From Soap Bubbles to Singularities: Exploring the Principle of Geometric Minimization

IAS - October 2023
Imagine that you are blowing a soap bubble using a simple children’s wand. Why does the bubble that you create take the form of a sphere? Why not an ovoid? Or a cylindrical shape? Or something totally random?
An answer is provided by the concept of “minimization.” In nature, certain phenomena tend toward minimization, meaning that they seem to “want” to take a path that uses minimal energy to return to an equilibrium state, or, in the case of the bubble, form a shape that has the least surface area possible within the system’s required constraints. The constraint for the bubble is, of course, the volume of air that has been placed inside. A sphere is the shape that accommodates the required amount of air with the minimal surface area. Thus, bubbles are spherical.
But what can you say about the geometric structure of more general objects that are governed by minimization principles? Such questions often occupy mathematicians working in the field of geometric measure theory. To demonstrate the difficulty inherent in providing an answer, imagine that, having admired your first bubble with its minimized surface area, you decide to blow a second. Your two bubbles join in mid-air, and you notice that they connect in a very specific configuration: the double bubble. This is the shape that minimizes the surface area of the two bubbles combined. But something fundamental has changed about them.
A single bubble is a smooth sphere, but when two bubbles connect, joining points are created, which have “corners.” At these points, described by mathematicians as “singularities,” the bubble ceases to be smooth. Geometric measure theorists are interested in what happens when such singularities arise. How complicated can singularities be? Can there be lots of singularities? Do the singularities themselves have some sort of structure? Paul Minter, Veblen Fellow (2022–27) at the Institute for Advanced Study’s School of Mathematics and Princeton University, is fascinated by such questions.
“In this area of research, the ultimate goal would be to provide a full geometric description of a singularity for a variety of different geometric minimization problems. Progress has been made in certain cases, and the problem has been completely resolved in others, but there are many key situations for which we do not yet know the answer,” Minter states. “Full geometric descriptions appear to be very difficult to achieve, but there are softer, related questions that one can ask.”
An important player in the development of this field was Frederick J. Almgren Jr., a frequent past Member in the Institute’s School of Mathematics. When confronted with geometric objects that have singularities, Almgren investigated whether he could quantify in some way how many singularities there were. He was able to show that if one fixes some geometric boundary, and then looks for the surface that spans that boundary with the minimum area (like a soap film stretched within a piece of circular wire), then singularities of that surface always exist in two dimensions fewer than the surface itself.
What does this mean? Let’s first return to the slightly different example of the bubble. In this case, one is not fixing a boundary, but rather fixing the total enclosed volume (i.e., the amount of air inside the bubble), and then finding the minimum surface. That surface is two-dimensional, and, for a double bubble, the singularity which occurs where two bubbles meet takes the form of a one-dimensional line. Thus, the singularity exists in one dimension less than the surface. Almgren’s work was analogous to this, but his starting point was different, since he was interested in “soap bubbles” with a given boundary rather than those that enclosed a given volume. As a result, his singularities always existed in two dimensions fewer than their associated surface. Unfortunately, this difference means the simplest “surface” that Almgren considered existed in four-dimensions, which is rather hard to visualize!
Almgren’s rule also applies to objects in far higher dimensions than just four. For example, in a 400-dimensional object in 1000-dimensional space, the singularities could be 398-dimensional. “Understanding the number of possible singularities through their dimension is a useful way of starting to describing their structure.” Minter continues, “But it remains something of a qualitative statement. It cannot, for example, tell you anything about the local structure of singularities and it does not help to classify them.” 
To study singularities in a more precise way, Minter and his colleagues Camillo De Lellis, von Neumann Professor in the Institute’s School of Mathematics, and Anna Skorobogatova, Ph.D. student in Mathematics at Princeton University, investigated their local geometric structure. In simple terms, put the singularity under a microscope and took a close look at it; in the case of the double bubble singularity, when one zooms in to the connection point between the two bubbles, one observes that the bubbles meet along a connection point in the form of a geometric structure, namely a line. Even more so, as one continues to zoom in, the surfaces begin to look like three flat planes that ultimately intersect at the singularity, forming a type of Y-shape. The technical name for these intersecting planes is a “tangent cone.” 
In the case of bubbles, only two different types of singularities are observed: Y-type singularities in double bubbles and another type which looks like a tetrahedron, seen in triple bubble clusters. But in other objects, especially in higher dimensions, singularities can look very different. Minter and his collaborators were able to show that these singularities in high dimensions are not arbitrary. Instead, they have a certain geometric structure. Moreover, they were able to prove that for the vast majority of singularities the tangent cone is unique. In other words, they demonstrated that when you put a typical singularity under a microscope, you see a fixed shape (analogous to the Y-shape in the double bubble) as you zoom in.[1]
To summarize, Minter and his colleagues have enhanced our understanding of what kind of structures form minimal surfaces. For Minter, this is an exciting result because minimization is a fundamental principle upon which the universe seems to operate. The principle of minimization governs everything from the motion of a ball around a roulette wheel to the trajectories of galaxies through spacetime. As Swiss mathematician Leonhard Euler put it 300 years ago, “Nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.”

[1] These conclusions were independently reached at the same time by another pair of scholars, Brian Krummel (University of Melbourne) and Neshan Wickramsekera (past Member in the Institute for Advanced Study’s School of Mathematics (2019), now University of Cambridge), using different methods.