Geometrization

When the manifold has aspherical prime factors, the flow with surgery need not become extinct. It runs forever, and the question is what it looks like as $t \to \infty$. Perelman's answer, in §7 of his second preprint, is the one Hamilton had found in 1999 for flows without singularities, now without Hamilton's extra hypothesis. Rescale by $\frac1t$. Where the rescaled manifold is thick (not collapsed), it becomes hyperbolic, converging to finite-volume hyperbolic pieces whose cusps are incompressible tori. Where it is thin (collapsed with a lower curvature bound), it is a graph manifold, made of Seifert fibred pieces. That is Thurston's geometrization conjecture (10A.5 Thurston’s Eight Geometries), and with it, the classification of closed 3-manifolds.

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

• explain why $\frac{g(t)}{t}$ is the right normalisation, and what the monotone quantities $V(t)(t + \frac14)^{-3/2}$ and $R_{\min}V^{2/3}$ say about it;
• define the thick and thin parts and state what happens to each;
• state the collapsing theorem and its history after Perelman;
• explain how incompressibility of the cusp tori is proved;
• assemble the geometrization theorem from the long-time picture, prime decomposition and the theory of graph manifolds.

4.1Telling knots apart

4.2The long-time normalisation

From now on, assume that $M$ has no metric of nonnegative scalar curvature (otherwise 12C.1 Reading Off the Topology and 12B.5 Ricci Flow with Surgery for All Time have already identified it), and that any component which acquires nonnegative scalar curvature is removed at once. For normalised initial data, the scalar curvature satisfies (II, §7.1)

so $V(t)(t + \frac14)^{-3/2}$ is nonincreasing; surgeries only decrease $V$. The scale-invariant quantity $\hat R = R_{\min}V^{2/3}$ is nondecreasing whenever $R_{\min} \leq 0$:

Both have limits as $t \to \infty$. If the limit $\bar V$ of $V(t)(t + \frac14)^{-3/2}$ is positive, these two monotone quantities force $R_{\min}(t) \sim -\frac{3}{2t}$, and in any region where the rescaled flow $\frac{g(t)}{t}$ converges, its limit has scalar curvature exactly $-\frac{3}{2t}$, and then, by the strong maximum principle, constant sectional curvature $-\frac{1}{4t}$. The hyperbolic metric $g(t) = (1 + 4t)g_0$, with $g_0$ of curvature $-1$, is the model (Exercise 4.4).

4.3Thick and thin

Let $\rho(x, t)$ be the radius at which $\inf\operatorname{Rm}$ on $B(x, t, \rho)$ equals $-\rho^{-2}$: the scale at which the curvature lower bound becomes significant. For $w > 0$, the thin part $M^-(w, t)$ is the set of points with

collapsed at the scale of their curvature bound, and the thick part $M^+(w, t)$ is its complement. Perelman shows (II, §7.3) that for large $t$ every point of the thick part has a ball of radius comparable to $\sqrt t$ on which the flow is nearly hyperbolic: $|2t\operatorname{Ric} + g| < \xi$ (Lemma 7.2), using the curvature estimates proved from volume bounds in §6, which replace Hamilton's assumption of bounded normalised curvature.

Parts 1 and 2 follow Hamilton's 1999 argument, which uses Mostow–Prasad rigidity of finite-volume hyperbolic manifolds (10A.6 Hyperbolic Three-Manifolds) to show that the copies are stable in time. Part 3 is a minimal surface argument, also Hamilton's, using a theorem of Meeks and Yau on least-area discs; it rests on the same evolution of the area of least-area discs that drives 12C.2 Finite Extinction. Perelman notes that the argument survives surgery, because surgery can only decrease the area of a least-area disc.

4.4The thin part

Applied to the thin part of $\frac{g(t)}{t}$, with the thick part removed, this says the thin part is a graph manifold. One exceptional case, when the whole manifold is thin but at a scale larger than itself, leads to a limit with nonnegative curvature, and then $M$ is flat. Perelman stated that the proof "has nothing to do with the Ricci flow", relies on critical point theory for distance functions from Alexandrov geometry, and would appear in a separate paper. It did not appear. Proofs were supplied by others:

• Takashi Shioya and Takao Yamaguchi, "Volume collapsed three-manifolds with a lower curvature bound" (Mathematische Annalen 333, 2005), using Alexandrov space theory, including Perelman's stability theorem;
• John Morgan and Gang Tian, in their completion of the geometrization proof (2008);
• Laurent Bessières, Gérard Besson, Michel Boileau, Sylvain Maillot and Joan Porti, for irreducible manifolds with nontrivial fundamental group (Inventiones Mathematicae, 2010), and in their book;
• Jianguo Cao and Jian Ge, "A simple proof of Perelman's collapsing theorem for 3-manifolds" (Journal of Geometric Analysis 21, 2011);
• Bruce Kleiner and John Lott, "Locally collapsed 3-manifolds" (Astérisque 365, 2014).

