What Shape is Best?

I study math at university and in particular, I study an area of mathematics called Riemannian geometry. For most people, geometry is the study of triangles (and other shapes) from high school. While what I study is (unfortunately) not quite that simple, I think the idea behind what I do study is genuinely that simple.

What I study on a day to day basis boils down to the following question: If I have an object I can mold, what is the best shape I can mold it into?

The reason we should care about this question is because of how good geometry is at describing things. For example, one of the most exciting applications of geometry is Einstein’s theory of general relativity. Einstein uses the language of geometry to describe the shape of the universe. The universe naturally wants to attain the best shape possible, so our question is fundamental to understanding our own universe.

There isn’t really a correct answer to the question I’ve posed, so instead what I’m going to talk about are the candidates I deal with most often. These are called Ricci solitons. In order to introduce these I’ll take a fairly scenic route. I’ll start with constant curvature shapes and talk about how this leads to the notion of Ricci solitons.

Before we go any further I’ll raise a few points:

  1. The word ‘best’ here is sometimes replaced by ‘distinguished’? There is a simple reason for this; ‘best’ is subjective. After all, what makes one shape better than another?
  2. Mathematically, we describe a shape with two pieces of information. The first is the underlying object, I’ll call this M. This will be fixed in our discussion. The second is the geometry of the object, which I’ll call g. We usually write these together as (M,g).

To make these two points more concrete lets consider a piece of play-doh. We can stretch and squish the play-doh to make different shapes, but these are same underlying object, they just has a different geometry!

Sphere are pretty nice!

Now if you hand some play-doh to a young child, they’ll often roll it into a lovely spherical shape. A sphere isn’t necessarily better than other shapes we can make with play-doh, but there’s clearly something special about it. What makes it special is it has constant curvature.

The mathematical name we give to shapes of constant curvature are Einstein manifolds. There are called Einstein as they are solutions of the vacuum Einstein field equations. The Einstein field equations are a set of equations proposed by Albert Einstein. They specify what geometry the universe has to satisfy. Mathematicians usually write them succinctly as

\text{ric} = \lambda g .

These equations roughly say that if you’re standing at any point on M decide to measure it’s curvature (which is the \text{ric} in the above equation) in a random direction then you always get the same number \lambda.

There is a great deal of interest in Einstein manifolds in modern geometry. So much so that a group of mathematicians wrote a book (and not a short book at that) solely about them under the pseudonym Arthur Besse. There are lots of reasons why Einstein manifolds are interesting, here are some bigs ones.

  1. As I said before, according to Einstein (who seemed to know his stuff), the universe is an Einstein manifold. Therefore, if we want to understand the universe then we need to understand Einstein manifolds.
  2. Among shapes with a fixed volume, the shapes with minimal total curvature are Einstein manifolds.
  3. Einstein manifold are geometric fixed points of the Ricci Flow.

I’ll elaborate on this last point in a moment. You might wonder here why we would want to go further. There seems to be a lot to suggest that Einstein manifolds are fantastic. Why not call it a day here? The short answer is that the Einstein condition is too restrictive in some situations. Sometimes we need cast a wider net. This is where point 3 above comes in. Not every geometric fixed point of the Ricci flow is an Einstein metric. This gives an excellent opportunity to define a slightly more general condition.

The Ricci flow is an equation introduction by a mathematician called Richard Hamilton in the 1980’s. For a shape M it looks like

\frac{\partial}{\partial t}g = -2\text{ric}.

This idea is that we are trying to improve to geometry of M by deforming it in the direction of its curvature. The equation is based on how heat behaves.

To understand why taking inspiration from heat flow is a good idea, lets think of an example of how heat behaves. The example I want you to think of is your fridge. Your fridge (hopefully) has fairly evenly distributed temperature, sitting around 4 degrees Celsius (apparently the ideal temperature is about 3.3 degrees Celsius). If you get hungry at some point in the day and open your fridge then the temperature near the door is going to spike as the outside air rushes in. After finding your snack of choice and closing the door, what happens to this spike? Well, it redistributes itself until the temperature is once again evenly distributed throughout the fridge.

An example of how a shape evolves under Ricci Flow

Hamilton’s Ricci flow takes this same idea and applies it to curvature, and in many cases it works wonders. You can see how this works in the image above, the curvature basically redistributes itself to become more uniform. Of course, there are difficulties (nothing works perfectly) but the Ricci flow is an extremely powerful concept. In fact, the Ricci flow was crucial to the solution of one of the 7 millennium problems posed by the Clay institute. The problem in question, the Poincaré conjecture, was resolve by Grigory Perelman by completing a program involving the Ricci flow that was initiated by Hamilton.

If the Ricci flow is supposed to somehow improve a shape, then an excellent way of finding a ‘best shape’ is to look at the fixed point of the flow. That is, why not just find the metrics which can’t be improved by our equation and anoint them as the best? This is essentially what we are going to do, but there a little bit to be careful of here.

You’ll notice that I said Einstein metrics are geometric fixed points of the flow (emphasis on the geometric here). They actually evolve by scaling. For example, a sphere will shrink under the Ricci flow. The reason I’m still calling this a fixed point is because we don’t count this as an essential change to the geometry of the shape. I mean, think about the difference between a sphere of radius 1 and and sphere of radius 2. Some distances between points have changed, sure, but they still look exactly the same.

There is another way which we can ‘change’ a shape in a way which the geometry essentially stay the same. These are metrics on M which are isometric (meaning same shape) to g. I’ll write these as \varphi^*g. Therefore, if we have a Ricci flow g(t) which stays geometrically the same as the initial g then we can write it as

g(t) = c(t) \varphi^*_tg.

We describe solutions of this form as being self similar up to the scaling by c(t). Taking a derivative of this gives the Ricci soliton condition (you’ll have to trust me on this one)

\text{ric} = \lambda g+ \mathscr{L}_Xg.

Whenever we have a metric g which satisfies this equation we know that it can’t be improved by the Ricci flow. So there we have it. This is how we arrive at a class of shapes which we consider to be a cut above the rest.

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