The Thick Part: Hyperbolic Pieces

12C.6 Geometrization stated what happens to a Ricci flow with surgery that does not become extinct: rescaled by $\frac1t$, it splits for large $t$ into a thick part, close to finitely many complete hyperbolic manifolds of finite volume whose cusp tori are incompressible, and a thin part, collapsed with a lower curvature bound, which is a graph manifold. This chapter proves the statement about the thick part, following Perelman's §§6–7.3 and Kleiner and Lott's detailed account; 12C.8 The Thin Part: Collapse and Graph Manifolds treats the thin part.

The argument has three layers.

1. Curvature from volume (Perelman II, §6). On a flow with surgery, a ball that is noncollapsed and has a lower curvature bound at its own scale has bounded curvature on a parabolic neighbourhood. This replaces the assumption of bounded normalised curvature in Hamilton's 1999 work, and it is where the canonical neighbourhood machinery of Book 12B is used once more.
2. Monotone quantities force hyperbolic limits (§7.1). Two scale-invariant quantities, $V(t)(t + \frac14)^{-3/2}$ and $R_{\min}V^{2/3}$, are monotone; in the limit they force every converging region of $\frac{g(t)}{t}$ to have constant scalar curvature $-\frac{3}{2t}$, hence constant sectional curvature $-\frac{1}{4t}$.
3. Hamilton's stabilization and incompressibility (§7.3). Mostow–Prasad rigidity makes the hyperbolic pieces stable in time; a minimal-disk argument with Gauss–Bonnet, of the kind used for extinction in 12C.2 Two-Spheres Disappear and 12C.4 Loops, Disks and Curve Shrinking, shows that their boundary tori are incompressible.

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

• state Perelman's curvature estimate from volume and lower curvature bounds (Corollary 6.8) and explain its role;
• prove the monotonicity of $V(t)(t + \frac14)^{-3/2}$ and of $\hat R = R_{\min}V^{2/3}$, and deduce that limits of $\frac{g(t)}{t}$ in the thick part are hyperbolic with curvature $-\frac14$;
• define the thick and thin parts through the scale $\rho(x, t)$;
• explain Hamilton's stabilization of the hyperbolic pieces through a rigidity lemma for $\xi$-approximations;
• prove that the cusp tori are incompressible by the least-area compressing disk argument.

7.1The setting

Assume the flow with surgery does not become extinct and, by removing at once any component that acquires nonnegative scalar curvature, that $R_{\min}(t) < 0$ for all $t$. Components with $R \geq 0$ are understood: positive scalar curvature leads to extinction (12B.13 Ricci Flow with Surgery for All Time), and nonnegative scalar curvature becomes positive or is flat. The initial metric is normalised, as in 12B.10 Surgery on a δ-Neck.

The model is a hyperbolic manifold. If $g_0$ has constant curvature $-1$, the Ricci flow is $g(t) = (1 + 4t)g_0$, and the rescaled metric $\frac{g(t)}{t} \to 4g_0$, of constant curvature $-\frac14$ (Exercise 7.9). So "hyperbolic" in this chapter means curvature $-\frac14$ after rescaling by $\frac1t$.

7.2Curvature from volume

The hyperbolic limit is obtained by taking limits of the rescaled flow, and for that one needs curvature bounds where the manifold is noncollapsed. In Hamilton's setting they were assumed. Perelman proves them.

The point is that the constants do not depend on $t_0$, while the noncollapsing and canonical neighbourhood parameters of the construction of 12B.13 Ricci Flow with Surgery for All Time deteriorate with time. Part 1 is a reduced-volume argument started from the good ball (Perelman: "an analog of Theorem I.8.2", the local noncollapsing theorem of 12A.8 Local Noncollapsing from the Reduced Volume), with a modified function $\hat L = \bar L + 2\sqrt\tau$ to handle $R \geq -3$ and an estimate (6.3) for curves that come near surgeries. Part 2 is the blow-up argument of 12B.12 Canonical Neighbourhoods After Surgery once more. Part 3 is bounded curvature at bounded distance (12B.6 Bounded Curvature at Bounded Distance) along a geodesic from $x_0$.

Here Perelman corrects a statement of his first preprint: Theorem I.12.2, as stated, was "incorrect"; the assertion he had in mind, and used, is that for $\varphi$-almost nonnegative curvature, a flow satisfying the hypotheses of the local noncollapsing theorem (Theorem I.8.2) at $(x_0, r_0)$ with $\varphi(r_0^{-2})$ small has $R(x, r_0^2) \leq K(A)r_0^{-2}$ at distance less than $Ar_0$ (II, §6.2).

