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<\/p>\nThey started by re-proving Hardt and Simon\u2019s 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.<\/p>\n
But that\u2019s 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 \u2212 8. That means that in eight dimensions, any singularity must be zero-dimensional: an isolated point. Lines aren\u2019t allowed. Chodosh, Mantoulidis and Schulze extended Federer\u2019s argument to apply to stacks of surfaces in eight dimensions as well. Yet in their proof, they\u2019d produced a stack of surfaces with just such a line. The contradiction showed that their original assumption was false \u2014 meaning that you can perturb the wire frame to get rid of the singularity after all.<\/p>\n
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Zhihan Wang and his colleagues proved that when singularities form on minimizing surfaces in 11-dimensional space, it\u2019s possible to wiggle them away.<\/p>\n
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.<\/p>\n
The mathematicians had proved generic regularity<\/a> for minimizing surfaces in dimension nine. They were able to use the same argument in dimension 10 \u2014 but in 11 dimensions, the singularities get even harder to deal with. Their techniques didn\u2019t work for a particular kind of three-dimensional singularity. \u201cThere is a zoo of singularity types,\u201d Mantoulidis said. \u201cAny successful argument must be broad enough to handle all of them.\u201d<\/p>\n 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\u2019d solved the problem<\/a> in dimension 11.<\/p>\n \u201cThe fact that they extended [our understanding] by a few dimensions is really fantastic,\u201d White said.<\/p>\n But they\u2019ll likely have to find a different approach to handle higher dimensions. \u201cWe need a new ingredient,\u201d Schulze said.<\/p>\n 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 \u2014 about the existence and behavior of shapes with certain curvature properties, for instance \u2014 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.<\/p>\n 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, Chodosh, Mantoulidis and Schulze\u2019s latest progress on Plateau\u2019s problem offers a new way to confirm the positive mass theorem in dimensions nine, 10 and 11. \u201cThey provide another, more intuitive way to do the extension,\u201d White said. \u201cDifferent proofs give different insights.\u201d<\/p>\n 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<\/a>. Mathematicians hope that the team\u2019s new methods will help deepen their understanding of these connections.<\/p>\n 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\u2019ll discover that beyond dimension 11, it\u2019s no longer possible to wiggle singularities away. That would be \u201ca bit of a miracle too,\u201d Schulze said \u2014 another mystery to unravel. \u201cEither way, it would be very exciting.\u201d<\/p>\n