Varifolds

How do we mathematically describe the intricate, branching structure of a soap film or the infinitely wrinkled surface of a turbulent fluid? Classical geometry, with its reliance on smooth surfaces and calculus, falls short when faced with the "wild" shapes found throughout nature and mathematics. This gap necessitates a more powerful and flexible definition of what a surface can be—one that embraces imperfections, singularities, and complex structures. The theory of varifolds provides precisely this new language. By shifting perspective from a rigid set of points to a dynamic distribution of tiny, oriented planes, varifolds offer a revolutionary framework for geometric analysis.

This article serves as an introduction to this profound idea. In the following sections, we will explore the core concepts that make this theory work and the transformative impact it has had on our understanding of shapes. First, in Principles and Mechanisms, we will build the concept of a varifold from the ground up, uncovering the ideas of stationarity, mean curvature, and the all-powerful monotonicity formula. Then, in Applications and Interdisciplinary Connections, we will witness how these tools are wielded to solve centuries-old problems, dissect the anatomy of singularities, and build bridges to other fields of science.

What is a surface? Your intuition probably conjures an image of a smooth, continuous sheet, like a piece of paper or the skin of an apple. For centuries, mathematicians studied these "nice" surfaces using the tools of calculus. But nature is not always so tidy. What about the complex, branching structure of a soap film meeting at an edge? Or the frothy, infinitely wrinkled surface of a turbulent fluid? How can we even begin to talk about the "area" or "curvature" of such objects? To venture into this wilder geometric territory, we need a new, more powerful way of seeing.

This new way is the concept of a ​​varifold​​. The genius of this idea is to shift our perspective. Instead of defining a surface by a smooth parameterization (like drawing it on a coordinate grid), we think of it as a distribution of tiny, flat patches. Imagine grinding a surface down into a fine dust. Each speck of dust is a point in space, $x$, and remembers the orientation of the surface it came from—the direction of its tangent plane, $S$. A varifold is, in essence, a mathematical description of this dust cloud. It's a rule that tells us how much "surface-dust" is located at any given position, with any given orientation.

Formally, a ​​rectifiable varifold​​ is a measure on the space of positions and plane orientations, $\mathbb{R}^n \times G(m,n)$, where $G(m,n)$ (the Grassmannian) is the collection of all possible ways an $m$-dimensional plane can sit inside $n$-dimensional space. This definition, born from geometric measure theory, has two profound advantages.

First, it naturally handles objects that are far from smooth. As long as we can approximate a shape with flat pieces almost everywhere (what mathematicians call a ​​rectifiable set​​), we can define a varifold for it.

Second, it introduces the crucial idea of ​​multiplicity​​, denoted by $\theta(x)$. Imagine two soap films lying on top of each other. A classical description of a surface as a simple set of points would just see one surface. A varifold, however, can assign a multiplicity of 2 to that region, correctly accounting for the total area. It simply adds up the mass. This is a radical departure from other geometric tools like currents, which are sensitive to orientation. For instance, consider two oppositely oriented surfaces brought infinitesimally close together. In the world of currents, their areas would cancel each other out, and in the limit, they would vanish into nothing! The associated varifolds, however, being blind to orientation, would simply add up. In the limit, we'd see a new surface with twice the density. This non-cancellation property is the superpower of varifolds; it ensures that we are always tracking the true geometric "stuff" of an object, which is exactly why it became the tool of choice in modern theories like the Almgren-Pitts min-max theory for finding minimal surfaces in arbitrary spaces.

Many of nature's most beautiful shapes, from the iridescent sphere of a soap bubble to the delicate membrane of a cell, are sculpted by a universal principle of economy: they minimize their surface area (or, more generally, their energy) for the boundaries they have to fill. Mathematicians call these shapes ​​minimal surfaces​​. How do we apply this principle to our new, generalized varifold surfaces?

We can't just use calculus to find where the derivative of the area is zero. Instead, we have to think like a physicist. We imagine "wobbling" our surface a tiny bit and observing how its total area changes. This conceptual wobble is described by a smooth vector field $X$, which provides a direction and magnitude for the deformation at every point in space. The initial rate of change of the area is called the ​​first variation of the varifold​​, denoted $\delta V(X)$.

If the first variation is zero for any possible small, localized wobble—that is, if $\delta V(X) = 0$ for all valid $X$—we say the varifold is ​​stationary​​. This is the deep and powerful generalization of a minimal surface. It is a surface in perfect equilibrium, a critical point of the area functional.

