The Canonical Neighbourhood Theorem

This is the theorem that makes surgery possible. It says that in any three-dimensional Ricci flow on a closed manifold, every point where the curvature is large enough has a neighbourhood that, after rescaling, looks like a piece of a $\kappa$-solution, and so (12B.2 The Structure of κ-Solutions) like a neck, a cap or a small closed component of known type. The threshold for "large enough" depends only on how close you want the picture to be, on the noncollapsing constant and on the pinching. It does not depend on the flow. The proof is the contradiction–compactness template that the guide has carried since 2A.5 Quantifiers and the Shape of a Proof, now with every box filled by a substantial theorem, including one hard new estimate: curvature is bounded at bounded distance.

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

• state Perelman's Theorem I.12.1 with every quantifier and hypothesis, and its consequence for flows with normalised initial data;
• write its negation and describe a sequence of counterexamples;
• outline the proof step by step, saying which earlier result supplies each step;
• explain the role of $\varphi$-almost nonnegative curvature, and why the threshold $r_0$ cannot be avoided.

3.1Identification at a distance

3.2The statement

Two hypotheses define the class of flows. Noncollapsing (12A.4 κ-Noncollapsing): the flow is $\kappa$-noncollapsed at scales below some $r_0$. Pinching: fix a decreasing function $\varphi$ with $\varphi(s) \to 0$ as $s \to \infty$; a flow has $\varphi$-almost nonnegative curvature if

at every point. This says that wherever the scalar curvature is large, the negative part of the curvature is small compared with it. For flows with normalised initial data, the Hamilton–Ivey estimate (11A.5 Hamilton–Ivey Pinching) gives this with $\varphi(s)$ behaving like $\frac{1}{\log s}$ (applied at $R(t + 1)$, which Perelman writes as $\operatorname{Rm} \geq -\varphi(R(t + 1))R$).

Combined with the structure of $\kappa$-solutions (12B.2 The Structure of κ-Solutions), this gives what is used. For a smooth Ricci flow on a closed oriented 3-manifold on a finite time interval, the pinching estimate holds and the flow is $\kappa$-noncollapsed at small scales (12A.4 κ-Noncollapsing), so there is $r = r(\varepsilon) > 0$ such that every point with $R \geq r^{-2}$ has a canonical neighbourhood: a strong $\varepsilon$-neck, an $\varepsilon$-cap, or a closed positively curved component (Perelman II, §3). The condition $t_0 \geq 1$ only ensures there is room to look back in time; it is harmless after rescaling the initial metric.

As Perelman writes it, the same $r_0$ serves as the noncollapsing scale and as the curvature threshold. In the proof, the threshold tends to zero while the rescaled flows must be noncollapsed at scales tending to infinity, so it is cleaner to fix the noncollapsing scale, say $\rho$, in advance and let only the threshold $r_0$ shrink. That is how the theorem is applied: in II, §3 the flow is $\kappa$-noncollapsed at scales below a fixed $r$ by 12A.4 κ-Noncollapsing, and Theorem 12.1 then supplies the threshold. Below, "noncollapsed" means at scales below such a fixed $\rho$.

Read the quantifiers as 2A.5 Quantifiers and the Shape of a Proof taught: $\forall\varepsilon, \kappa, \varphi\ \exists r_0\ \forall$ flows in the class $\forall(x_0, t_0)$ with $t_0 \geq 1$: $R(x_0, t_0) \geq r_0^{-2} \implies$ the rescaled flow near $(x_0, t_0)$ is $\varepsilon$-close to a $\kappa$-solution. The threshold $r_0$ is universal for the class. It is not a property of one flow.

3.3The proof

The negation reads: there are $\varepsilon$, $\kappa$ and $\varphi$, and a sequence $r_0 \to 0$, with flows in the class and points $(x_0, t_0)$ with $R(x_0, t_0) \geq r_0^{-2}$ at which no $\kappa$-solution is $\varepsilon$-close (Exercise 3.2). The proof rescales at these points and shows that the rescaled flows converge to a $\kappa$-solution, which contradicts the choice of the points. Here is the architecture, step by step (Figure 3.1).

