After Perelman

The proof of the Poincaré conjecture was finished by 2006, and the field did not stop. Some of the questions Perelman left open have been answered, and the answers turned the Ricci flow into a tool for problems that have nothing to do with the Poincaré conjecture. This chapter is a guided tour, not a course. It states the main results since Perelman in three dimensions and beyond, says how each connects to a chapter of the guide, and points to the topic guides where the current state is tracked. Every result is stated as its authors state it; none is proved here.

By the end of this chapter you will be able to:

• explain what a Ricci flow through singularities is, and what Kleiner–Lott and Bamler–Kleiner proved about it in dimension three;
• state the generalized Smale conjecture and how the Ricci flow proved it;
• state the complete classification of three-dimensional $\kappa$-solutions;
• describe the main ideas of Bamler's theory of Ricci flows in all dimensions;
• say what is known in higher dimensions, and name some open problems.

5.1A living field

5.2Flow through singularities

Perelman's surgery depends on choices: the scale $h$, the parameters $\delta(t)$, where exactly to cut. He wrote that his aim had been a canonical Ricci flow, defined on as large a part of space-time as possible, which he had not achieved (12B.4 Surgery). It now exists in dimension three.

Bruce Kleiner and John Lott defined singular Ricci flows: Ricci flow space-times, smooth except where curvature is infinite, subject to conditions near the singularities modelled on Perelman's canonical neighbourhoods ("Singular Ricci flows I", Acta Mathematica 219, 2017). They proved that Ricci flows with surgery from a fixed initial metric subconverge to a singular Ricci flow as the surgery scale tends to zero. Richard Bamler and Kleiner then proved that the singular Ricci flow from a given initial metric is unique and depends continuously on the initial metric ("Uniqueness and stability of Ricci flow through singularities", Acta Mathematica 228, 2022). Together these confirm Perelman's conjecture that a canonical Ricci flow through singularities exists for every compact Riemannian 3-manifold. The surgeries of Book 12B become approximations to a single well-defined object. See the topic guide on weak flows.

5.3Diffeomorphism groups

Because the singular flow is canonical and continuous, it can be run on a whole family of metrics at once, continuously in the family. That turns the Ricci flow into a tool for spaces of metrics and groups of diffeomorphisms.

The Smale conjecture, proved by Allen Hatcher in 1983, says that the diffeomorphism group of $S^3$ deformation retracts onto its isometry group $O(4)$. The generalized Smale conjecture asks the same for every spherical space form $S^3/\Gamma$ (10A.1 A Zoo of Three-Manifolds). Bamler and Kleiner proved it with the Ricci flow, first for all cases except $\mathbb{RP}^3$ ("Ricci flow and diffeomorphism groups of 3-manifolds", arXiv 2017), and then in full, together with the result that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible ("Ricci flow and contractibility of spaces of metrics", arXiv 2019). The second statement sharpens the classification of 12B.5 Ricci Flow with Surgery for All Time: not only do we know which 3-manifolds carry such metrics, but the metrics on each form a contractible space. The topic guide on diffeomorphism groups explains how different dimension four is.

5.4The classification of singularity models

Perelman's proof needed only the neck–cap structure of $\kappa$-solutions (12B.2 The Structure of κ-Solutions), but he conjectured more, and the list is now complete. Simon Brendle proved that every noncompact three-dimensional $\kappa$-solution is a family of shrinking round cylinders, a quotient of one, or the Bryant soliton (Acta Mathematica 225, 2020). For compact ones, Sigurd Angenent, Brendle, Panagiota Daskalopoulos and Nataša Šešum determined the asymptotics, and Brendle, Daskalopoulos and Šešum proved that a $\kappa$-noncollapsed ancient solution on $S^3$ is a family of shrinking round spheres or Perelman's compact example of 12B.1 κ-Solutions (Inventiones Mathematicae 226, 2021). See the topic guide on ancient solutions.

5.5Ricci flow in all dimensions

Much of Perelman's toolkit works in every dimension, but his use of it depended on dimension three: Hamilton–Ivey pinching, the classification of $\kappa$-solutions, necks of the form $S^2\times\mathbb{R}$. Two lines of work extend it.

