They began by re-proving Hardt and Simon’s decades-old result in eight dimensions, this time using a different methodology they wanted to test. First, he assumed the opposite of what he wanted to show: that when you slightly perturb the wire frame defining its surface, a singularity (single point) always remains. Every time you perturb, you get a new minimal surface that still contains a singularity. You can then superimpose all these minimal surfaces on top of each other, so that the points where the singularities occur form a line.
but it’s impossible. In 1970, mathematician Herbert Federer discovered that any singularity on a minimal surface nThe dimension of the -dimensional space can be maximum n − 8. This means that in eight dimensions, any singularity must be zero-dimensional: an isolated point. Lines are not allowed. Chodosh, Mantoulidis and Schultz extended Federer’s argument to also apply to stacks of surfaces in eight dimensions. Yet in his proof, he created a set of surfaces along one such line. The paradox showed that his original assumption was wrong – meaning that you could perturb the wire frame to get rid of the singularity.
He now felt ready to tackle the problem in nine dimensions. He started his proof the same way: he assumed the worst, made a series of perturbations, and ended up with an infinite pile of minimal surfaces that all had singularities. They then introduced a new tool called the separation function, which measures the distance between these singularities. If no perturbations can interfere with the singularity, this separation function should always remain small. But the trio was able to show that sometimes the function can be large: a few perturbations can make the singularity disappear.
Mathematicians had proved the general regularity for minimizing surfaces in dimension nine. They were able to use the same logic in dimension 10 – but in dimension 11, dealing with singularities becomes even more difficult. Their techniques do not work for a particular type of three-dimensional singularity. “There’s a zoo of the Singularity type out there,” Mantoulidis said. “Any successful argument must be broad enough to handle them all.”
The team decided to collaborate with Zihan Wang, who had extensively studied this type of singularity. Additionally, they improved their separation function to work in this case as well. They have solved the problem in Dimension 11.
“The fact that they expanded [our understanding] It’s really fantastic on some dimensions,” White said.
But they will probably have to find a different approach to handle higher dimensions. “We need a new component,” Schultz said.
In the meantime, mathematicians hope the new result will help them make progress on other problems in mathematics and physics. Proofs of many conjectures in geometry and topology – for example, about the existence and behavior of shapes with certain curvature properties – depend on the smoothness of minimal surfaces. As a result, these conjectures have been proven only 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, that the total energy of the universe must be positive. In the 1970s, Richard Schon and Shing-Tung Yau used minimal surfaces to prove this statement in dimensions seven and below. In 2017, they extended their results to all dimensions. Now, the latest progress on the plateau problem provides a new way to confirm the positive mass theorem in dimensions nine, 10 and 11. “They provide another, more intuitive way to expand,” White said. “Different evidence gives different insights.”
Work can also have many unexpected consequences. The plateau problem has been used to study all sorts of other questions, including questions about how ice melts. Mathematicians hope the team’s new methods will help deepen their understanding of these connections.
As far as the plateau problem is concerned, there are now two ways forward: either mathematicians will continue to prove general regularity in higher and higher dimensions, or they will find that beyond dimension 11, it is no longer possible to remove singularities. It would also be “a bit of a miracle,” Schultz said — another mystery yet to be solved. “Either way, it will be very exciting.”
editor’s Note: Jim Simmons founded the Simmons Foundation, which also funds the editorially independent magazineThe activities of the Simons Foundation have no influence on our coverage,