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.
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 generic regularity for minimizing surfaces in dimension nine. They were able to use the same argument in dimension 10 — but in 11 dimensions, the singularities get even harder to deal with. Their techniques didn’t work for a particular kind of three-dimensional singularity. “There is a zoo of singularity types,” Mantoulidis said. “Any successful argument must be broad enough to handle all of them.”
The team decided to collaborate with Zhihan Wang, who had studied this kind of singularity extensively. Together, they honed their separation function to work in this case, too. They’d solved the problem in dimension 11.
“The fact that they extended [our understanding] by a few dimensions is really fantastic,” White said.
But they’ll likely have to find a different approach to handle higher dimensions. “We need a new ingredient,” Schulze said.
In the meantime, mathematicians expect the new result to help them make progress on other problems in math and physics. The proofs of many conjectures in geometry and topology — about the existence and behavior of shapes with certain curvature properties, for instance — rely on the smoothness of minimizing surfaces. As a result, these conjectures have only been proved up to dimension eight. Now many of them can be extended to dimensions nine, 10 and 11.
The same is true for an important statement in general relativity called the positive mass theorem, which claims, loosely speaking, that the total energy of the universe must be positive. In the 1970s, Richard Schoen and Shing-Tung Yau used minimizing surfaces to prove this statement in dimensions seven and below. In 2017, they extended their result to all dimensions. Now, the latest progress on Plateau’s problem offers a new way to confirm the positive mass theorem in dimensions nine, 10 and 11. “They provide another, more intuitive way to do the extension,” White said. “Different proofs give different insights.”
The work could also have plenty of unforeseen consequences. The Plateau problem has been used to study all sorts of other questions, including one related to how ice melts. Mathematicians hope that the team’s new methods will help deepen their understanding of these connections.
As for the Plateau problem itself, there are now two paths forward: Either mathematicians will continue to prove generic regularity in higher and higher dimensions, or they’ll discover that beyond dimension 11, it’s no longer possible to wiggle singularities away. That would be “a bit of a miracle too,” Schulze said — another mystery to unravel. “Either way, it would be very exciting.”
Editor’s Note: Jim Simons founded the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation activities have no influence on our coverage.