∂g/∂t = −2 Ric[g]. Ricci flow deforms a Riemannian metric by its Ricci curvature — regions of positive curvature shrink, negative curvature expands. Perelman used it (with surgery) to prove the Poincaré conjecture.
Left: 2D surface visualized via Gaussian curvature K(x,y) color map (red=positive/mountain, blue=negative/saddle). Ricci flow in 2D is equivalent to the normalized equation ∂g/∂t = (r̄−R)g where r̄ is average scalar curvature. The surface homogenizes: curvature spreads like heat. Right: curvature histogram evolving toward uniform distribution — the "round sphere" fixed point of normalized Ricci flow.