Prerequisites: A Rough Understanding of Calculus
Reading Difficulty: 4/5

Today's post is a real treat! But before we dig in, I want to make sure we're all on the same page on an important term that we're going to use: "Smooth".

For the layman reader, an object is "smooth" if it is perfectly round. Things like spheres and ovals are smooth. Squares and triangles are not smooth because the corners are sharp. The graphs of nice functions such as and are also smooth.

For the somewhat more knowledgeable reader, we say a function is smooth if you can differentiate it as many times as you want. That's why is smooth, but the function is not because you can't take the derivative of at . For multivariate functions like , you'd be invoking the partial derivatives to assess smoothness.

I won't define precisely how you assess smoothness of things that can't really be modeled as the graph of a function, such as 2D surfaces in 3D space or wacky 1D curves in 2D space, but a bunch of the intuition for what it means to be "smooth" carries over.

Ok cool. Now we can have fun.

Minimal Surfaces

Let and be two distinct points in a plane. What is the shortest curve whose endpoints are and ?

The answer, of course, is the line segment between and . Thus, this segment is an example of a minimal surface. As you should expect from the terminology, we usually speak of "minimal surfaces" when in higher dimensions. The definition of a minimal surface in 3D space follows essentially the same idea: If you fix a 1D "boundary" or "edge", such as this closed loop,

then what is the surface with smallest surface area that has that boundary? The answer to this question is called a minimal surface.


A very illustrative example of a minimal surface in 3D is a catenoid:

The catenoid is the 2D surface of smallest surface area whose boundary/edge is given by two fixed circles. This is why a bubble will appear to curve inward when you put it between two loops:


Are minimal surfaces in 3D space always smooth? The answer is yes! That's not surprising whatsoever. After all, have you ever seen a bubble with sharp corners? That would be really strange!

Even Higher Dimensions

To recap what a minimal surface in 3D space is:


We can also try thinking about minimal surface in higher dimensions, like ! Though, it's pretty hard to visualize. The definition of a minimal surface in super high dimensions is essentially the same: For any positive integer ...


Again, don't worry if you cannot picture what that would look like. I can't really imagine what this looks like either. Anyways, now we can ask the same question!


In 4D space, it turns out that every (3D) minimal surface is smooth.

In 5D space, it turns out that every (4D) minimal surface is smooth.

In 6D space, it turns out that every (5D) minimal surface is smooth.

In 7D space, it turns out that every (6D) minimal surface is smooth.

If there is any justice in the world, surely this pattern will continue! Obviously, any minimal surface should be smooth... I mean, come on, have you ever seen an -dimensional bubble that has corners? That would be insa-

The Curse of Dimension 8

WHAT? THERE'S A 7 DIMENSIONAL MINIMAL SURFACE EMBEDDED IN 8 DIMENSIONAL SPACE THAT HAS A CORNER???

In 1968, mathematician Simons invented the Simons Cone, which is defined as the set of all points which satisfy:
I would love to draw a picture of this, but drawing 7-dimensional objects is pretty hard. Here is an incredibly inaccurate diagram: To make this more accurate, you need to mentally replace the plane with a 4-dimensional space, and then you need to replace the axis with another 4-dimensional space. As the inaccurate picture suggests, this is not a smooth surface because there is a sharp corner at the origin!

Simons figured out that if you change this surface slightly, then the surface area changes very little, so he conjectured that the Simons Cone is a minimal surface. In 1969, Bombieri, De Giorgi, and Giusti proved his claim, hence demonstrating for the first time that there exists a non-smooth minimal surface. In 2009, G. De Philippis (an NYU professor!) and E. Paolini came up with a simpler proof, which you can find at https://ems.press/content/serial-article-files/6405.

For the advanced reader: It turns out that any -dimensional minimal surface in is smooth when , and for , it can be shown that the singularities of a minimal surface have Hausdorff dimension at most .

But where does come from?

The exact reason is kinda above my paygrade, but I did some digging. A proof that minimal surfaces are always smooth in and lower dimensions is buried somewhere in Minimal Varieties in Riemannian Manifolds (1968)... and it's quite a big read. From what I understand, here is why counterexamples like the Simons Cone seem to fail in dimensions and below:

• We lose a dimension by considering only the base of the cone, so now we're considering dimension and below.
• We go down to dimensions because in dimensional space, (codimension 1) surfaces are dimensional.
• So the dimension of a certain thing is with . The key quantity, for some reason, is the expression
  and this happens to be negative exactly when . When this quantity is negative, it can be shown that cones can't be minimal surfaces. But when , this quantity is positive, so the argument breaks.

Now, if my understanding is right, the result above about cones can be used to argue that minimal surfaces in are always smooth. I think the argument goes something like this:

• It suffices to consider "locally minimizing", i.e. just need to look at the part of a surface near a given point.
• From a theorem by De Giorgi, if the surface is minimizing, then it should look like a plane if you zoom in far enough into the point.
• From this "blow-up argument", you can thus instead think about "globally minimizing" surfaces, since you can blow up a surface to fill the entirety of space or something. Federer was probably involved at some point here.
• Finally, when you're in , you can prove that the type of minimal surface in question needs to be a plane, and to get the desired contradiction here, it turns out that using the idea of cones is very useful.

That's all for today. I hope you are now slightly more disturbed about the mathematical world!