Under curvature conditions. Brendle extended surgery to manifolds of dimension $n \geq 12$ with positive isotropic curvature, proving curvature pinching estimates that make blow-up limits uniformly PIC, a canonical neighbourhood theorem, and a classification of such manifolds that contain no nontrivial incompressible space forms of dimension $n - 1$ ("Ricci flow with surgery on manifolds with positive isotropic curvature", Annals of Mathematics 190, 2019). This continues the four-dimensional work of Hamilton, Chen and Zhu, and Chen, Tang and Zhu (11B.5 Hamilton’s Program in 2002).

Without curvature conditions. Bamler developed a theory of Ricci flows in all dimensions built on the conjugate heat kernel (12A.6 Pseudolocality): sharp heat-kernel and entropy bounds ("Entropy and heat kernel bounds on a Ricci flow background", arXiv 2020), using Perelman's entropy in a localised form, the Nash entropy; a compactness theory for super Ricci flows with a new notion of convergence, $\mathbb{F}$-convergence (Inventiones Mathematicae 233, 2023); and a structure theory for the limits ("Structure theory of non-collapsed limits of Ricci flows", arXiv 2020). Noncollapsed limits are smooth away from a set of parabolic codimension at least four, and their tangent flows at every point are gradient shrinking solitons. This is the higher-dimensional analogue of the blow-up analysis of 11B.4 Singularities and 12B.1 κ-Solutions, with the shrinkers of 12A.3 The 𝓦-Entropy as the models. See the topic guide on limit spaces.

Figure 5.2 places these results on a timeline, by thread.

5.6Open problems

The topic guides list current open problems in their "Where it stands" sections. A few, as they are stated there:

• In dimensions above three, a well-posed notion of weak Ricci flow through general singularities is the central open problem; Bamler's limit flows are the leading candidate, but uniqueness and continuation remain open (weak flows).
• Classifying four-dimensional gradient shrinkers, at least under geometric assumptions; understanding which singularities form generically; whether the Ricci flow can say anything about the smooth four-dimensional Poincaré conjecture (four-manifolds).
• Ancient solutions in higher dimensions, and collapsed ones, which can be much richer than the three-dimensional list (ancient solutions).

5.7History

Each result above is dated in the text. The pattern is clear in hindsight. The first decade after Perelman was spent verifying and writing out his proof; the second turned his methods (canonical neighbourhoods, the entropy, the conjugate heat kernel) into general tools, and settled several of his explicit conjectures: the canonical flow through singularities, the uniqueness of the Bryant soliton among noncompact $\kappa$-solutions, and the generalized Smale conjecture.

5.8Exercises

Solution

Surgery involves choices (where to cut, at which scale and time) that cannot in general be made continuously in $s$: a neck that is cut for $g_s$ may not exist for nearby $s$, and the surgery times jump. A canonical flow has no choices, and if it depends continuously on $g_s$, the whole family flows together. The generalized Smale conjecture is a statement about families: to show that a space of metrics or diffeomorphisms is contractible, one must deform every family continuously, which requires a flow that acts continuously on families.

Solution

Every metric on $S^2$ is conformal to a round one (uniformization), and the normalised Ricci flow on $S^2$ converges to a round metric (11A.7 Ricci Flow on Surfaces), depending continuously on the initial metric. So the space of metrics deformation retracts onto the space of round metrics (of fixed area), which is $\operatorname{Diff}(S^2)/O(3)$ up to scaling. Since the space of all metrics is contractible (it is convex), $\operatorname{Diff}(S^2)/O(3)$ is contractible, which is the content of Smale's theorem. Bamler and Kleiner's proof in dimension three follows the same pattern, with the singular Ricci flow in place of the normalised flow.

Solution

(a) $n + 2$. (b) $n + 2 - 4 = n - 2$, which is $1$ for $n = 3$. (c) The central sphere's radius tends to zero, so its diameter does too, and in the limiting metric space at the singular time the whole sphere becomes a single point. The singular set seen by a metric limit is that point, of parabolic dimension $0 \leq 1$, not the 2-sphere it came from.

Solution

The shrinking round cylinder $S^2\times\mathbb{R}$, its $\mathbb{Z}_2$ quotient, and the Bryant soliton. The Bryant soliton is not a shrinker; it is a steady gradient soliton (11B.1 Ricci Solitons).

Solution

From 110 to 161: an increase of $51$, about $46\%$, or about $17$ more papers each year on average. Overstatement: the search matches any paper whose abstract mentions the Ricci flow, including papers where it is peripheral. Understatement: papers on the topic that do not use the search phrases, or that are not posted on the arXiv, are missed; and classification rules may assign borderline papers to the kindred flows.