Ricci Flow Visualization

Discrete Ricci flow on a 2D triangulated mesh: ∂g/∂t = -2 Ric. Curvature concentrations spread out until the metric becomes uniformly curved. Color shows Gaussian curvature K.

Surface

Grid N 20 Flow rate 0.01 View angle 30 Tilt 40

χ = 2

∫K dA ≈ 0.00

Ricci flow (Hamilton 1982):
∂gᵢⱼ/∂t = -2 Ricᵢⱼ

Sphere: K=+1 everywhere → roundifies
Torus: flat, K=0
Saddle: K<0 → hyperbolic

Discrete version: Ollivier-Ricci curvature κ for each edge, adjust edge weights.

Perelman (2003) used Ricci flow + surgery to prove Poincaré conjecture (Fields Medal 2006).

Gauss-Bonnet: ∫K dA = 2πχ