Editing Ricci flow (section)

==Convergence theorems==
Complete expositions of the following convergence theorems are given in {{harvtxt|Andrews|Hopper|2011}} and {{harvtxt|Brendle|2010}}.
{{quote|Let {{math|(''M'', ''g''<sub>0</sub>)}} be a smooth [[closed manifold|closed]] Riemannian manifold. Under any of the following three conditions:
* {{mvar|M}} is two-dimensional
* {{mvar|M}} is three-dimensional and {{math|''g''<sub>0</sub>}} has positive Ricci curvature
* {{mvar|M}} has dimension greater than three and the product metric on {{math|(''M'', ''g''<sub>0</sub>) × ℝ}} has positive isotropic curvature
the normalized Ricci flow with initial data {{math|''g''<sub>0</sub>}} exists for all positive time and converges smoothly, as {{mvar|t}} goes to infinity, to a metric of constant curvature.}}
The three-dimensional result is due to {{harvtxt|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|harmonic map heat flow]],<ref>{{cite journal |last1=Eells |first1=James Jr. |last2=Sampson |first2=J.H. |title=Harmonic mappings of Riemannian manifolds |journal=Amer. J. Math. |date=1964 |volume=86 |issue=1 |pages=109–160|doi=10.2307/2373037 |jstor=2373037 }}</ref> included many novel features, such as an extension of the [[maximum principle]] to the setting of symmetric 2-tensors. His paper (together with that of Eells−Sampson) is among the most widely cited in the field of differential geometry. There is an exposition of his result in {{harvtxt|Chow|Lu|Ni|2006|loc=Chapter 3}}.

In terms of the proof, the two-dimensional case is properly viewed as a collection of three different results, one for each of the cases in which the [[Euler characteristic]] of {{mvar|M}} is positive, zero, or negative. As demonstrated by {{harvtxt|Hamilton|1988}}, the negative case is handled by the maximum principle, while the zero case is handled by integral estimates; the positive case is more subtle, and Hamilton dealt with the subcase in which {{math|''g''<sub>0</sub>}} has positive curvature by combining a straightforward adaptation of [[Peter Li (mathematician)|Peter Li]] and [[Shing-Tung Yau]]'s gradient estimate to the Ricci flow together with an innovative "entropy estimate". The full positive case was demonstrated by Bennett {{harvtxt|Chow|1991}}, in an extension of Hamilton's techniques. Since any Ricci flow on a two-dimensional manifold is confined to a single [[conformal geometry|conformal class]], it can be recast as a partial differential equation for a scalar function on the fixed Riemannian manifold {{math|(''M'', ''g''<sub>0</sub>)}}. 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.

The higher-dimensional case has a longer history. Soon after Hamilton's breakthrough result, [[Gerhard Huisken]] extended his methods to higher dimensions, showing that if {{math|''g''<sub>0</sub>}} 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. {{harvtxt|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]]. As a consequence, he was able to settle the case in which {{mvar|M}} is four-dimensional and {{math|''g''<sub>0</sub>}} has positive curvature operator. Twenty years later, Christoph Böhm and Burkhard Wilking found a new algebraic method of constructing "pinching sets", thereby removing the assumption of four-dimensionality from Hamilton's result ({{harvnb|Böhm|Wilking|2008}}). [[Simon Brendle]] and [[Richard Schoen]] showed that positivity of the isotropic curvature is preserved by the Ricci flow on a closed manifold; by applying Böhm and Wilking's method, they were able to derive a new Ricci flow convergence theorem ({{harvnb|Brendle|Schoen|2009}}). Their convergence theorem included as a special case the resolution of the [[differentiable sphere theorem]], which at the time had been a long-standing conjecture. The convergence theorem given above is due to {{harvtxt|Brendle|2008}}, which subsumes the earlier higher-dimensional convergence results of Huisken, Hamilton, Böhm & Wilking, and Brendle & Schoen.

===Corollaries===
The results in dimensions three and higher show that any smooth closed manifold {{mvar|M}} which admits a metric {{math|''g''<sub>0</sub>}} 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
* Suppose that {{math|''M''}} is a smooth closed 3-dimensional manifold which admits a smooth Riemannian metric of positive Ricci curvature. If {{math|''M''}} is simply-connected then it must be diffeomorphic to the 3-sphere.
So if one could show directly that any smooth [[Closed manifold|closed]] [[Simply connected space|simply-connected]] 3-dimensional manifold admits a smooth Riemannian metric of positive [[Ricci curvature]], then the [[Poincaré conjecture]] would immediately follow. However, as matters are understood at present, this result is only known as a (trivial) corollary of the Poincaré conjecture, rather than vice versa.

===Possible extensions===
Given any {{mvar|n}} larger than two, there exist many closed {{mvar|n}}-dimensional smooth manifolds which do not have any smooth Riemannian metrics of constant curvature. So one cannot hope to be able to simply drop the curvature conditions from the above convergence theorems. It could be possible to replace the curvature conditions by some alternatives, but the existence of compact manifolds such as [[complex projective space]], which has a metric of nonnegative curvature operator (the [[Fubini-Study metric]]) but no metric of constant curvature, makes it unclear how much these conditions could be pushed. Likewise, the possibility of formulating analogous convergence results for negatively curved Riemannian metrics is complicated by the existence of closed Riemannian manifolds whose curvature is arbitrarily close to constant and yet admit no metrics of constant curvature.<ref>{{cite journal |last1=Gromov |first1=M. |last2=Thurston |first2=W. |title=Pinching constants for hyperbolic manifolds |journal=Invent. Math. |date=1987 |volume=89|issue=1 |pages=1–12|doi=10.1007/BF01404671|bibcode=1987InMat..89....1G |s2cid=119850633 }}</ref>