In ordinary space, these are the only possibilities. But mathematicians and scientists often have occasion to consider other types of worlds than the infinite three-dimensional space we’re used to thinking about — worlds that are curved or finite in size, such as the three-dimensional analogues of a sphere or doughnut surface. Within these shapes, a new, intriguing possibility emerges: minimal surfaces that curve back upon themselves and close up into a complete, finite shape that needs no wire support.

In relativity theory, these finite minimal surfaces appear as the event horizons of black holes. And when they can be found within some shape, they help mathematicians understand its geometry in a variety of ways: They provide a template for cutting the shape (or “manifold”) into potentially simpler pieces, and they point toward areas of positive curvature within the manifold — regions that curve inward, like a sphere or black hole, instead of spreading outward.

“We don’t know very much about manifolds of positive curvature,” Schoen said.

But proving that finite minimal surfaces exist inside a shape is often no easy matter. To see why, consider the two-dimensional version of this problem. The question of finding minimal surfaces makes sense in any dimension: Mathematicians simply consider a “surface” to be a shape whose dimension is one lower than the space it lives in. So in a two-dimensional world, the minimal surfaces are “geodesic” curves, built up from the shortest paths between nearby points.

For some two-dimensional shapes, it is easy to find geodesics that close up into a finite loop. Take, for example, a doughnut surface — not even necessarily a nicely rounded, symmetric doughnut, but one with bumps and irregularities. If we wrapped a rubber band around the doughnut, passing through its central hole, we could imagine pulling it tight and then sliding it about the surface into all possible positions. One of these configurations must be the shortest, and that has to be a geodesic, since otherwise we could shorten it further.

But if our shape is a sphere instead of a doughnut, this approach fails. In a perfectly round sphere, it is easy to identify the geodesics: They are the equator and the other “great circles.” But on a bumpy sphere like Earth’s surface, it isn’t clear where the geodesics go, or whether any of them close up into a finite loop. You could imagine wrapping a rubber band around Earth, as we did with the doughnut. But if you slide it around trying to shorten it, it will shrink to a single point, since unlike the doughnut, the sphere has no hole on which the rubber band can get snagged.

However, this rubber band failure actually holds the seeds for success. In a round sphere, if we imagine a rubber band sitting on the equator, one way of shifting it about — introducing wiggles in it — will make it longer. Another way of moving it — sliding it straight up or down to a new latitude — will make it shorter. So the equator is the shortest curve from one point of view, and the longest curve from another.

That makes the equator analogous to the saddle point at the top of a mountain pass, which is the highest point in one direction (the route over the pass) and the lowest point in another direction (the route up to the surrounding peaks). This is not just a vague analogy: As a rule, minimal surfaces really are saddle points, but their mountain range lives in a world much harder to visualize than our ordinary one.

When we’re looking for the minimal surfaces within a shape, we can consider a new world that consists of all possible finite surfaces that live in the shape — let’s call this world “surface space.” Each point in surface space corresponds to an entire surface back in the original shape. Next, we can think of each surface’s area as determining the altitude of its corresponding point in surface space, so that our new world has a natural topography. Looking for minimal surfaces in our original shape translates into looking for saddle points in surface space.

In 1917, George Birkhoff used this approach to show that any sphere, bumpy or round, must have at least one closed geodesic. And about six decades later, in a tour-de-force extension of Birkhoff’s ideas, Almgren and Pitts mapped out the topography of surface space for all finite shapes of dimensions three through seven, and then used that topography to prove that such shapes always have at least one closed minimal surface. Pitts’ 1981 dissertation on this “minmax” theory — so named because a saddle point is both a minimum and a maximum — was “absolutely outstanding,” Neves said.

But it was also extraordinarily difficult. Few people understood the theory’s nuances, and some mathematicians studying it made claims that were never fully verified, Schoen said. “I don’t think there was ever any doubt that it was extremely interesting and important,” he said. But “it wasn’t clear how much of it was completely rigorous.”

Work on minmax theory gradually petered out. “[Pitts’] work was forgotten from the mathematical community for maybe 30 years,” Neves said. It was not resurrected until after Neves and Marques met each other in 2006 in the elevator of Fine Hall, Princeton University’s mathematics building.

Over the Mountain Pass

At the time, Marques was visiting Princeton to give a talk; Neves had recently begun a postdoctoral position there. Both native Portuguese speakers (Marques is from Brazil and Neves, from Portugal), they slipped easily into conversation. “It was the first time I’d talked to him, but he talked to me as if we were friends for 10 years,” recalled Marques, now a professor at Princeton.

And they found that discussing mathematics together came just as naturally. The two had different styles: Marques is patient, Neves more intense. But this, if anything, was a plus. “It’s very rare, I would say, that you find someone that complements you so much,” Marques said.

Both were eager to find some mathematical challenge they could really sink their teeth into. For several years, the pair threw ideas around whenever their paths crossed, to “see what sticks,” Neves said. “We’d have millions of ideas, and eventually one filters through and just becomes something formed.”

The big challenge that eventually filtered through was a famous problem called the Willmore conjecture. This asks for the doughnut shape that minimizes a quantity called Willmore energy, which, roughly speaking, measures how different the shape is from a round sphere. Willmore had conjectured in 1965 that the roundest doughnut is an especially symmetric shape called the Clifford torus, but no one had been able to prove this, despite many attempts.