Hamilton's Ricci flow ∂g/∂t = -2Ric smooths Riemannian metrics — the heat equation for curvature.
Curvature Field Evolution
Settings
∂g_{ij}/∂t = -2R_{ij} (Ricci flow)
2D: R_{ij} = K·g_{ij} (Gauss)
∂g/∂t = -2K·g → ∂u/∂t = Δu
Uniformization: K→const as t→∞
Step: 0
Max |K|: —
Min |K|: —
∫K dA (Euler): —
Total variation: —
In 2D, the Ricci tensor is Ric=K·g (proportional to the metric via Gaussian curvature K). Ricci flow reduces to the log-Laplacian equation for the conformal factor. By the uniformization theorem, all surfaces flow to constant curvature metrics.