内容来源：https://www.quantamagazine.org/new-proofs-probe-soap-film-singularities-20251112/

内容总结：

【本报综合报道】历经近四十年停滞，数学家终于在高维空间肥皂膜研究领域取得重大突破。斯坦福大学数学家奥蒂斯·乔多什领衔的国际团队，成功证明在九至十一维空间中，光滑的最小表面积结构具有普遍性，这一成果为理解黑洞、生物分子设计等科学问题提供了全新数学工具。

19世纪中叶，比利时物理学家约瑟夫·普拉托通过金属丝框与肥皂液实验发现，皂膜总能自动形成最小表面积的曲面。这一现象被归纳为"普拉托问题"，直到20世纪30年代才被数学家严格证明。此类最小曲面不仅在几何学中具有美学价值，更在细胞生物学、广义相对论等领域具有重要应用。

随着研究深入，数学家发现当空间维度超过七维时，最小曲面可能出现折叠、收缩等奇异点，严重影响对其性质的分析。1985年，数学家证实八维空间的奇异点可通过微调边界消除，但更高维度的研究长期陷入僵局。

乔多什团队创新性地引入"分离函数"等工具，通过反证法成功突破维度壁垒。他们的研究表明，在九至十一维空间中，适当调整边界框架即可获得光滑的最小曲面，这一特性被数学家称为"通有正则性"。该突破使得众多原本局限于八维以下的几何猜想得以向更高维度拓展，其中包括广义相对论中描述宇宙总能量必须为正的"正质量定理"。

研究团队成员曼图里迪斯指出："奇异点类型复杂如动物园，任何成功论证都必须能处理所有类型。"目前团队正寻求新方法攻克十二维及以上空间的挑战。无论最终发现更高维度能否消除奇异点，都将为人类理解高维空间本质开启全新篇章。

此项研究不仅推进了基础数学前沿，其方法还有望应用于冰层融化等物理过程模拟，展现理论数学与现实世界的深刻关联。

中文翻译：

新证探秘皂膜奇点结构

19世纪中叶，比利时物理学家约瑟夫·普拉托——这位自幼便痴迷设计科学实验的学者——将金属线圈浸入皂液，潜心研究形成的液膜形态。当他把铁丝弯成圆环时，皂膜在框架上延展成平坦圆盘；而将两个平行圆环浸入溶液时，皂膜则在环间勾勒出沙漏般的曲面——数学家称之为悬链面。各式金属框架催生出千姿百态的皂膜：有的形似马鞍，有的状如螺旋坡道，更有某些复杂结构难以用语言描述。

普拉托提出假设：这些皂膜总能实现最小表面积，即数学家所称的“极小曲面”。近一个世纪后，数学家才证实其真知灼见。20世纪30年代初，杰西·道格拉斯与蒂博尔·拉多分别独立证明“普拉托问题”的肯定答案：对于三维空间中任意闭合曲线（金属框架），总能找到具有相同边界的二维极小曲面（皂膜）。这项证明后来为道格拉斯赢得了首枚菲尔兹奖。

此后数学家不断拓展普拉托问题，试图深化对极小曲面的认知。这类曲面遍布数学与科学领域：从几何拓扑重要猜想的证明，到细胞与黑洞的研究，乃至生物分子设计。“它们是极具美感的研究对象，”斯坦福大学奥蒂斯·乔多什表示，“既自然优美又引人入胜。”

目前数学界已确认普拉托猜想在七维及以下维度完全成立。但在更高维度中存在例外：形成的极小曲面未必保持圆盘或沙漏般的光滑形态，可能在某些位置出现折叠、收缩或自交，形成所谓“奇点”。一旦极小曲面出现奇点，研究难度便急剧增加。

因此数学家亟需了解这类非光滑极小曲面的普遍性及其特性。若奇点在特定维度中极为罕见，仅出现在特殊构造情形下，则通过适当扰动金属框架即可消除奇点。如此便能获得易于研究的光滑极小曲面，为深入理解该维度曲面特性创造契机。

1985年，数学家证实在八维空间中确实可通过扰动消除奇点。但在更高维度中，“情况变得错综复杂”，乔多什坦言。奇点分析难度陡增，近四十年来该领域研究始终停滞不前。

如今僵局终被打破。2023年，乔多什与莱斯大学的克里斯托斯·曼图利迪斯、华威大学的费利克斯·舒尔策共同证明，在九维与十维空间中，光滑极小曲面占据主导地位。今年初，这支团队在康奈尔大学王知涵加入后，进一步将结论拓展至十一维。

