In differential geometry, Ricci flow is the flow of Riemannian metrics given by the equation

where g is the metric and Ric is the Ricci curvature.

Richard Hamilton first considered this flow in 1981, showing that any 3-manifold which admits a metric of positive Ricci curvature, admits a metric of constant curvature as well. More recent work in analysis has focused on the question of how metrics evolve under the flow, and what types of parametric singularities may form. For instance, a certain class of solutions to the Ricci flow demonstrates that neckpinch singularities will form on an evolving n-dimensional metric of positive Euler characteristic as the flow approaches some characteristic time t₀. In certain cases such neckpinches will even fix around a special class of solution known as the Ricci Soliton .

The Ricci flow can be used formally to prove various important results, like the uniformization theorem or possibly the geometrization conjecture, which includes the famous Poincaré conjecture.