The Thin Part: Collapse and Graph Manifolds

12C.7 The Thick Part: Hyperbolic Pieces showed that, for large $t$, the thick part of a Ricci flow with surgery that does not become extinct is covered by almost isometric copies of finitely many truncated hyperbolic manifolds, with incompressible boundary tori. What is left is the thin part: rescaled by $\frac1t$, every point has a ball, at the scale $\rho$ where its curvature lower bound $-\rho^{-2}$ is reached, whose volume is a small fraction of $\rho^3$. Perelman's final step (II, §7.4) is that such a manifold is a graph manifold: a union of circle bundles over surfaces (Seifert fibred pieces) glued along tori. With the thick part, the surgeries, and Waldhausen's theory of graph manifolds, this gives Thurston's geometrization conjecture.

The step is unusual in two ways. First, as Perelman put it, it "has nothing to do with the Ricci flow": it is a theorem in Riemannian geometry about manifolds that collapse with a lower curvature bound. Second, its proof was promised for "a separate paper" which never appeared, and it was supplied by others: Shioya and Yamaguchi (2005), Morgan and Tian (2008), Bessières, Besson, Boileau, Maillot and Porti (2010, an alternative that avoids the full theorem), Cao and Ge (2011) and Kleiner and Lott (2014). This chapter explains the statement, why the thin part of the flow satisfies it, the mechanism of its proof, and how the pieces of Book 12C assemble into geometrization.

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

• state Perelman's collapsing theorem in the versions of Kleiner and Lott, with each hypothesis explained;
• compute the collapse of homogeneous examples under the rescaled Ricci flow, including the explicit Nil solution;
• explain why the thin part of the flow satisfies the hypotheses, including the derivative bounds, and how the exceptional case leads to flat manifolds;
• describe the local models of collapse (to surfaces, to intervals and circles, to points) and how they assemble into a graph manifold;
• assemble the geometrization theorem from extinction, the thick part and the thin part.

8.1Graph manifolds

A graph manifold is a compact three-manifold that can be cut along finitely many disjoint embedded tori into pieces each of which is a circle bundle over a surface (more generally, Seifert fibred) (10A.7 Seifert Spaces and the JSJ Decomposition). The name comes from the graph whose vertices are the pieces and whose edges are the tori. Examples: $S^1\times\Sigma$; torus bundles over the circle (Euclidean, Nil and Sol manifolds); all Seifert fibred spaces, which carry six of Thurston's eight geometries; and gluings of these along tori. Waldhausen classified graph manifolds (1967): each is a connected sum of irreducible graph manifolds, and an irreducible one splits along incompressible tori into Seifert fibred pieces. So proving that a manifold is a graph manifold proves that it satisfies the geometrization conjecture.

8.2The collapsing theorem

Reading the hypotheses. Condition 1 is the collapse: at each point, at the scale where the curvature lower bound kicks in, the manifold is thin in the sense that a ball has much less volume than a Euclidean ball. Condition 2 handles the boundary, which in the application is the union of the cusp tori of the thick part: small flat tori with nearly hyperbolic collars, exactly what a hyperbolic cusp looks like. Condition 3 says that wherever the manifold is not collapsed at some smaller scale, it is smooth with controlled derivatives. Perelman added it because it "allows to simplify the proof quite a bit, by avoiding 3-dimensional Aleksandrov spaces, and in particular, the non-elementary Stability Theorem"; Kleiner and Lott note that a bound on finitely many derivatives is what the flow actually provides, and suffices (Remark 92.4). Perelman's statement also had $\rho < 1$, which Kleiner and Lott remark seems unnecessary (Remark 92.2).

Why the diameter condition matters. Without it, the whole manifold could be collapsed at a scale larger than itself: for instance a flat three-torus, small compared with $\rho = \infty$. Then the Alexandrov limit is a point and one learns only that $M$ is close to a nonnegatively curved space (Shioya–Yamaguchi, Theorem 1.1(2)), which is not directly a graph manifold statement (Remark 92.6).