Now, here is where the new theory beautifully connects back to the old. For a classical smooth surface, being stationary in the varifold sense is precisely equivalent to having its ​​mean curvature vector​​ $\mathbf{H}$ be zero everywhere. The mean curvature measures how much a surface is bending, on average, at a point. So, a stationary surface is one that is, on average, perfectly flat at every point.

The theory of varifolds pushes this idea into the non-smooth world. For any varifold with a reasonably well-behaved first variation, we can define a ​​generalized mean curvature vector​​ $\mathbf{H}$. This vector field lives on the surface and measures its "tendency to move" at each point. The first variation can then be expressed wonderfully simply as an integral involving this vector:

where $\|V\|$ is the weight measure, or area, of the varifold. From this, the grand conclusion is immediate: a varifold is stationary if and only if its generalized mean curvature $\mathbf{H}$ is zero almost everywhere. This single, elegant equation, $\mathbf{H}=0$, now describes everything from a perfect plane to the fantastically complex, singular structures formed by intersecting soap films.

If stationarity is the defining principle of minimal surfaces, then the ​​monotonicity formula​​ is the secret key that unlocks their deepest properties. It is one of the most beautiful and powerful results in all of geometric analysis, and it reveals a startlingly rigid structure hidden within these objects.

The formula states that for any stationary $m$-dimensional varifold $V$, the "average density" of the surface around any point $x_0$ can never decrease as you zoom out. More precisely, the function

is a non-decreasing function of the radius $r$. Here, $\|V\|(B_r(x_0))$ is the area of the surface contained within a ball of radius $r$ centered at $x_0$, and $\omega_m$ is just a normalizing constant (the volume of a unit $m$-dimensional ball).

Think about what this means. It's as if you have a "cosmic zoom lens" centered on a point on a minimal surface. As you zoom out (increase $r$), the ratio of the surface area you see to the volume of your view ($r^m$) can only stay the same or go up. This simple rule holds universally, whether your surface is a smooth plane in $\mathbb{R}^3$ or a non-orientable, singular object in a higher-dimensional space. The derivation of this formula relies on nothing more than the definition of stationarity and a clever application of calculus, yet its consequences are profound.

The first magical consequence is the existence of the ​​density​​ of the surface at a point, $\Theta^m(\|V\|, x_0)$. Because the density ratio is non-decreasing as $r \to 0$, it must converge to a specific value. This number tells you, in a precise way, how "heavy" the surface is at that exact point.

The second, even more magical, consequence is the existence of ​​tangent cones​​. The monotonicity formula provides the uniform control needed to "blow up" the surface at a point. Imagine pointing your zoom lens at $x_0$ and increasing the magnification indefinitely (letting $r \to 0$). The monotonicity formula guarantees that the sequence of images you see will converge to a well-defined limiting shape. This limit is the tangent cone. And what is special about this cone? The same monotonicity formula tells us that if the density ratio is perfectly constant, not just non-decreasing, then the surface must be a ​​cone​​. Since the blow-up process creates a scale-invariant limit, its density ratio must be constant. Thus, the limiting object—the tangent cone—is always a stationary cone.

The monotonicity formula is the engine that guarantees that every minimal surface, no matter how complex, has a self-similar, conical structure at infinitesimal scales.

What can these tangent cones tell us about the surface itself? They are, in a sense, the fundamental building blocks, the local DNA of the surface at a point. By analyzing the shape of the tangent cone, we can understand the point's character.

If you zoom into a point $x_0$ and the tangent cone you find is just a simple, flat $m$-dimensional plane with multiplicity one, then the point $x_0$ is called a ​​regular point​​. This means that in a small neighborhood of $x_0$, your surface is a perfectly smooth minimal submanifold, just like the classical surfaces from introductory geometry. In fact, a deep theorem states that the converse is also true: if a point is regular, the blow-up limit is unique and is precisely the classical tangent plane.

But what if the tangent cone is not a plane? What if it's, for example, two planes crossing at an angle, or the famous seven-dimensional Simons cone which is a minimal cone. Then the point is a ​​singularity​​. This is where the surface is not smooth—it might have a corner, an edge, or a more exotic structure. The theory of varifolds gives us a rigorous way to classify and study these singularities by analyzing the geometry of their tangent cones.

The story gets even more subtle and fascinating. For the "best-behaved" minimal surfaces—those that are truly ​​area-minimizing​​—theory guarantees that the tangent cone is unique at every single point. But for surfaces that are merely ​​stationary​​, this is not always true! It's possible to construct a stationary varifold where, depending on the sequence of magnifications you use to zoom in, you see two completely different tangent cones flickering in and out of existence.

