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Ricci flow

Hamilton succeeded in proving that any smooth closed three-manifold which admits a metric of positive Ricci curvature also admits a unique Thurston geometry, namely a spherical metric, which does indeed act like an attracting fixed point under the Ricci flow, renormalized to preserve volume. (Under the unrenormalized Ricci flow, the manifold collapses to a point in finite time.) This doesn't prove the full geometrization conjecture because the most difficult case turns out to concern manifolds with negative Ricci curvature and more specifically those with negative sectional curvature. (A strange and interesting fact is that all closed three-manifolds admit metrics with negative Ricci curvatures! This was proved by L. Zhiyong Gao and Shing-Tung Yau in 1986.) Indeed, a triumph of nineteenth century geometry was the proof of the [uniformization theorem] , the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.

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