The proof combines two lemmas. Lemma 6.5, extracted from the proof of Lemma I.11.6: a smooth flow on $P(x_0, 0, 1, -\tau)$ with sectional curvature at least $-1$ and an almost Euclidean ball at the final time has $R \leq K_0\tau^{-1}$ on a smaller neighbourhood, and the ball at the earlier time is still at least a tenth as large as Euclidean. Lemma 6.6, "an elementary fact from the theory of Aleksandrov spaces": a unit ball of volume at least $w$ with curvature at least $-1$ contains a ball of radius $\theta_0(w)$ all of whose sub-balls are almost Euclidean. Perelman runs a contradiction argument at the first counterexample time and smallest counterexample radius, stepping backward in time with Lemma 6.5(b) and Lemma 6.6 until Lemma 6.5(a) gives the bound. Kleiner and Lott (Corollary 81.3) restate the result in this form: a lower bound on volume and on curvature at a scale gives an upper bound on curvature, on a backward parabolic neighbourhood of that scale, free of surgeries. It is the three-dimensional Ricci flow analogue of the volume–curvature estimate for $\kappa$-solutions in 12B.4 Compactness and Universal Estimates for κ-Solutions.

7.3Two monotone quantities

Here $\bar V$ measures how much of the manifold is "hyperbolic at the scale $\sqrt t$": for a hyperbolic manifold of volume $V_0$ (curvature $-1$), $V(t) = (1 + 4t)^{3/2}V_0$, so $\bar V = 8V_0$.

The argument is a "rigidity from monotonicity" argument of the same kind as the equality case of the reduced volume (12B.2 The Asymptotic Soliton and Three-Dimensional Shrinkers): a monotone quantity with a finite limit cannot keep increasing by definite amounts, so in the limit the inequality that drives it must be an equality, and equality characterises the model.

7.4Thick and thin

The pointwise statement is a contradiction argument: Proposition 7.2 gives curvature bounds, so a sequence of counterexamples has a rescaled limit, which is hyperbolic by Proposition 7.4. The statement on large balls uses Proposition 7.1 to rule out high curvature, and the forward statement uses the fact that an almost hyperbolic region evolves almost homothetically until the closeness fails, which it cannot at a first failure time.

Now the decomposition. Let $\rho(x, t)$ be the radius at which $\inf_{B(x, t, \rho)}\operatorname{Rm} = -\rho^{-2}$ (Kleiner–Lott, Definition 89.1): the scale at which the lower curvature bound becomes significant. For the hyperbolic model, $\rho \approx 2\sqrt t$. Given $w > 0$, the thin part $M^-(w, t)$ is the set of $x$ with $\operatorname{Vol}B(x, t, \rho(x, t)) < w\rho(x, t)^3$, and the thick part $M^+(w, t)$ its complement. For large $t$ (depending on $w$), points with $\rho(x, t)$ small compared with $\sqrt t$ are automatically thin (Perelman II, (7.6); Kleiner–Lott, Lemma 89.2), so every thick point satisfies the hypotheses of Proposition 7.5 with $r$ comparable to $1$.

7.5Hamilton's stabilization

Rescaled by $\frac1t$, every sequence of thick points has a subsequence converging to a complete hyperbolic manifold of finite volume (curvature $-\frac14$). The limits might depend on the sequence; Hamilton's achievement was to show that for large $t$ they do not, and that the thick part is parametrised by a fixed finite collection of hyperbolic manifolds.

Thick–thin for hyperbolic manifolds. By the Margulis lemma there is $\mu_0 > 0$ such that for a complete finite-volume hyperbolic three-manifold $X$ and $\mu \leq \mu_0$, the set $X_\mu$ of points with injectivity radius at least $\mu$ is compact and connected, and its complement is a finite union of tubes around short closed geodesics and cusps, each a quotient of a horoball by $\mathbb{Z}^2$, bounded by a flat torus whose principal curvatures are all $\frac12$ (with curvature $-\frac14$) (10A.9 Hyperbolic Three-Manifolds).

The hypothesis on cusps is essential: by Thurston's hyperbolic Dehn filling, a cusped hyperbolic manifold is a limit of closed ones, which are $\xi$-approximated by it but not isometric to it. Under the hypothesis, the image of the thick part of $X$ is close to the thick part of $X'$, the cusps of $X$ go into thin components of $X'$ which must also be cusps (there are at least as many), so $X$ and $X'$ are diffeomorphic, and Mostow–Prasad rigidity makes them isometric.

Perelman's description of the construction: take a limit $H_1$ with the least number of cusps; by rigidity, for small $w$ a standard truncation $H_1(w)$ embeds almost isometrically in $M^+(\frac w2, t)$ for all large $t$, moving by isotopy; if thick points remain outside, repeat with another limit $H_2$, and so on. The process stops because the volume is bounded: each $H_j$ contributes a definite amount to $\bar V$. Hamilton's original proof used harmonic maps to make the embeddings move smoothly; Kleiner and Lott give a proof with compactness and smoothing. Either way, the result is that for large $t$ the thick part is covered by almost isometric copies of truncated $H_j$, the same $H_j$ for all large $t$.

