eroding sandcastle — changes over time.
To analyze this phenomenon, they study how more abstract mathematical surfaces and shapes evolve according to a particular set of rules. This set of rules defines a process called mean curvature flow, which simultaneously smooths out a surface — even a highly irregular one — and shrinks it.
But as the surface evolves, singularities can form: points where our mathematical descriptions break down. The surface might jut out sharply, or it might thin to a point where the curvature “blows up” to infinity. Many common kinds of surfaces — such as those that are closed up, like a sphere — are guaranteed to exhibit singularities during mean curvature flow.
If these singularities are too complicated, it becomes impossible for the flow to continue.
Mathematicians want to ensure that even after a singularity forms, they can still analyze how the surface will continue to evolve. In 1995, the mathematician Tom Ilmanen, now at the Swiss Federal Institute of Technology Zurich, proposed the “multiplicity-one” conjecture. It stated that any singularities that do form during the process of mean curvature flow must be relatively simple. “Bad” behavior should be limited to individual points: You should never see, for instance, multiple regions (whether from the same surface or different surfaces) stacked on top of each other.
If true, the multiplicity-one conjecture would affirm that singularities are not a showstopper when it comes to mean curvature flow. Should a singularity appear, the flow can go on — making it possible for mathematicians to assess the surface’s evolution.
Tom Ilmanen posed the multiplicity-one conjecture to rule out “nightmare scenarios” that might otherwise occur during a process known as mean curvature flow.
Michel Büchel/ETH Zurich
Over the past few decades, mathematicians have made many advances in characterizing the behavior of surfaces as they move through mean curvature flow. “But a lot of the results achieved so far were contingent on the multiplicity-one conjecture being true,” said Richard Bamler, a mathematician at the University of California, Berkeley. “Somehow, the main stumbling block was always the multiplicity-one conjecture.”
Now, he and Bruce Kleiner of New York University have finally proved that the conjecture is in fact true.
“It’s a great breakthrough,” said Brian White of Stanford University. Not only does the work now enable mathematicians to better understand mean curvature flow, but it may also have important applications throughout geometry and topology.
Full-Throttle Flow
The notion of mean curvature flow was introduced in the 1950s as a way to explain various phenomena that occurred in metals as they cooled. In 1978, Kenneth Brakke, now an emeritus professor at Susquehanna University in Pennsylvania, formalized the concept mathematically. His model ultimately gave a more general mathematical description that could also be applied to abstract surfaces and shapes of any dimension.
The multiplicity-one conjecture concerns closed two-dimensional surfaces, like spheres or doughnuts, that live in three-dimensional space. At any point on such a surface, you can compute its curvature in a given direction — a measure of how quickly the surface bends in that direction. There are infinitely many directions you might consider. But mathematicians often only need to look at the directions that give the biggest and smallest curvature values. They then take the average of those two numbers. This average is called the mean curvature, and it can provide a lot of useful information about the surface at that point.
Mark Belan/Quanta Magazine
Mean curvature flow uses that information to decrease the area of a surface as quickly and efficiently as possible. During mean curvature flow, each point on the surface will move at a speed equal to its mean curvature — and in a direction perpendicular to its “tangent” plane, a two-dimensional plane that best approximates the surface at that point. (There are two such perpendicular directions, one pointing inward, the other outward. If the surface bulges outward at that point, then the flow moves inward. If the surface curves inward, the flow is directed outward.)
Take a sphere. Mean curvature flow will shrink it toward its center at a faster and faster rate. (That’s because as the sphere shrinks, the mean curvature at each point will get larger: Smaller spheres bend more than larger ones.) Eventually, all that’s left will be a single point, where the sphere’s center once was.
Now suppose your surface is a partially dented sphere, like a soccer ball that’s bashed in at certain spots. During mean curvature flow, the dented portions will get pushed out, while the rest of the surface moves inward. It will look more and more like a perfect sphere, and then it will shrink down to a point.
The same process reduces a cylinder to a line and a doughnut to a circle. But what about more complicated shapes, like a barbell whose handle narrows at the center? During mean curvature flow, the thinnest spot on the handle will shrink to a point, creating a singularity. This singularity will resemble the “pinch point” where a soap bubble separates from a plastic wand or a drop of water separates from a faucet. At that pinch point, the surface of the barbell is no longer smooth, and the curvature gets infinitely large.
That’s a problem. You can’t plug infinity into your equation for mean curvature flow. The equation will break down; it can no longer predict the surface’s future evolution. But if you remove the singularity, you’ll get two distinct teardrop-shaped pieces. It’s now possible to continue studying how mean curvature flow will affect these pieces. They’ll gradually get smoother and rounder, almost becoming perfect spheres before contracting to two separate points.
Mark Belan/Quanta Magazine
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.
Bruce Kleiner and his colleague recently proved a major conjecture about how singularities form on evolving surfaces.
Courtesy of Bruce Kleiner
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.”
