Ricci flow
/ REE-chee /
Heat spreads out: drop a hot poker in cold water and after a while the temperature evens out everywhere. Ricci flow does the same thing to the shape of a space instead of to its temperature. You hand it a curved manifold with a Riemannian metric, and it deforms that metric forward in time, ironing out lumps of curvature the way heat irons out hot spots — bumpy regions tend to round off and the geometry tries to become uniform. Richard Hamilton introduced it in 1982 precisely to try to flow an arbitrary metric toward a canonical one, and Perelman finished using it to prove the Poincaré and geometrization conjectures.
Precisely, Ricci flow is the evolution equation d/dt g(t) = -2 Ric(g(t)), where g(t) is a one-parameter family of Riemannian metrics on a fixed manifold M and Ric is the Ricci curvature of the current metric. Read it term by term: the left side is the rate of change of the metric; the right side feeds back the metric's own Ricci curvature, with a minus sign so that positively curved directions shrink and negatively curved directions expand. It is a nonlinear analogue of the heat equation d/dt u = Delta u: in harmonic (or DeTurck-gauge) coordinates the leading term of -2 Ric is literally the Laplacian of the metric, so the metric's components diffuse. Hamilton proved short-time existence and uniqueness on any closed manifold — given a smooth initial metric there is an epsilon > 0 and a unique solution on [0, epsilon). On the round sphere the flow simply shrinks the sphere homothetically to a point in finite time; on a flat torus it does nothing.
Why it matters: Ricci flow turns a topological question ('what shape is this manifold?') into a PDE you can analyze, and the long-time behaviour of the flow reads off canonical geometric structure. But the honest headline is that the flow does NOT just smooth everything out forever. On most manifolds curvature concentrates and the metric degenerates in finite time — the flow develops singularities. Understanding, classifying, and surgically removing those singularities is the entire difficulty, and pretending Ricci flow 'evolves any metric to a nice one' badly misrepresents both Hamilton's program and Perelman's work. The normalized flow (rescaling to keep total volume fixed) is often used so that round spheres become fixed points rather than vanishing.
Start with a sphere that is slightly squashed like an egg. Under Ricci flow the more sharply curved ends (high Ric) contract faster than the gently curved middle, the egg rounds out toward a round sphere, and then the whole round sphere shrinks self-similarly to a point in finite time — a clean illustration of curvature-driven smoothing followed by an unavoidable singularity.
An egg-shaped metric rounds toward a sphere under Ricci flow, then collapses to a point: smoothing first, singularity second.
Ricci flow is only weakly parabolic as written, because diffeomorphism invariance leaves a degenerate direction; Hamilton's original proof of short-time existence used the Nash-Moser theorem, and DeTurck's trick later fixes the gauge to make the equation strictly parabolic and the existence proof routine.