7.6Incompressibility of the cusp tori

Let $Z$ be a horospherical torus in a cusp of some $H_i$ and $\hat Z_t = f(t)(Z)$ its image in $M_t$. The claim is that $\hat Z_t$ is incompressible: $\pi_1(Z) \to \pi_1(M_t)$ is injective. This is what makes the hyperbolic pieces part of a geometric decomposition along incompressible tori (10A.7 Seifert Spaces and the JSJ Decomposition).

This is the same argument as the extinction arguments of 12C.2 Two-Spheres Disappear and 12C.4 Loops, Disks and Curve Shrinking: an area defined by topology, a Gauss–Bonnet computation giving a loss of at least $2\pi$ per unit time, curvature terms controlled by pinching, and a contradiction with positivity. Historically it came first: Hamilton invented it for exactly this purpose in 1999, before Perelman used it for extinction.

7.7History

Hamilton's "Non-singular solutions of the Ricci flow on three-manifolds" (1999) proved the thick-part results for flows without singularities and with $|\operatorname{Rm}| \leq \frac Ct$: the hyperbolic limits, their stabilization via rigidity and harmonic maps, and incompressibility via Meeks–Yau and minimal disks. Perelman (II, §§6–7, 2003) removed both hypotheses, the second through the curvature-from-volume estimates of §6, and observed that surgery does not disturb the minimal-disk argument. He also withdrew a claim from his first preprint (end of I, §13.2) that the volumes of maximal horns could be bounded below, which would have made the solution smooth after some time. Kleiner and Lott (§§81–91) gave complete proofs, replacing Hamilton's harmonic maps by a compactness argument and adding an alternative proof of incompressibility (their §93). Perelman's §8 gives a second route to the decomposition through the first eigenvalue of $-4\Delta + R$, connecting it with Anderson's conjectures on the sigma constant.

7.8Exercises

Solution

(a) $\operatorname{Ric}(g_0) = -2g_0$, so $-2\operatorname{Ric}(g(t)) = -2\operatorname{Ric}(g_0) = 4g_0 = \partial_tg$. (b) Sectional curvature $-\frac{1}{1 + 4t}$, $R = -\frac{6}{1 + 4t} = -\frac32(t + \frac14)^{-1}$: equality in 1. $V = (1 + 4t)^{3/2}V_0 = 8(t + \frac14)^{3/2}V_0$, so $V(t + \frac14)^{-3/2} = 8V_0$, constant. $\hat R = -\frac32(t + \frac14)^{-1}\cdot4(t + \frac14)V_0^{2/3} = -6V_0^{2/3}$, constant. (c) $\frac{g(t)}{t} = (4 + \frac1t)g_0 \to 4g_0$, of curvature $-\frac14$.

Solution

$\frac{d}{dt}(-\frac1y) = \frac23$, so $-\frac1y = \frac16 + \frac23t = \frac23(t + \frac14)$, $y = -\frac32(t + \frac14)^{-1}$. Since $\frac{d}{dt}R_{\min} \geq \frac23R_{\min}^2$ and $R_{\min}(0) \geq y(0)$, ODE comparison gives $R_{\min} \geq y$. Normalised means $|\operatorname{Rm}| \leq 1$, so each sectional curvature is at least $-1$ and $R \geq -6$.

Solution

(a) In dimension three the Weyl tensor vanishes, so $\operatorname{Rm}$ is determined by $\operatorname{Ric}$ and $R$. (b) If $\operatorname{Ric} = \lambda g$, then the sectional curvature of a plane with unit normal $\nu$ is $\frac R2 - \operatorname{Ric}(\nu, \nu) = \frac{3\lambda}{2} - \lambda = \frac\lambda2$, the same for all planes. (c) With $R = -\frac{3}{2t}$ spatially constant, $\Delta R = 0$ and $\partial_tR = \frac{3}{2t^2} = \frac23R^2$; the evolution equation then gives $2|\operatorname{Ric}^\circ|^2 = 0$.

Solution

(a) On $z = c$ the induced metric is $\frac{dx^2 + dy^2}{c^2}$, flat. The unit normal is $z\partial_z$, and the shape operator is the identity (horospheres are umbilic with curvature $1$). (b) The lattice vectors have Euclidean length fixed; their hyperbolic length on $z = c'$ is that divided by $c'$. (c) Scaling the metric by $4$ halves curvatures of hypersurfaces: principal curvatures $\frac12$, as Kleiner and Lott state.

Solution

With $B = A(t + \frac14)^{-3/4}$: $B' = (t + \frac14)^{-3/4}(A' - \frac{3A}{4(t + \frac14)}) \leq -(2\pi - D)(t + \frac14)^{-3/4}$, and $\int^\infty(t + \frac14)^{-3/4}dt = \infty$, so $B \to -\infty$. If $D \geq 2\pi$, the boundary terms could cancel the topological $2\pi$ and the area might stay positive. Making $D$ small is why the torus is pushed far down the cusp: there the boundary of the disk is short and the boundary terms are small.