This brings us to a final, crucial distinction. Stationarity ($\mathbf{H}=0$) means a surface is at a critical point of area—it is in equilibrium. But this equilibrium might be unstable. Think of a pencil balanced on its tip. It's at a critical point of potential energy, but it's not a true minimum. A classical example in mathematics is the ​​catenoid​​, the shape a hanging chain makes. A catenoid is a minimal surface, so its varifold is stationary. However, if you stretch it too far between two rings, it becomes unstable. Its area is no longer a local minimum; a pair of flat disks spanning the rings has less area. The catenoid is stationary, but it is not area-minimizing.

The theory of varifolds provides the precise framework to explore these deep and subtle questions, giving us a language to describe not just the smooth, idealized world of classical geometry, but the wild, singular, and beautiful reality of shapes as they truly are.

In our previous discussion, we laid the groundwork for varifolds, building them up from the intuitive idea of a surface to a rigorous measure-theoretic object. You might be tempted to think this is merely a case of mathematical housekeeping—tidying up fussy details and definitions. But nothing could be further from the truth. The real power and beauty of varifolds, much like any profound scientific concept, are revealed in what they allow us to do. They provide us with a new language and a new set of tools to explore the universe of shapes, and in doing so, they have revolutionized our understanding of geometry, analysis, and beyond. This chapter is a journey through that revolution. We will see how varifolds solve age-old problems with breathtaking generality, how they allow us to perform surgery on the infinitely small, and how they build bridges to other fields of science.

For centuries, mathematicians have been captivated by the problem of minimal surfaces—the shapes, like soap films, that nature forms to minimize area for a given boundary. This is Plateau's problem. The classical approach, pioneered by Jesse Douglas and Tibor Radó, was a triumph of parametric thinking: they imagined stretching a rubber sheet (the unit disk in the plane) to span a wire loop. The solution is the shape taken by the sheet that minimizes its elastic energy. This method is brilliant, but it's like forcing nature to speak a very specific dialect. The solution is always, by construction, topologically a disk. It cannot form the fantastic, multi-layered, and branching structures that real soap films do.

This is where varifolds offer a liberating change of perspective. In the varifold framework, we don't prescribe the topology. We simply look for the object, among an incredibly vast collection of "generalized surfaces," that has the given boundary and the smallest possible area. This collection includes surfaces with multiple sheets, weird topologies, and even different densities from point to point. A varifold that wins this competition is called area-minimizing. The results are astonishing. The theory not only guarantees a solution exists for very general boundaries, but the solutions it finds can have precisely the kinds of singularities seen in nature. While area-minimizing currents in $\mathbb{R}^3$ are beautifully smooth, a more general class of stationary varifolds—which model physical soap films—can exhibit the famous "Y" junctions (three surfaces meeting at $120^\circ$) and "T" junctions predicted by the physicist Joseph Plateau and rigorously classified by the mathematician Jean Taylor. The language of integral currents, by enforcing orientability, cannot capture these junctions, but the unoriented nature of varifolds handles them perfectly.

This idea of "stationarity" is the varifold's generalization of minimality. Just as a ball at the bottom of a valley is at a minimum of potential energy, a ball balanced on a saddle point is also at an equilibrium—it's stationary. A varifold is stationary if its area doesn't change, to first order, under any small wiggle. For a smooth surface, this condition of being stationary is precisely equivalent to the classical condition of having zero mean curvature everywhere. Thus, the varifold framework doesn't discard the classical theory; it embraces it, extends it, and places it on a more solid and general foundation.

Perhaps the most profound application of this new framework is in proving the existence of minimal surfaces that are not area-minimizing. The Almgren-Pitts min-max theory is a machine for doing just that. Imagine a vast, mountainous landscape where the height of each point represents the area of a possible surface. An area-minimizing surface is a point in the lowest valley. But what about the mountain passes? A mountain pass is the lowest point on a ridge connecting two peaks. It's not a minimum overall, but it's a critical point—it's stationary. The min-max principle gives us a way to find these mountain passes. By considering entire families of surfaces that sweep out the ambient space, we can find the "path of highest resistance" and identify the surface at its peak, which turns out to be a minimal surface. This incredibly powerful idea, built on the robust topology of spaces of varifolds (or their relatives, flat chains), has been used to prove the existence of countless new minimal surfaces in manifolds of any dimension, settling long-open conjectures in geometry.

