Editing Ricci flow (section)

==Examples==

===Constant-curvature and Einstein metrics===
Let <math>(M,g)</math> be a Riemannian manifold which is [[Einstein manifold|Einstein]], meaning that there is a number <math>\lambda</math> such that <math>\text{Ric}^g=\lambda g</math>. Then <math>g_t=(1-2\lambda t)g</math> is a Ricci flow with <math>g_0=g</math>, since then
:<math>\frac{\partial}{\partial t}g_t=-2\lambda g=-2\operatorname{Ric}^g=-2\operatorname{Ric}^{g_t}.</math>
If <math>M</math> is closed, then according to Hamilton's uniqueness theorem above, this is the only Ricci flow with initial data <math>g</math>. One sees, in particular, that:
* if <math>\lambda</math> is positive, then the Ricci flow "contracts" <math>g</math> since the scale factor <math>1-2\lambda t</math> is less than 1 for positive <math>t</math>; furthermore, one sees that <math>t</math> can only be less than <math>1/2\lambda</math>, in order that <math>g_t</math> is a Riemannian metric. This is the simplest examples of a "finite-time singularity". 
* if <math>\lambda</math> is zero, which is synonymous with <math>g</math> being Ricci-flat, then <math>g_t</math> is independent of time, and so the maximal interval of existence is the entire real line.
* if <math>\lambda</math> is negative, then the Ricci flow "expands" <math>g</math> since the scale factor <math>1-2\lambda t</math> is greater than 1 for all positive <math>t</math>; furthermore one sees that <math>t</math> can be taken arbitrarily large. One says that the Ricci flow, for this initial metric, is "immortal".
In each case, since the Riemannian metrics assigned to different values of <math>t</math> differ only by a constant scale factor, one can see that the normalized Ricci flow <math>G_s</math> exists for all time and is constant in <math>s</math>; in particular, it converges smoothly (to its constant value) as <math>s\to\infty</math>.

The Einstein condition has as a special case that of constant curvature; hence the particular examples of the sphere (with its standard metric) and hyperbolic space appear as special cases of the above.

===Ricci solitons===

{{anchor|cigar_soliton_solution}}
[[Ricci soliton]]s are Ricci flows that may change their size but not their shape up to diffeomorphisms.
* Cylinders '''S'''<sup>''k''</sup> × '''R'''<sup>''l''</sup> (for ''k'' ≥ 2) shrink self similarly under the Ricci flow up to diffeomorphisms
*A significant 2-dimensional example is the '''cigar soliton''', which is given by the metric (''dx''<sup>2</sup>&nbsp;+&nbsp;''dy''<sup>2</sup>)/(''e''<sup>4''t''</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>) on the Euclidean plane. Although this metric shrinks under the Ricci flow, its geometry remains the same. Such solutions are called steady Ricci solitons.
* An example of a 3-dimensional steady Ricci soliton is the '''[[Robert Bryant (mathematician)|Bryant]] soliton''', which is rotationally symmetric, has positive curvature, and is obtained by solving a system of ordinary differential equations.  A similar construction works in arbitrary dimension.
* There exist numerous families of Kähler manifolds, invariant under a ''U''(''n'') action and birational to ''C<sup>n</sup>'', which are Ricci solitons. These examples were constructed by Cao and Feldman-Ilmanen-Knopf. (Chow-Knopf 2004)
* A 4-dimensional example exhibiting only torus symmetry was recently discovered by Bamler-Cifarelli-Conlon-Deruelle. 


A '''gradient shrinking Ricci soliton''' consists of a smooth Riemannian manifold (''M'',''g'') and ''f''&nbsp;∈&nbsp;''C''<sup>∞</sup>(''M'') such that
:<math>\operatorname{Ric}^g+\operatorname{Hess}^gf=\frac{1}{2}g.</math>
One of the major achievements of {{harvtxt|Perelman|2002}} was to show that, if ''M'' is a closed three-dimensional smooth manifold, then finite-time singularities of the Ricci flow on ''M'' are modeled on complete gradient shrinking Ricci solitons (possibly on underlying manifolds distinct from ''M''). In 2008, [[Huai-Dong Cao]], Bing-Long Chen, and [[Xi-Ping Zhu]] completed the classification of these solitons, showing:
* Suppose (''M'',''g'',''f'') is a complete gradient shrinking Ricci soliton with dim(''M'')&nbsp;=&nbsp;3. If ''M'' is simply-connected then the Riemannian manifold (''M'',''g'') is isometric to <math>\mathbb{R}^3</math>, <math>S^3</math>, or <math>S^2\times\mathbb{R}</math>, each with their standard Riemannian metrics. This was originally shown by {{harvtxt|Perelman|2003a}} with some extra conditional assumptions. Note that if ''M'' is not simply-connected, then one may consider the universal cover <math>\pi:M'\to M,</math> and then the above theorem applies to <math>(M',\pi^\ast g,f\circ\pi).</math>

There is not yet a good understanding of gradient shrinking Ricci solitons in any higher dimensions.