Philosophy behind the Ricci flow

Question

I don't know if my question is too simple for this forum but let me proceed.

In Ricci flow one equips a smooth manifold $M$ with a Riemannian metric $g_0$ and evolves the metric with "time": giving rise to $g_t$. Then one interprets the singularities, formed in finite time, in $g_t$ as indicating that $M$ has to be changed (via surgery) to prolong the evolution.

If I think of $M$ as a surface and embedded in Euclidean 3-space, and I think of the metric as giving it a shape (maybe as the pullback of the flat metric), then I can be convinced that a singularity in the metric makes the surface singular in the sense that it may pinch off somewhere or completely shrink to a point.

But I don't understand how this should follow from the definitions of $M$ as a smooth manifold. In other words I am asking: why do we interpret singularities in the metric as singularities in the underlying manifold? After all, we can specify $M$ without any reference to a metric.

Answer 1

The Ricci flow behaves like a heat equation for the curvature with some additional reaction terms. Intuitively, the diffusion term acts to spread the curvature out and make the geometry more homogeneous. The catch is that the reaction terms tend to make the curvature more positive, so the curvature can concentrate until the space becomes singular.

So it is natural to ask why singularities must emerge. Some of them are fairly natural, such as what happens when a sphere shrinks to a point. In this case, the curvature is growing without bound but spreads evenly through so the limiting spaces are round (though very small). And for surfaces, this is the only way a singularity can emerge. If one starts with a surface of genus one or more, the limiting metric is either flat or hyperbolic (after rescaling), so the Ricci flow converges to the canonical metric on the surface.

But three-dimensional manifolds don’t always admit metrics of constant curvature. However, they can be decomposed into multiple pieces which admit one of eight canonical geometries. Very roughly speaking, the singularities of Ricci flow in three dimensions occur in ways to make this structure more apparent. (Technically speaking, the surgeries process on a 3-fold does not give its Geometrization, but it does break it into pieces where the decomposition can be understood, which is how Perelman proved the Geometrization conjecture.)

In higher dimensions, the singularities can be even more complicated, and there isn’t a simple interpretation for them in general. However, in many cases they are detecting regions where the surgery must occur to find a canonical geometry.

Here are some additional points which might help clarify things:

1. Interpreting the Ricci flow as a heat-type equation is a bit subtle because of a property known as diffeomorphism invariance. However, there’s an argument by Deturck which addresses this issue.
2. Ricci flow only makes sense for a Riemannian manifold, and it’s helpful to not think about it as being embedded in some higher dimensional Euclidean space but rather just existing as an intrinsic geometric space.
3. As a simple model for how singularities to Ricci flow can emerge, it is helpful to consider the non-linear reaction-diffusion equation $\frac{\partial}{\partial t} u = \Delta u + u^2$, which behaves quite similarly.