ricci_flow

# Ricci flow

Geometric heat equation

Ricci flow is a central object in geometric analysis. It is the geometric analogue of the heat equation, used to change the shape of spaces.

Ricci flow converging to a round sphere, from Khan's paper

It is the main ingredient in Perelman's proof of Thurston's geometrization conjecture, with special case the Poincarè conjecture. For a Riemannian manifold $(M,g)$, the Ricci flow is given by $$\frac{\partial g}{\partial t}=-2\,\mathrm{Ric}(g).$$

## Material

A very elementary introduction with nice pictures:

• Curvature
• Thurston geometrization
• Poincaré conjecture