8.3Collapse in examples

Homogeneous manifolds show what collapse under the rescaled flow $\frac{g(t)}{t}$ looks like (Kleiner–Lott, Remark 92.7).

• $S^1(\mu)\times\Sigma$ with $\Sigma$ hyperbolic: the flow is $S^1(\mu)\times(\Sigma, (c_0 + 2t)g_{\rm hyp})$. Rescaled by $\frac1t$, the surface tends to curvature $-\frac12$ while the circle has length $\mu t^{-1/2} \to 0$. Balls of fixed radius are circle bundles over hyperbolic disks, with $\rho^{-3}\operatorname{Vol} \sim t^{-1/2}$: collapse to a surface.
• Nil manifolds: with left-invariant forms $\theta_1, \theta_2, \theta_3$, $d\theta_3 = \theta_1\wedge\theta_2$, and $g = \alpha^2\theta_1^2 + \beta^2\theta_2^2 + \gamma^2\theta_3^2$, the Ricci flow is solved by $\alpha^2 = A(t + t_0)^{1/3}$, $\beta^2 = B(t + t_0)^{1/3}$, $\gamma^2 = \frac{AB}{3}(t + t_0)^{-1/3}$ (Exercise 8.3). Rescaled, the diameter goes like $t^{-1/3}$ and the volume like $t^{-4/3}$, with bounded curvature: collapse to a point.
• More generally (11A.13 Homogeneous Flows): $\mathbb{H}^2\times\mathbb{R}$ and $\widetilde{SL_2\mathbb{R}}$ manifolds collapse to hyperbolic surfaces of curvature $-\frac12$, Sol manifolds to a circle, and Euclidean and Nil manifolds to a point. Hyperbolic manifolds do not collapse; spherical and $S^2\times\mathbb{R}$ manifolds become extinct.

These collapses do not contradict Perelman's no local collapsing theorem (12A.4 κ-Noncollapsing), which bounds volume from below only on finite time intervals, at scales below a fixed constant: here the collapse is at scales comparable to $\sqrt t$, as $t \to \infty$.

8.4The thin part of the flow satisfies the hypotheses

Let $M_{\rm thin}(t)$ be what remains of $M_t$ after removing the truncated hyperbolic pieces $f(t)(\hat H_1(t)\cup\dots\cup\hat H_k(t))$ of 12C.7 The Thick Part: Hyperbolic Pieces, with horoballs cut off at distance about $\frac{1}{2\alpha(t)}$ from the base points (Kleiner–Lott, (92.11)).

Proof outline. Suppose not, along $t_\alpha \to \infty$, and let $M^\alpha$ be $M_{\rm thin}(t_\alpha)$ with closed nonnegatively curved components discarded (they are graph manifolds by Hamilton's classification), with metric $g^\alpha = t_\alpha^{-1}g(t_\alpha)$ and radius $\rho^\alpha = t_\alpha^{-1/2}\rho(\cdot, t_\alpha)$.

• Condition 1 holds with $w^\alpha \to 0$, by the definition of the thin part and the fact, from 12C.7 The Thick Part: Hyperbolic Pieces, that points not in the hyperbolic pieces are $w$-thin for every $w$ once $t$ is large.
• Condition 2 holds because the boundary is a union of horospherical tori far down the cusps of the hyperbolic pieces, of small diameter, with nearly hyperbolic collars.
• Condition 3 (Lemma 92.13): if a ball $B(x, r)$ in the rescaled metric has volume at least $w'r^3$ and curvature at least $-r^{-2}$, then the curvature-from-volume estimate of 12C.7 The Thick Part: Hyperbolic Pieces (Perelman's Corollary 6.8) bounds the curvature on a backward parabolic neighbourhood free of surgeries, and Shi-type estimates bound the derivatives. If the scale is below the surgery scales, the canonical neighbourhood assumption gives universal derivative bounds $|\nabla^m\operatorname{Rm}| \leq C(m)R^{(m + 2)/2}$; and a ball that is noncollapsed at its own scale cannot be filled with canonical neighbourhoods of much smaller scale, because by Perelman's Lemma 6.6 it contains an almost Euclidean sub-ball, while a neck or cap at a small scale is far from Euclidean at larger scales.

