Editing Ricci flow (section)

== Relationship to uniformization and geometrization ==
Hamilton's first work on Ricci flow was published at the same time as [[William Thurston]]'s [[geometrization conjecture]], which concerns the [[homeomorphism|topological classification]] of three-dimensional smooth manifolds.<ref>{{cite book | author=Weeks, Jeffrey R. | title=The Shape of Space: how to visualize surfaces and three-dimensional manifolds| location=New York | publisher=Marcel Dekker | year=1985 | isbn=978-0-8247-7437-0| author-link=Jeffrey Weeks (mathematician)}}.  A popular book that explains the background for the Thurston classification program.</ref> Hamilton's idea was to define a kind of nonlinear [[heat equation|diffusion equation]] which would tend to smooth out irregularities in the metric. Suitable canonical forms had already been identified by Thurston; the possibilities, called '''Thurston model geometries''', include the three-sphere '''S'''<sup>3</sup>, three-dimensional Euclidean space '''E'''<sup>3</sup>, three-dimensional hyperbolic space '''H'''<sup>3</sup>, which are [[homogeneous space|homogeneous]] and [[isotropic]], and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic.  (This list is closely related to, but not identical with, the [[Bianchi classification]] of the three-dimensional real [[Lie algebra]]s into nine classes.)

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.) However, this doesn't prove the full geometrization conjecture, because of the restrictive assumption on curvature.

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.

Note that the term "uniformization" suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" suggests placing a geometry on a smooth manifold.  ''Geometry'' is being used here in a precise manner akin to [[Felix Klein|Klein]]'s [[Erlangen program|notion of geometry]] (see [[Geometrization conjecture]] for further details).  In particular, the result of geometrization may be a geometry that is not [[isotropic]].  In most cases including the cases of constant curvature, the geometry is unique.  An important theme in this area is the interplay between real and complex formulations.  In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.