Step 1: choose the counterexample well. Among the bad points of a given flow, Perelman chooses one with nearly the smallest curvature $Q$, in this sense: the conclusion does hold at every $(x, t)$ with $R(x, t) > 2Q$ and $t_0 - HQ^{-1} \leq t \leq t_0$, where $H \to \infty$ as $r_0 \to 0$. This is a form of point picking (11B.4 Singularities). It means that, in the region that matters, every point of much higher curvature already has a canonical neighbourhood, and so (by 12B.2 The Structure of κ-Solutions) satisfies the derivative estimates $|\partial_tR| \leq CR^2$ and $|\nabla R| \leq CR^{3/2}$.

Step 2: local control backwards in time (Claim 1). For each $(\bar x, \bar t)$ in the time window, with $\bar Q = Q + R(\bar x, \bar t)$, the curvature is at most $4\bar Q$ on a backward parabolic neighbourhood of size $c\bar Q^{-1/2}$ in space and $c\bar Q^{-1}$ in time, with $c = c(\kappa)$. This follows from the derivative estimates at high-curvature points, as in 12B.2 The Structure of κ-Solutions's rehearsal exercise.

Step 3: bounded curvature at bounded distance (Claim 2). This is the hardest step. If the curvature of the rescaled flows were unbounded at bounded distance from $x_0$, follow a shortest geodesic out from $x_0$ towards the high curvature. Either the curvature along it stays bounded at bounded distances, and in the limit the geodesic runs out along an almost cylindrical region, so the high-curvature point escapes to infinity; or a limit along the geodesic has a first singular point $o$ at finite distance. Near $o$ the limit is cylindrical, with radius at most $\varepsilon$ times the distance to $o$, so $o$ has nonnegative curvature in Aleksandrov's sense and the metric near $o$ is cone-like. Rescaling at points approaching $o$ gives a piece of a non-flat metric cone, and by Step 2 it is a limit of Ricci flows on a time interval, not just of metrics. A Ricci flow with nonnegative curvature cannot contain a piece of a non-flat cone: that contradicts Hamilton's strong maximum principle, which forces such a flow to split or to have positive curvature, and a cone does neither.

Step 4: a limit at the final time. With curvature bounded at bounded distance and $\kappa$-noncollapsing, which bounds the injectivity radius (9B.3 Collapsing and Noncollapsing), Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows) gives a smooth limit of the rescaled metrics at time $t_0$. Pinching makes its curvature nonnegative (Exercise 3.3). Its curvature is bounded, since otherwise it would contain round necks of radii tending to zero, impossible in a complete manifold of nonnegative curvature. Step 2 then extends the limit to a short interval of earlier times.

Step 5: the limit is ancient. Let $t'$ be the earliest time to which the limit can be taken. Hamilton's Harnack inequality (11B.2 Ancient Solutions and the Harnack Inequality) gives $\partial_tR \geq -\frac{R}{t - t'}$, hence $R(x, t) \leq \tilde Q\frac{t_0 - t'}{t - t'}$, with $\tilde Q$ the maximum of $R$ at $t_0$ (Exercise 3.6). This bounds how fast distances change (Perelman's Lemma 8.3(b)). For a noncompact limit, a geometric argument with long geodesics then bounds the curvature uniformly outside a compact set of bounded diameter, and Step 3 bounds it everywhere. With the curvature bounded on $(t', t_0]$, the limit extends past $t'$: a contradiction. So the limit exists for all earlier times. It is ancient, complete, of bounded nonnegative curvature, non-flat (its curvature is $1$ at the base point), and $\kappa$-noncollapsed at all scales, because the rescaled flows are noncollapsed at scales below $\rho\sqrt Q$, which tends to infinity since $Q \geq r_0^{-2}$ and $r_0 \to 0$. It is a $\kappa$-solution.

Step 6: the contradiction. The rescaled flows converge smoothly on compact sets to a $\kappa$-solution, so for large $k$ the region in the statement is $\varepsilon$-close to it. That contradicts the choice of $(x_0, t_0)$.