So Theorem 8.1 applies, and $M^\alpha$ is a graph manifold for large $\alpha$: a contradiction. Kleiner and Lott also give a version using the first form of the theorem, where the diameter condition can fail only if some $\rho^\alpha(x)$ exceeds the diameter $d^\alpha$; then the whole component is collapsed at a scale larger than itself, and Perelman's argument applies: the volume grows at most like $t^{3/2}$, so the diameter grows at most like $\sqrt t$, and rescaling to diameter one produces, in the limit, a smooth Ricci flow with nonnegative sectional curvature but not strictly positive scalar curvature, so $M$ is flat.

8.5Why collapse gives a graph manifold

The collapsing theorem itself is proved with the geometry of spaces with a lower curvature bound (9B.4 Convergence of Manifolds). The intuition is a local one.

Rescale at the curvature scale. Around $x$, rescale by $\rho(x)^{-2}$, so the ball $B(x, 1)$ has curvature at least $-1$ and tiny volume. By Gromov's compactness theorem for lower curvature bounds, such balls subconverge in the Gromov–Hausdorff sense to a ball in an Alexandrov space $X$ of curvature at least $-1$ and dimension $0$, $1$ or $2$: the collapse lowers the dimension.

Local models. The dimension of $X$ decides the local topology.

• $X$ two-dimensional. At regular points of $X$ (a surface, possibly with boundary), Yamaguchi's fibration theorem gives a fibration of the region of $M$ over the region of $X$ with fibre a closed manifold of nonnegative curvature in some sense, here a circle: locally $M$ is a circle bundle over a surface. Near the singular points of $X$ (cone points) and near its boundary the circle fibration degenerates, and the proofs identify the corresponding regions of $M$ as standard pieces, such as solid tori with an exceptional Seifert fibre.
• $X$ one-dimensional (an interval or a circle). Over interior points the fibres are closed surfaces of small diameter with an approximately nonnegative curvature bound, so by Gauss–Bonnet tori, Klein bottles, spheres or projective planes. In the torus and Klein bottle cases $M$ is locally a $T^2$- or Klein bottle-bundle over an interval, a graph manifold piece; the other cases, and the regions near the endpoints of the interval, give further standard pieces which the proofs treat separately.
• $X$ a point. Then the whole region is small at its own scale, and closeness to a nonnegatively curved Alexandrov space (with condition 3, a smooth nonnegatively curved manifold) identifies it: for a closed component, a manifold of nonnegative curvature, which by Hamilton's classification is a graph manifold.

Gluing. The local fibrations are then made compatible on overlaps and glued: circle fibrations over surface regions to each other and to torus fibrations over intervals, along tori. The result is a decomposition of $M$ into Seifert fibred pieces and pieces fibred by tori, glued along tori: a graph manifold. The gluing is the technical heart of each proof. Shioya and Yamaguchi use Perelman's stability theorem and his critical point theory for distance functions on Alexandrov spaces; Morgan and Tian construct local models and glue them, also with Alexandrov space theory; Cao and Ge simplify with Perelman's multiple conic singularity theory and fibration theory, describing the theorem as "an extension of implicit function theorem"; Kleiner and Lott work with the smoothness supplied by condition 3 and construct the decomposition from local models at many scales.

An alternative. Bessières, Besson, Boileau, Maillot and Porti avoid the full collapsing theorem for the purpose of geometrization: for closed, orientable, irreducible, non-simply connected $M$, they prove a weaker collapsing result, that if $M$ admits metrics with locally controlled curvature whose thick part becomes asymptotically hyperbolic with small volume, then $M$ is Seifert fibred or contains an incompressible torus. An irreducible manifold containing an incompressible torus is Haken, and Haken manifolds were geometrized by Thurston.