4.5Geometrization

Perelman summarises (II, §7.4): for large $t$, every component of the flow with surgery is a graph manifold, or a closed hyperbolic manifold, or splits along finitely many disjoint incompressible tori into pieces that are graph manifolds or complete finite-volume hyperbolic manifolds. Graph manifolds are understood (Waldhausen): each is a connected sum of irreducible graph manifolds, each of which splits along incompressible tori into Seifert fibred pieces (10A.4 Seifert Spaces and the JSJ Decomposition). Figure 4.1 shows the picture.

Combining this with the bookkeeping of surgery (12C.1 Reading Off the Topology) gives Thurston's conjecture.

The connected sum decomposition is produced by the surgeries, and the pieces they discard are spherical or $S^2\times S^1$ (12C.1 Reading Off the Topology). What survives forever decomposes into hyperbolic pieces (the thick part) and graph manifolds (the thin part), and graph manifolds are made of Seifert fibred pieces, which carry the six Seifert geometries, or of torus bundles carrying Sol (10A.5 Thurston’s Eight Geometries). Figure 4.2 labels each arrow of the "fates of pieces" diagram of 10A.7 Geometrization and Ricci Flow with the theorem that now proves it.

4.6History

Thurston formulated the geometrization conjecture around 1980 and proved it for Haken manifolds. Hamilton's 1999 paper carried out the long-time analysis for flows without singularities and with bounded normalised curvature. Perelman's second preprint (March 2003) removed both assumptions, deferred the collapsing theorem, and withdrew a claim from his first preprint about the smoothness of the solution after some time, which he called unjustified and irrelevant to the other conclusions. Morgan and Tian completed the proof in detail in 2008 and published it as The Geometrization Conjecture (Clay Mathematics Monographs 5, 2014).

4.7Exercises

Solution

(a) $\operatorname{Ric}(g_0) = -2g_0$, unchanged by scaling, so $\partial_tg = 4g_0 = -2\operatorname{Ric}$. Sectional curvature $-\frac{1}{1 + 4t}$, so $R = -\frac{6}{1 + 4t} = -\frac32\cdot\frac{1}{t + \frac14}$, equality in the bound for $R_{\min}$. (b) $V(t) = (1 + 4t)^{3/2}V_0 = 8(t + \frac14)^{3/2}V_0$. (c) Scaling a metric by $\frac1t$ multiplies sectional curvature by $t$: $-\frac{t}{1 + 4t}$.

Solution

$R_{\min}(cg) = c^{-1}R_{\min}(g)$ and $V(cg) = c^{3/2}V(g)$, so $V^{2/3}$ scales by $c$. For the model, $\hat R = -6V_0^{2/3}$ at all times, since the flow is just scaling.

Solution

(a) $\Sigma$ expands like $t$, the circle stays fixed, so after rescaling by $\frac1t$ the circle factor shrinks to zero length: collapsed, thin; it is Seifert fibred ($\mathbb{H}^2\times\mathbb{R}$ geometry). (b) The rescaled flow converges to a hyperbolic metric: thick. (c) Nil collapses after rescaling, as in 11A.8 Homogeneous Flows: thin, a Seifert fibred (Nil) graph manifold.

Solution

The boundary torus of a solid torus in $S^3$ is compressible: the meridian bounds a disc. Cutting along it gives a solid torus and a knot complement, which says nothing about $S^3$, and any such torus could be chosen in infinitely many ways. Incompressible tori are essential: they are detected by $\pi_1$ (they inject), and the JSJ theorem makes a minimal family of them canonical (10A.4 Seifert Spaces and the JSJ Decomposition). Hyperbolic pieces of finite volume have cusp tori that are incompressible, so the decomposition along incompressible tori is the one that geometry sees.

Solution

Suppose, as one expects for an initial metric with a thin neck between the summands, that this neck pinches first. By the canonical neighbourhood theorem (12B.3 The Canonical Neighbourhood Theorem) the region is made of necks, and surgery (12B.4 Surgery) cuts along one, leaving a lens space component and a component diffeomorphic to $N$. The lens space has no aspherical prime factors, so its component becomes extinct in finite time (12C.2 Finite Extinction). The $N$ component runs forever (12B.5 Ricci Flow with Surgery for All Time); rescaled by $\frac1t$, it has a nonempty thick part, which converges to the hyperbolic metric on $N$ (this chapter, Theorem 4.1). The ledger records $M \cong L(3, 1)\#N$.