a geometric heat flow
Drop a cold bar into a warm room and the temperature smooths out: heat flows from hot to cold until everything is even. A geometric heat flow does the same thing to a geometric object instead of a temperature — it lets a curve, a surface, a map, or a whole metric evolve in time so that its irregularities diffuse away, ideally relaxing toward an optimal shape: a geodesic, a minimal surface, a harmonic map, or a round, symmetric metric. It is the parabolic (time-dependent, diffusive) sibling of the elliptic critical-point equations of the calculus of variations.
The recipe is uniform: take a geometric energy or curvature functional, and evolve the object by the negative gradient of that functional, partial_t (object) = - grad(Energy). Because the gradient of a Dirichlet-type energy is a Laplacian-like operator, the resulting equation is a (often nonlinear) heat equation, partial_t u = Delta u + lower-order terms, hence 'heat flow'. The harmonic-map heat flow partial_t f = tau(f) flows a map toward a harmonic one (this is exactly the Eells-Sampson method). Curve-shortening flow moves a plane curve in the direction of its curvature vector, partial_t gamma = kappa, shrinking and rounding it; mean-curvature flow does the same for a hypersurface, partial_t x = H, the steepest descent of area. Ricci flow, partial_t g = -2 Ric, evolves the metric itself toward a more uniform curvature. In each case parabolic theory gives short-time existence and smoothing, and the hope is convergence to a critical point of the underlying functional.
Geometric flows are one of the most powerful ideas in modern geometry: they turn the static problem 'find the optimal shape' into the dynamic problem 'flow until you reach it', and they delivered the Poincare and geometrization conjectures (via Ricci flow) and deep results on minimal surfaces (via mean-curvature flow). The essential honesty, and the reason the subject is hard, is that flows do not simply smooth everything out forever. They can develop singularities in finite time — a curve-shortening loop shrinks to a point, a neck can pinch under mean-curvature or Ricci flow — and understanding and surgically removing these singularities is the whole technical drama. Saying a geometric flow 'just relaxes the object to its best shape' is false in general; the singularities are the heart of the theory, and the research program of Ricci flow with surgery belongs to the frontier, not to this introductory picture.
Curve-shortening flow evolves a simple closed plane curve by partial_t gamma = kappa; Grayson's theorem says any embedded closed curve stays embedded, becomes convex, and shrinks to a round point in finite time — a clean instance of a flow relaxing a shape, singularity and all.
Curve-shortening flow rounds any embedded closed curve before collapsing it to a point.
Geometric flows do not 'just smooth everything forever': they generically form finite-time singularities (neck-pinches, collapse to a point), and handling those singularities — not the smoothing — is where the real difficulty and the research frontier lie.