In the mid-19th century, the Belgian physicist Joseph Plateau submerged loops of wire in a soapy solution. When he used a single circular ring, the film formed a flat disk. With two parallel rings, it created an hourglass shape—a surface mathematicians call a catenoid. Plateau observed a profound pattern: these films always seem to take up the smallest possible area. They are what’s known as minimizing surfaces.

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Soap films stretch within wire frames to form area-minimizing surfaces.

It took nearly a century for mathematicians to formally prove Plateau’s hunch. In the 1930s, Jesse Douglas and Tibor Radó independently solved the “Plateau problem,” showing that for any closed curve in 3D space, a corresponding minimal surface with that boundary always exists. Douglas’s work even earned him the first-ever Fields Medal.

Since then, the quest has been to understand these surfaces not just in our familiar three dimensions, but in higher-dimensional spaces. These abstract constructs are not mere mathematical curiosities; they underpin proofs in geometry and topology, inform our understanding of black holes, and even inspire the design of advanced materials like the gyroid, used in drug delivery and photonics.

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The gyroid is an area-minimizing surface that has been used in materials design and drug delivery.

The Problem of Singularities

While minimizing surfaces are always smooth in dimensions up to seven, mathematics gets stranger in higher dimensions. Surfaces can develop pinch points, folds, or self-intersections, creating what are known as singularities. These irregularities make the surfaces incredibly difficult to analyze with standard mathematical tools.

The critical question became: Are these singularities rare, or are they the norm? If they are rare, a slight “wiggle” of the boundary curve could smooth them out, allowing mathematicians to study the well-behaved versions of the surface. This property, known as generic regularity, was proven for eight-dimensional space in 1985 by Robert Hardt and Leon Simon. But for nearly 40 years, this was as far as anyone could go.

A New Approach to an Old Problem

That barrier has finally been shattered. In a series of breakthroughs, a team of mathematicians—Otis Chodosh of Stanford University, Christos Mantoulidis of Rice University, Felix Schulze of the University of Warwick, and Zhihan Wang of Cornell University—has extended the proof of generic regularity to dimensions nine, ten, and eleven.

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Christos Mantoulidis (left), Felix Schulze (center) and Otis Chodosh showed that in dimensions nine and 10, most minimizing surfaces are smooth.

Their approach was both ingenious and audacious. To prove that singularities can be smoothed out, they first assumed the opposite: that no matter how you wiggle the boundary, a singularity always remains. They then imagined creating an infinite stack of these singular surfaces. This led to a contradiction based on a deep result from the 1970s, which dictates that singularities in higher dimensions are highly constrained. By showing their initial assumption was impossible, they proved that singularities can be eliminated.

For dimensions nine and ten, the core argument held. Dimension eleven proved trickier, requiring a more nuanced tool called a “separation function” and a deeper dive into the “zoo of singularity types.” By collaborating with Wang, an expert on the specific three-dimensional singularities plaguing their proof, they successfully completed the puzzle.

The Ripple Effect of a Breakthrough

This isn’t just a victory for pure mathematics; it has profound implications across scientific disciplines. Many theorems in geometry and topology have been proven only up to dimension eight, as they rely on the smoothness of minimizing surfaces. The new results now extend these powerful theorems into dimensions nine, ten, and eleven.

Perhaps most notably, the work offers a new and more intuitive way to confirm a cornerstone of general relativity: the positive mass theorem. First proven by Richard Schoen and Shing-Tung Yau, the theorem states that the total energy of the universe must be positive. Their original proof used minimizing surfaces and was limited to lower dimensions. While they later generalized it, the new approach from the Plateau problem team provides an alternative, more insightful path to the same conclusion in the higher dimensions they’ve now conquered.

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The catenoid (left) and the Costa surface are examples of area-minimizing surfaces.

The team’s methods are also expected to deepen connections between the Plateau problem and other fields, such as the study of phase transitions in physics and the behavior of crystals.

The Road to Higher Dimensions

With dimensions nine, ten, and eleven now settled, the mathematical community is abuzz with anticipation. The team suspects their current techniques will not work for dimension twelve and beyond, hinting that a fundamentally new idea may be required to continue this journey. The path forward is unknown, but as Schulze notes, the outcome will be exciting regardless. Either generic regularity holds in even higher dimensions, revealing more hidden order, or it fails, opening the door to an entirely new class of exotic, singular geometries.

In the world of mathematics, as in the world of soap films, the most beautiful and profound discoveries often lie just beyond the next dimension.