8.6Geometrization assembled

Putting Books 12B and 12C together, for any closed orientable three-manifold $M$ with a normalised metric:

1. The Ricci flow with surgery exists for all time (12B.13 Ricci Flow with Surgery for All Time).
2. Surgeries change the topology only by connected sum decompositions and the removal of spherical space forms and $S^2$-bundles over $S^1$ (12B.10 Surgery on a δ-Neck, 12C.1 Reading Off the Topology).
3. Components with $\pi_2 \neq 0$ disappear or lose $\pi_2$ in finite time (12C.2 Two-Spheres Disappear); components whose fundamental group is a free product of finite and cyclic groups become extinct (12C.4 Loops, Disks and Curve Shrinking).
4. If the flow does not become extinct, then for large $t$ each component is a closed hyperbolic manifold, or a graph manifold, or is cut by finitely many incompressible tori into finite-volume hyperbolic pieces and graph manifolds (12C.7 The Thick Part: Hyperbolic Pieces, this chapter). Perelman's summary: "for large $t$ every component of the solution is either diffeomorphic to a graph manifold, or to a closed hyperbolic manifold, or can be split by a finite collection of disjoint incompressible tori into parts, each being diffeomorphic to either a graph manifold or to a complete noncompact hyperbolic manifold of finite volume."
5. Graph manifolds are connected sums of irreducible ones, each split by incompressible tori into Seifert fibred pieces (Waldhausen).

Running the bookkeeping of step 2 backwards to time $0$ gives Thurston's geometrization conjecture for $M$ (12C.6 Geometrization): $M$ is a connected sum of prime manifolds, each cut along incompressible tori into geometric pieces.

8.7History

Perelman stated the collapsing theorem in II, §7.4 (2003), promising a separate paper based on critical point theory for distance functions. Shioya and Yamaguchi had already worked on collapse in dimension three with a lower curvature bound; their paper "Volume collapsed three-manifolds with a lower curvature bound" (arXiv 2003, Mathematische Annalen 2005) determined the topology. Morgan and Tian's "Completion of the proof of the geometrization conjecture" (arXiv 2008) and their book The Geometrization Conjecture (2014) glue local models built from Alexandrov space theory. Bessières, Besson, Boileau, Maillot and Porti gave the weak-collapsing alternative (arXiv 2007, Inventiones 2010) and a book, Geometrisation of 3-Manifolds (2010). Cao and Ge gave a simplified proof (arXiv 2010, Journal of Geometric Analysis 2011), and Kleiner and Lott a proof using the smoothness hypothesis, "Locally collapsed 3-manifolds" (arXiv 2010, Astérisque 2014).

8.8Exercises

Solution

(a) $\frac{d}{dt}\alpha^2 = -2\operatorname{Ric}_{11} = \frac{\gamma^2}{\beta^2}$, $\frac{d}{dt}\beta^2 = \frac{\gamma^2}{\alpha^2}$, $\frac{d}{dt}\gamma^2 = -\frac{\gamma^4}{\alpha^2\beta^2}$ (coefficients of $\theta_i^2$). (b) $\frac{d}{dt}\alpha^2 = \frac A3(t + t_0)^{-2/3}$ and $\frac{\gamma^2}{\beta^2} = \frac{A}{3}(t + t_0)^{-2/3}$; similarly for $\beta$; $\frac{d}{dt}\gamma^2 = -\frac{AB}{9}(t + t_0)^{-4/3}$ and $\frac{\gamma^4}{\alpha^2\beta^2} = \frac{A^2B^2}{9}(t + t_0)^{-2/3}\cdot\frac{1}{AB(t + t_0)^{2/3}} = \frac{AB}{9}(t + t_0)^{-4/3}$. (c) $\operatorname{Vol}(g(t)) \propto \alpha\beta\gamma \propto (t + t_0)^{1/6 + 1/6 - 1/6} = (t + t_0)^{1/6}$, so $\operatorname{Vol}(\frac{g}{t}) \propto t^{1/6 - 3/2} = t^{-4/3}$. The largest length scale is $\alpha \sim t^{1/6}$, so the diameter of $\frac gt$ is $\sim t^{1/6 - 1/2} = t^{-1/3}$. The curvature scale of $g(t)$ is $\frac{\gamma^2}{\alpha^2\beta^2} \sim \frac{1}{3}(t + t_0)^{-1}$, so for $\frac gt$ it is multiplied by $t$: bounded. The rescaled manifold has bounded curvature and shrinks to a point.