这项突破性工作为了解高维空间中奇异极小曲面开辟了新途径。数学界现在可运用该成果解决众多长期局限于八维及以下的研究难题，显著提升相关定理的普适性。

奇点研究历程

1962年，数学家温德尔·弗莱明证明所有二维极小曲面——即普拉托可能研究的所有皂膜——必定光滑，根本不存在带奇点的二维极小曲面。这些二维曲面存在于熟悉的三维空间，但当我们进入更难以直观想象的高维空间时，情况如何？斯坦福大学布莱恩·怀特举例说明：“在四维空间，金属框架的类比物是二维曲面，而普拉托问题要求我们寻找填充该曲面的最小体积三维形状。这种形状可能极其怪异——类似分形或极度不规则。”

弗莱明证明之后数十年间，数学家陆续证实四至七维空间中极小曲面始终光滑。但1968年，数学家吉姆·西蒙斯在八维空间中构造出仅含单个奇点的七维形状。翌年，数学界证明该形状确为极小曲面，确立了八维空间存在带奇点极小曲面的事实。

随之而来的疑问是：这些奇点的严重程度如何？是普遍存在还是偶然现象？能否通过微调框架形状予以消除？“若要分析曲面特性，奇点会极大增加研究难度，”怀特指出。但若奇点仅偶然出现，且可通过微扰获得光滑曲面，研究者就能运用微积分等工具大幅简化工作。

1985年，罗伯特·哈特与莱昂·西蒙证明八维空间的极小曲面具有“通有正则性”这一优良特性。但无人能将其方法推广至更高维度。这一停滞状态持续数十年，直至乔多什团队介入研究。

探索未知维度

三位数学家决心探索未知的高维领域，如同生物学家探察新发现岛屿的生态体系般，揭示其中极小曲面的本质特性。他们从用新方法重新验证哈特与西蒙的八维结论入手：先假设与目标结论相反的情形——当微扰定义曲面的框架时，奇点（单个点）始终存在。每次扰动都会产生带奇点的新极小曲面，将这些曲面叠加后，奇点位置将连成直线。

但这与既定结论矛盾。1970年数学家赫伯特·费德雷尔发现，n维空间极小曲面的奇点维度至多为n-8。这意味着八维空间的奇点必为零维孤立点，不允许出现线状奇点。乔多什团队将费德雷尔论证拓展至八维曲面叠加情形，却在其证明中构造出含线状奇点的曲面叠加，由此证伪初始假设——奇点确实可通过扰动框架消除。

攻克八维问题后，团队转向九维情形。沿用相同论证思路：假设最坏情况，通过系列扰动获得带奇点的无穷极小曲面叠加，继而引入测量奇点间距的新工具“分离函数”。若扰动无法影响奇点，该函数值应始终较小；但团队证明函数值可能增大，意味着某些扰动确实能消除奇点。

由此他们证得九维极小曲面的通有正则性，并将论证成功推广至十维。但在十一维空间中，特定三维奇点构成新挑战。“奇点类型犹如动物园，”曼图利迪斯比喻，“任何成功论证必须足够普适以处理所有类型。”团队邀请专攻此类奇点的王知涵加入，共同优化分离函数工具，最终攻克十一维难题。

“他们将认知边界拓展数个维度，这确实了不起，”怀特评价。但若要进军更高维度，“我们需要新方法，”舒尔策坦言。数学界期待新成果推动数学与物理领域其他问题的研究进展。许多几何拓扑猜想（如特定曲率形状的存在性与行为）的证明依赖极小曲面的光滑性，因此这些猜想目前仅证至八维。如今多数结论可拓展至九至十一维。

广义相对论中的“正质量定理”亦然——该定理粗略表述为宇宙总能量必为正。1970年代，理查德·舍恩与丘成桐运用极小曲面在七维及以下空间证明该定理，2017年将其推广至全维度。乔多什团队在普拉托问题上的最新突破，为验证九至十一维正质量定理提供了新路径。“他们给出了更直观的推广方式，”怀特指出，“不同证明能带来不同视角。”

这项研究可能孕育诸多意外成果。普拉托问题已被用于探索各类课题（包括冰层融化机制），数学界期望团队的新方法能深化对这些关联的认知。

对于普拉托问题本身，未来存在两条路径：要么持续证明更高维度的通有正则性，要么发现十一维以上无法通过扰动消除奇点。“后者同样堪称奇迹，”舒尔策表示，“这将是待解的新谜题。无论哪种结果，都令人振奋不已。”