One of the greatest challenges in geometry is understanding points where a surface is not smooth—where it might pinch, cross itself, or otherwise misbehave. Before varifolds, singularities were largely untamable. But the varifold framework gives us a microscope to zoom in on these points and understand their structure. This process is called a "blow-up" analysis.

Imagine you have a minimal surface with a singularity at a certain point. We can't use calculus there, so what do we do? We zoom in, magnifying the picture around the singularity more and more. If we do this in the right way, a remarkable thing happens. The sequence of ever-more-magnified pictures converges, in the varifold sense, to a new object called a tangent cone. This tangent cone is the infinitesimal model of the singularity. And it is always a cone—a surface made up of straight lines emanating from the origin. This is an astounding result: no matter how complicated a singularity is, when viewed from infinitely close, its structure is always conical. Furthermore, since the original surface was minimal, the limiting cone is also a minimal surface. If the original surface possessed a stronger property called stability (meaning its area is a true local minimum, not just a saddle), this stability is also passed down to the tangent cone in the limit.

This "soft" geometric information about the existence of tangent cones leads to incredibly "hard" analytical results about smoothness. This is the magic of regularity theory. A key player is the monotonicity formula, a direct consequence of stationarity. It tells us that the "density" of a stationary varifold—a measure of how much area is packed into a small ball—always increases as the ball gets bigger. The density of a smooth, flat plane is exactly $1$. The density of two planes passing through the same point would be $2$.

Allard's celebrated $\varepsilon$-regularity theorem uses this to make a startling claim: if the density of a stationary varifold at a point is very close to $1$, then the varifold must be a smooth surface near that point. In fact, more quantitatively, if the varifold is "flat enough" in an averaged sense (meaning its density is close to $1$ and almost constant over a small range of scales), then it is locally the graph of a smooth function. It's as if the varifold framework tells us, "If it looks like a single smooth sheet in a blurry, measure-theoretic sense, then it is a single smooth sheet." This theorem is the engine that drives nearly all modern regularity results. It tells us that singularities can only form where the density jumps above $1$—for instance, where multiple sheets of a surface come together. Moreover, these regularity estimates are robust; they are stable under varifold convergence, meaning that if we have a sequence of smooth minimal surfaces with uniform curvature bounds, those bounds pass to the limiting varifold, ensuring the limit is also smooth.

The language of varifolds is not an isolated dialect. It connects deeply with other areas of mathematics and physics, providing both powerful tools and inspiring new perspectives.

One of the most fundamental tools is the ​​coarea formula​​. It's a kind of calculus for varifolds, a generalization of the fundamental theorem of calculus and Fubini's theorem. It tells us how to relate an integral over a $k$-dimensional varifold to an integral over its $(k-1)$-dimensional "level sets" or "slices." This allows us to dissect a varifold and study it piece by piece, a technique that is indispensable in the proofs of many deep theorems.

Another beautiful illustration of the power of varifold thinking is the ​​reflection principle​​. Suppose you have a minimal surface living in a half-space, say, the region above the floor, and it meets the floor at a perfect right angle. This is a "free boundary" problem. The reflection trick is to create a "doubled" surface by taking the original surface and adding its mirror image in the floor. The new surface is now a complete surface in all of space, with no boundary. The free boundary condition on the original surface miraculously ensures that the doubled surface is a stationary varifold everywhere, even across the reflection plane. Along that plane, the new varifold has a multiplicity of two. We have converted a difficult boundary problem into a more manageable interior one, where powerful tools like the monotonicity formula can be applied directly. This elegant trick is reminiscent of the "method of images" in electrostatics and showcases the creative power of geometric reasoning.

Finally, varifolds provide the language to describe surfaces not just as static objects, but as things that evolve in time. The mean curvature flow is a process where a surface moves in the direction of its mean curvature vector, like a hot metal sheet radiating heat and smoothing itself out. It's the geometric analogue of the heat equation. A smooth surface evolving this way can develop singularities—it can pinch off into separate pieces, for instance. A classical description breaks down at the moment of the singularity. ​​Brakke flow​​, a measure-theoretic generalization defined using varifolds, allows the flow to continue right through the singularities. The surface can break, lose area, and reform, all governed by a beautiful variational inequality. This connects the world of varifolds to the rich field of geometric partial differential equations and has applications in understanding physical processes like the growth of crystals or grains in a metal.

From solving classical problems to taming singularities and describing geometric evolution, varifolds have proven to be an indispensable concept. They are a testament to the fact that sometimes, to make progress, you must be willing to abandon old assumptions—like smoothness or simple topology—and build a new, more flexible world. In that world, armed with the right language, you can see the underlying unity and beauty that governs the shape of things.