Ricci Flow — Surface Uniformization

Ricci flow evolves a Riemannian metric by ∂g/∂t = −2 Ric(g). In 2D this simplifies to ∂g/∂t = −R·g where R is the scalar curvature. The surface deforms until curvature becomes uniform — uniformizing the metric toward a constant-curvature representative (sphere, flat torus, or hyperbolic). Here a conformal factor u(x,y) evolves via ∂u/∂t = Δu − K₀·e^u driving the surface to constant Gaussian curvature.