Ricci flow converging to a round sphere

This depicts an approximation of the normalized Ricci flow as it converges to a round sphere.

Since numerically approximating the Ricci flow is difficult, this visualization was obtained by considering the linearization of the flow 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 C^1 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 the flow behaves.

There is another interesting phenomena here, which is that the lowest-frequency spherical harmonics correspond to linearized Mobius transformations, which are fixed under the flow. In practice, what that means is that if you deform the radius by a spherical harmonic Y_{1,m}, the linearized flow will converge to a sphere that is not centered at the origin.

Видео Ricci flow converging to a round sphere канала Gabe Khan