An Illustrated Introduction to the Ricci Flow – Third Version

Here is a third (and hopefully final) version of the introduction to Ricci flow, which has been expanded to include higher quality images and more exposition.

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(Edit 1/28/2022: This is a slightly updated version addressing some comments from the previous version. Thanks for the suggestions!)

Visualizing the Geometrization conjecture

One of the biggest challenges for me was understanding and picturing the Geometrization conjecture. Three-dimensional topology is not my area of research, and the attempts to accurately depict the eight Thurston geometries are fascinatingly complex (and well outside my drawing ability).

After reading several helpful MathOverflow answers on the topic, this was the diagram I drew for the eight geometries.

The next challenge was to try to draw a three-dimensional space which decomposes into several pieces. I wasn’t really sure how to “draw” a three-dimensional manifold, so ended up taking a somewhat abstract approach.

A three-manifold with its geometric decomposition

I took several pieces of artistic license with this figure. The most obvious is that the space as drawn is not closed, so doesn’t satisfy the hypotheses of the Geometrization conjecture.

However, I thought it was necessary to include at least one depiction of a hyperbolic knot complement. The image of the knot is adapted from a figure from Thurston’s The geometry and topology of three-manifolds. The central component of this manifold is intended to show a structure on the connected sum of two copies of . Since I didn’t know how to draw three-dimensional projective space, I instead drew a connected sum of Boy’s surfaces, which are immersions of the real projective plane in Euclidean 3-space.

The right-most piece has Nil geometry. Instead of trying to draw what this geometry looks like, I drew a portion of the Cayley graph of the Heisenberg group, which is the most well known example of a space admitting this geometry. If you want to see what it’s like to walk around in Nil geometry, ZeroRogue has made some fascinating Youtube videos.

As a final note, the Geometrization conjecture states that three-manifolds can be cut along spheres as well as tori to obtain the canonical geometries. I didn’t know a good way to show a space being cut along a torus, which is why I didn’t try to show details on how the geometries are joined.

Visualizing the Ricci Flow

In the paper, I included a link to a video which depicts the Ricci flow on a surface of revolution. There is a paper by J. Hyam Rubinstein and Robert Sinclair on numerically solving Ricci flow on such a surface, which I presume was used for that animation. I wanted to create a visualization for a less symmetric initial condition, and so made the following video.

The Ricci flow converging to a round sphere

Since numerically approximating the Ricci flow is difficult (and something I have no background in), I created this video by linearizing the equation at the round sphere. Doing so, it is possible to realize the flow as one-parameter family of graphs (in spherical coordinates) where the radius evolves by a linear reaction-diffusion equation. Then one can deform the radius by some low-frequency spherical harmonics and solve the flow explicitly. The initial metric is not actually close to a sphere, so the approximation during the first second or so might not be particularly accurate, but still hopefully gives some intuition for how Ricci flow behaves on surfaces.

There is another interesting phenomena here, which is that the lowest-frequency spherical harmonics correspond to linearized Mobius transformations, which are gauge transformations. In practice, what that means is that if you deform the radius by a spherical harmonic , the linearized flow will converge to a sphere that is not centered at the origin.