编者注：吉姆·西蒙斯创立了西蒙斯基金会，本刊亦获其资助。西蒙斯基金会的活动不影响本刊编辑独立性。

英文来源：

New Proofs Probe Soap-Film Singularities
Introduction
In the mid-19th century, the Belgian physicist Joseph Plateau — who had been designing and conducting scientific experiments since he was a child — submerged loops of wire in a soapy solution and studied the films that formed. When he bent his wire into a circular ring, a soap film stretched across it, creating a flat disk. But when he dipped two parallel wire rings into the solution, the soap stretched between them to form an hourglass shape instead — what mathematicians call a catenoid. Different wire frames produced all sorts of different films, some shaped like saddles or spiraling ramps, others so complicated they defied description.
These soap films, Plateau posited, should always take up the smallest area possible. They’re what mathematicians call area-minimizing surfaces.
It would take nearly a century for mathematicians to prove him right. In the early 1930s, Jesse Douglas and Tibor Radó independently showed that the answer to the “Plateau problem” is yes: For any closed curve (your wire frame) in three-dimensional space, you can always find a minimizing two-dimensional surface (your soap film) that has the same boundary. The proof later earned Douglas the first-ever Fields Medal.
Since then, mathematicians have expanded on the Plateau problem in hopes of learning more about minimizing surfaces. These surfaces appear throughout math and science — in proofs of important conjectures in geometry and topology, in the study of cells and black holes, and even in the design of biomolecules. “They’re very beautiful objects to study,” said Otis Chodosh of Stanford University. “Very natural, appealing and intriguing.”
Mathematicians now know that Plateau’s prediction is categorically true up through dimension seven. But in higher dimensions, there’s a caveat: The minimizing surfaces that form might not always be nice and smooth, like the disk or hourglass. Instead, they might fold, pinch or intersect themselves in places, forming what are known as singularities. When minimizing surfaces have singularities, it becomes much harder to understand and work with them.
Mathematicians consequently want to know how common such non-smooth minimizing surfaces are, and what properties they might have. If singularities are rare in a given dimension, appearing only under contrived circumstances, then they’ll disappear if you wiggle your wire frame just right. You’ll be left with a smooth minimizing surface that you can study more easily, which will give you the chance to develop a thorough understanding of such surfaces in that dimension.
In 1985, mathematicians proved that in eight-dimensional space, singularities can indeed be wiggled away. But in higher dimensions, “all hell breaks loose,” Chodosh said. The singularities become much more difficult to analyze. For nearly 40 years, no one could make much progress on the problem.
That barrier has finally been broken. In 2023, Chodosh — along with Christos Mantoulidis of Rice University and Felix Schulze of the University of Warwick — showed that in dimensions nine and 10, smooth minimizing surfaces are the norm. And earlier this year, the team, joined by Zhihan Wang of Cornell University, showed that the same is true in dimension 11.
The work marks a major advance toward understanding the strange kinds of minimizing surfaces that can arise in higher and higher dimensions. And mathematicians can now use the result to resolve a host of other math problems that have long been limited in scope to dimension eight or below — making those theorems even more powerful.
A Singular History
In 1962, the mathematician Wendell Fleming proved that all minimizing two-dimensional surfaces — any possible soap film that Plateau might have tried to study — must be smooth. Minimizing surfaces with singularities simply don’t exist.
These 2D surfaces exist in our familiar three-dimensional space. But what happens when we move to higher dimensions, where the problem gets harder to visualize? In four dimensions, for instance, the analogue of our wire frame is a 2D surface, and the Plateau problem asks us to find the 3D shape that fills that surface with the smallest possible volume. What might that shape look like? For all we knew, said Brian White of Stanford, “it could be very horrible — fractal-like or extremely irregular.”
In the years that followed Fleming’s proof, mathematicians showed that this never happens in four, five, six or seven dimensions. Minimizing surfaces are always smooth. But in 1968, the mathematician Jim Simons constructed a seven-dimensional shape in eight dimensions that had a singularity at just one point. The following year, mathematicians proved that this shape was a minimizing surface, establishing that minimizing surfaces in eight-dimensional space could, in fact, be singular.
The question then became: Just how bad are these singularities, really? Are they rare or common? And can you get rid of them by changing the shape of your wire frame just a bit, in just the right way? “If you want to figure out things about a surface, singularities make it much harder to analyze,” White said. But if singularities arise only rarely, and you can easily nudge them away to get a smooth surface, life becomes much easier — you can use the tools of calculus, for example.
In 1985, Robert Hardt and Leon Simon proved that minimizing surfaces in eight dimensions have this nice property, which mathematicians call generic regularity. But no one could figure out how to adapt their techniques to show whether it exists in higher dimensions.
That’s where things stood for decades — until Chodosh, Mantoulidis and Schulze stepped in.
Into Unfamiliar Domains
The three mathematicians wanted to explore uncharted higher-dimensional realms and understand the nature of their minimizing surfaces, the way a biologist might seek to understand the flora and fauna of a newly discovered island. And so they set out to see whether they could wiggle these singularities away.
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, Chodosh, Mantoulidis and Schulze’s 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.