Solution

(a) $\operatorname{Ric} = -g_\Sigma$ on the surface factor (curvature $-1$ for $g_\Sigma$, and Ricci is scale invariant) and $0$ on the circle, so $\partial_t((1 + 2t)g_\Sigma) = 2g_\Sigma = -2\operatorname{Ric}$. (b) Sectional curvatures are $-\frac{1}{1 + 2t}$ and $0$, so $\inf\operatorname{Rm}$ on any ball is $-\frac{1}{1 + 2t}$ and $\rho = \sqrt{1 + 2t}$ once the ball is large enough. The ball of radius $\rho$ is roughly the circle times a disk of radius $\rho$ in the surface: volume $\approx 2\pi\mu\cdot\pi\rho^2c$ (with $c$ from hyperbolic area growth at that radius, a constant since the surface's curvature scale is $\rho$). So the ratio is $\sim\frac{\mu}{\rho} \sim\frac{\mu}{\sqrt{2t}} \to 0$. (c) A ball in a hyperbolic plane of curvature $-\frac12$ (in the metric $\frac gt$): two-dimensional collapse, with circle fibres.

Solution

(a) $d^3 \leq CV \leq C't^{3/2}$. (b) After rescaling by $d^{-2}$ the lower bound $-\rho^{-2}$ becomes $-\frac{d^2}{\rho^2}$. In the exceptional case $\rho$ is much larger than $d$ (Perelman: "$\rho(x, t)$ ... is much larger than the diameter"), so this tends to $0$. (c) With volume bounded below and condition 3 giving smoothness, the rescaled flows converge to a smooth Ricci flow with nonnegative sectional curvature on a closed manifold whose scalar curvature is not strictly positive (since $R_{\min} < 0$ throughout); by Hamilton's strong maximum principle such a flow is flat. So $M$ is diffeomorphic to a flat manifold: one of the Euclidean geometries.

Solution

(a) By Gauss–Bonnet, $\chi \geq 0$: the sphere, the projective plane, the torus and the Klein bottle. (b) The torus and the Klein bottle: a torus bundle over an interval is $T^2\times I$, and a Klein bottle bundle over an interval is $K\times I$, both Seifert fibred, so graph manifold pieces; over a circle they give torus or Klein bottle bundles over $S^1$. (c) Mapping tori of $A \in GL_2(\mathbb{Z})$ acting on $T^2$: Euclidean if $A$ has finite order, Nil if $A$ is parabolic (trace $\pm2$, infinite order), Sol if $A$ is hyperbolic ($|\operatorname{tr}A| > 2$). All are graph manifolds.

Solution

An irreducible graph manifold is cut by incompressible tori into Seifert fibred pieces (Waldhausen, and the JSJ decomposition, 10A.7 Seifert Spaces and the JSJ Decomposition); each Seifert piece carries one of the geometries $S^3$, $S^2\times\mathbb{R}$, $\mathbb{E}^3$, Nil, $\mathbb{H}^2\times\mathbb{R}$, $\widetilde{SL_2\mathbb{R}}$, and Sol torus bundles carry Sol. That is exactly the conclusion of the geometrization conjecture for the manifold. The flow reduces the general case to: spherical and $S^2\times S^1$ pieces (extinction), hyperbolic pieces with incompressible cusps (thick part), and the complement, which the collapsing theorem identifies as a graph manifold. Nothing else can occur.