Ricci flow
Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the resolution of the differentiable sphere conjecture by Simon Brendle and Richard Schoen.
In 2002 and 2003, Grigori Perelman presented a number of fundamental new results about the Ricci flow, including a novel variant of some technical aspects of Hamilton's program.
Since the Ricci tensor of a Riemannian metric also assigns to each p a symmetric bilinear form on TpM, the following definition is meaningful.
The analysis of the ellipticity of the local coordinate formula provides the foundation for the existence of Ricci flows; see the following section for the corresponding result.
For this reason, the use of −2 can be regarded as an arbitrary convention, albeit one which essentially every paper and exposition on Ricci flow follows.
The parameter t is usually called time, although this is only as part of standard informal terminology in the mathematical field of partial differential equations.
In fact, in the standard quantum field theoretic interpretation of the Ricci flow in terms of the renormalization group, the parameter t corresponds to length or energy, rather than time.
[2] His work is essentially a simpler Riemannian version of Yvonne Choquet-Bruhat's well-known proof and interpretation of well-posedness for the Einstein equations in Lorentzian geometry.
the normalized Ricci flow with initial data g0 exists for all positive time and converges smoothly, as t goes to infinity, to a metric of constant curvature.The three-dimensional result is due to Hamilton (1982).
Hamilton's proof, inspired by and loosely modeled upon James Eells and Joseph Sampson's epochal 1964 paper on convergence of the harmonic map heat flow,[3] included many novel features, such as an extension of the maximum principle to the setting of symmetric 2-tensors.
As such, the Ricci flow in this setting can also be studied by purely analytic methods; correspondingly, there are alternative non-geometric proofs of the two-dimensional convergence theorem.
Soon after Hamilton's breakthrough result, Gerhard Huisken extended his methods to higher dimensions, showing that if g0 almost has constant positive curvature (in the sense of smallness of certain components of the Ricci decomposition), then the normalized Ricci flow converges smoothly to constant curvature.
Hamilton (1986) found a novel formulation of the maximum principle in terms of trapping by convex sets, which led to a general criterion relating convergence of the Ricci flow of positively curved metrics to the existence of "pinching sets" for a certain multidimensional ordinary differential equation.
The results in dimensions three and higher show that any smooth closed manifold M which admits a metric g0 of the given type must be a space form of positive curvature.
Since these space forms are largely understood by work of Élie Cartan and others, one may draw corollaries such as So if one could show directly that any smooth closed simply-connected 3-dimensional manifold admits a smooth Riemannian metric of positive Ricci curvature, then the Poincaré conjecture would immediately follow.
[4] Making use of a technique pioneered by Peter Li and Shing-Tung Yau for parabolic differential equations on Riemannian manifolds, Hamilton (1993a) proved the following "Li–Yau inequality".
The terms on the right hand side of Perelman's Li–Yau inequality motivates the definition of his "reduced length" functional, the analysis of which leads to his "noncollapsing theorem".
Hamilton's Li–Yau inequality can then be applied to see that the scalar curvature is, at each point, a nondecreasing (nonnegative) function of time.
Hamilton's first work on Ricci flow was published at the same time as William Thurston's geometrization conjecture, which concerns the topological classification of three-dimensional smooth manifolds.
[6] Hamilton's idea was to define a kind of nonlinear 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 S3, three-dimensional Euclidean space E3, three-dimensional hyperbolic space H3, which are 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 algebras 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.
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.
[7] In general, however, due to the highly non-linear nature of the Ricci flow equation, singularities form in finite time.
For instance, analyzing the geometry of singular regions that may develop in 3d Ricci flow, is the crucial ingredient in Perelman's proof of the Poincare and Geometrization Conjectures.
The easiest way to compute the Ricci tensor and Laplace-Beltrami operator for our Riemannian two-manifold is to use the differential forms method of Élie Cartan.
That is, Taking another Hodge dual gives which gives the desired expression for the Laplace/Beltrami operator To compute the curvature tensor, we take the exterior derivative of the covector fields making up our coframe: From these expressions, we can read off the only independent spin connection one-form where we have taken advantage of the anti-symmetric property of the connection (
This computation suggests that, just as (according to the heat equation) an irregular temperature distribution in a hot plate tends to become more homogeneous over time, so too (according to the Ricci flow) an almost flat Riemannian manifold will tend to flatten out the same way that heat can be carried off "to infinity" in an infinite flat plate.
Some recent work has focused on the question of precisely how higher-dimensional Riemannian manifolds evolve under the Ricci flow, and in particular, what types of parametric singularities may form.