Two remarks on the architecture. Step 3, bounded curvature at bounded distance, is the genuinely new ingredient; Steps 4–6 are the template. And the argument is about the first bad point in a suitable sense: the points of higher curvature are already known to be good, and their good behaviour is what controls the limit. 12B.5 Ricci Flow with Surgery for All Time runs the same argument again for flows with surgery, where it must also handle surgeries that happen nearby in space and time.

3.4History

Theorem 12.1 is in §12 of Perelman's first preprint (2002), under the heading "Almost nonnegative curvature in dimension three". The same section contains Theorems 12.2 and 12.3, which control curvature from volume; Perelman corrected the statement of 12.2 in §6.2 of his second preprint. The proof of 12.1 is two pages in the original; the expositions by Kleiner and Lott, Morgan and Tian, and Cao and Zhu take considerably longer, and filling in the details of Claim 2 and the backward extension is where much of their work went.

3.5Exercises

Solution

$\exists\varepsilon, \kappa, \varphi\ \forall r_0\ \exists$ a flow in the class ($\varphi$-pinched, $\kappa$-noncollapsed below the fixed scale $\rho$) $\exists(x_0, t_0)$ with $t_0 \geq 1$, $R(x_0, t_0) \geq r_0^{-2}$, such that no $\kappa$-solution is $\varepsilon$-close to the rescaled region. With $r_0 = \frac1k$: flows $g_k$ and points with $R \geq k^2$ at which the rescaled geometry is not $\varepsilon$-close to any $\kappa$-solution. The flows are noncollapsed at scales below $\rho$, which after rescaling by $R \geq k^2$ means at scales below at least $k\rho \to \infty$.

Solution

Divide $\operatorname{Rm} \geq -\varphi(R)R$ by $Q$, with $R = Q\tilde R$. If $c \leq \tilde R \leq C$, then $\varphi(Q\tilde R) \leq \varphi(Qc) \to 0$, so $\widetilde{\operatorname{Rm}} \geq -\varphi(Qc)C \to 0$. Where $\tilde R$ is small, use the Hamilton–Ivey estimate itself (11A.5 Hamilton–Ivey Pinching): for normalised initial data it bounds the negative part of the curvature by a constant $C$ wherever $R$ is bounded, and after rescaling that bound becomes $\frac CQ \to 0$. In the limit the curvature operator is nonnegative everywhere.

Solution

Compactness: Step 4 (and the backward extension in Step 5). Cheeger–Gromov–Taylor: Step 4, turning noncollapsing into an injectivity radius bound. Hamilton–Ivey: through $\varphi$, Step 4 (nonnegative curvature of the limit). Harnack: Step 5, and in the compactness of 12B.2 The Structure of κ-Solutions. Strong maximum principle: Step 3. Derivative estimates: Steps 1–2. Noncollapsing: a hypothesis, used in Steps 4 and 5; for actual flows it is supplied by 12A.4 κ-Noncollapsing.

Solution

(a) Rescaling keeps sectional curvatures negative, while $\kappa$-solutions have nonnegative curvature; closeness in $C^2$ would make them nearly equal. (b) $R = -\frac{6}{1 + 4t} < 0 < r_0^{-2}$. (c) The theorem describes only the high-curvature part of a flow. Regions of low or negative curvature can look like anything, and in 12C.4 Geometrization they turn out to carry the hyperbolic pieces of geometrization.

Solution

(a) $\partial_t\log R \geq -\frac{1}{t - t'}$; integrate from $t$ to $t_0$: $\log R(t_0) - \log R(t) \geq -\big(\log(t_0 - t') - \log(t - t')\big)$. (b) Exponentiate. (c) For $t \geq \frac{t' + t_0}{2}$, $\frac{t_0 - t'}{t - t'} \leq 2$. As $t \to t'$ the factor blows up, so the bound is only useful away from $t'$; Perelman combines it with the distance estimate and Step 3 to get a bound up to $t'$, and then extends past it.