Editing Ricci flow (section)

==Singularities==
[[Richard S. Hamilton|Hamilton]] showed that a compact Riemannian manifold always admits a short-time Ricci flow solution. Later [[Wanxiong Shi|Shi]] generalized the short-time existence result to complete manifolds of bounded curvature.<ref>{{cite journal |first=W.-X. |last=Shi |title=Deforming the metric on complete Riemannian manifolds |journal=Journal of Differential Geometry |volume=30 |year=1989 |pages=223–301 |doi=10.4310/jdg/1214443292 |doi-access=free }}</ref> In general, however, due to the highly non-linear nature of the Ricci flow equation, singularities form in finite time. These singularities are curvature singularities, which means that as one approaches the singular time the norm of the [[Riemann curvature tensor|curvature tensor]] <math>|\operatorname{Rm}|</math> blows up to infinity in the region of the singularity. A fundamental problem in Ricci flow is to understand all the possible geometries of singularities. When successful, this can lead to insights into the topology of manifolds. For instance, analyzing the geometry of singular regions that may develop in 3-D Ricci flow, is the crucial ingredient in Perelman's proof of the Poincaré and Geometrization conjectures.

===Blow-up limits of singularities===

To study the formation of singularities it is useful, as in the study of other non-linear differential equations, to consider blow-ups limits. Intuitively speaking, one zooms into the singular region of the Ricci flow by rescaling time and space. Under certain assumptions, the zoomed in flow tends to a limiting Ricci flow <math> (M_\infty, g_\infty(t)), t \in (-\infty, 0] </math>, called a '''singularity model'''. Singularity models are ancient Ricci flows, i.e. they can be extended infinitely into the past. Understanding the possible singularity models in Ricci flow is an active research endeavor.

Below, we sketch the blow-up procedure in more detail: Let <math> (M, g_t), \, t \in [0,T), </math> be a Ricci flow that develops a singularity as <math>t \rightarrow T</math>. Let <math>(p_i, t_i) \in M \times [0,T) </math> be a sequence of points in spacetime such that 
:<math>K_i := \left|\operatorname{Rm}(g_{t_i})\right|(p_i) \rightarrow \infty  </math>
as <math>i \rightarrow \infty</math>. Then one considers the parabolically rescaled metrics
:<math>g_i(t) = K_i g\left(t_i + \frac{t}{K_i}\right), \quad t\in[-K_i t_i, 0]</math>
Due to the symmetry of the Ricci flow equation under parabolic dilations, the metrics <math>g_i(t)</math> are also solutions to the Ricci flow equation. In the case that
:<math> |Rm| \leq K_i  \text{ on } M \times [0,t_i],</math>
i.e. up to time <math>t_i</math> the maximum of the curvature is attained at <math>p_i</math>, then the pointed sequence of Ricci flows <math>(M, g_i(t), p_i)</math> subsequentially converges smoothly to a limiting ancient Ricci flow <math> (M_\infty, g_\infty(t), p_\infty)</math>. Note that in general <math> M_\infty </math> is not diffeomorphic to <math>M</math>.

===Type I and Type II singularities===
Hamilton distinguishes between '''Type I and Type II singularities''' in Ricci flow. In particular, one says a Ricci flow <math> (M, g_t), \, t \in [0,T) </math>, encountering a singularity a time <math>T</math> is of Type I if
:<math> \sup_{t < T} (T-t)|Rm| < \infty </math>.
Otherwise the singularity is of Type II. It is known that the blow-up limits of Type I singularities are gradient shrinking [[Ricci soliton]]s.<ref>{{cite journal |first1=J. |last1=Enders |first2=R. |last2=Mueller |first3=P. |last3=Topping |s2cid=968534 |title=On Type I Singularities in Ricci flow |journal=Communications in Analysis and Geometry |volume=19 |issue=5 |year=2011 |pages=905–922 |doi=10.4310/CAG.2011.v19.n5.a4 |arxiv=1005.1624 }}</ref> In the Type II case it is an open question whether the singularity model must be a steady Ricci soliton—so far all known examples are.

===Singularities in 3d Ricci flow===
In 3d the possible blow-up limits of Ricci flow singularities are well-understood. From the work of Hamilton, Perelman and Brendle, blowing up at points of maximum curvature leads to one of the following three singularity models:
* The shrinking round spherical space form <math> S^3/\Gamma </math>
* The shrinking round cylinder <math> S^2 \times \mathbb{R} </math> 
* The Bryant soliton
The first two singularity models arise from Type I singularities, whereas the last one arises from a Type II singularity.

===Singularities in 4d Ricci flow===
In four dimensions very little is known about the possible singularities, other than that the possibilities are far more numerous than in three dimensions. To date the following singularity models are known
*<math>S^3 \times \mathbb{R} </math>
*<math>S^2 \times \mathbb{R}^2 </math>
*The 4d Bryant soliton
*Compact Einstein manifold of positive scalar curvature
*Compact gradient Kahler–Ricci shrinking soliton
*The FIK shrinker (discovered by [[M. Feldman]], [[Tom Ilmanen|T. Ilmanen]], [[D. Knopf]]) <ref>{{cite journal |first=D. |last=Maximo |s2cid=17651053 |title=On the blow-up of four-dimensional Ricci flow singularities |journal=J. Reine Angew. Math. |volume=2014 |issue=692 |year=2014 |pages=153–171 |doi=10.1515/crelle-2012-0080 |arxiv=1204.5967 }}</ref>
*The BCCD shrinker (discovered by [[Richard Bamler]], [[Charles Cifarelli]], [[Ronan Conlon]], and [[Alix Deruelle]])<ref>{{cite arXiv |last1=Bamler |first1=R. | last2=Cifarelli | first2=C. |last3=Conlon | first3=R. |last4=Deruelle |first4=A.|
date=2022 |title=A new complete two-dimensional shrinking gradient Kähler-Ricci soliton |eprint=2206.10785 |class=math.DG}}</ref> 
Note that the first three examples are generalizations of 3d singularity models. The FIK shrinker models the collapse of an embedded sphere with [[Intersection number|self-intersection number]]&nbsp;−1.