Canonical Neighbourhoods After Surgery

The second half of the induction that builds Ricci flow with surgery is the canonical neighbourhood assumption: on the next time step, every point of curvature at least $r_{i+1}^{-2}$ has a strong $(C, \varepsilon)$-canonical neighbourhood, for a suitable $r_{i+1}$, provided the surgeries are done with $\delta$ small enough. For a smooth flow this was Perelman's Theorem I.12.1 (12B.5 The Canonical Neighbourhood Theorem), proved by a blow-up argument whose limits are $\kappa$-solutions. In a flow with surgery the same argument runs, with one new difficulty: the parabolic region behind a blow-up point may run into a recent surgery, so that the limit cannot be extended backward. Perelman's answer (II, §5.4) is that in that case the point is near a surgery cap that has not yet disappeared, the flow there is close to the standard solution (12B.8 The Standard Solution), and the standard solution's own canonical neighbourhoods supply one for the point. Morgan and Tian's chapter 17 carries this out in detail. This chapter follows it.

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

• state the canonical neighbourhood proposition for flows with surgery (Morgan–Tian, Proposition 17.1) and explain why $r_{i+1}$ and $\delta_{i+1}$ are found by a double contradiction sequence;
• prove that a first counterexample time exists, including the case of a strong neck whose backward evolution runs into a surgery cap (Lemma 17.2);
• prove that if a backward flow line from near a blow-up point reaches a surgery cap along bounded curvature, the point has a canonical neighbourhood, by comparison with the standard solution and gluing of evolving necks (Lemma 17.7);
• finish the blow-up argument: either surgeries interfere, and Lemma 17.7 applies, or the limit extends backward forever and is a $\kappa$-solution;
• explain the order in which $\kappa_{i+1}$, $r_{i+1}$ and $\delta_{i+1}$ are chosen.

12.1The statement and the order of quantifiers

The quantifiers explain the shape of the proof. The noncollapsing constant $\kappa$ of 12B.11 Noncollapsing After Surgery does not depend on $r_{i+1}$, only on the previous step. So fix $\kappa$ first. Then suppose the proposition fails: for every choice of $r$ and $\delta$ there is a bad flow. Take $r_a \to 0$, and for each $a$ surgery controls $\delta_{a, b} \to 0$ as $b \to \infty$ with $\delta_{a, b} \leq \delta(r_a)$, and flows $(\mathcal M_{(a, b)}, G_{(a, b)})$ that satisfy every hypothesis but violate the conclusion with parameter $r_a$. Because $\delta_{a, b} \leq \delta(r_a)$, the flows are $\kappa$-noncollapsed, with the same $\kappa$ for all $(a, b)$. In the end a diagonal subsequence $b = b(a)$ is taken, so that both $r_a \to 0$ (the blow-up scale goes to zero, so limits are ancient) and $\delta_{a, b(a)} \to 0$ (the surgeries become arbitrarily precise, so near surgery caps the flows converge to the standard solution).

12.2The first counterexample

The content is that the set of good points is "open" in the right sense, so that the infimum of the times of bad points is attained by a bad point. Take bad points $x_n$ with times decreasing to the infimum $t'$, and a limit point $x$ at time $t'$ (time slices are compact, surgery times are discrete). Then $R(x) \geq r_a^{-2}$. If $x$ had a canonical neighbourhood of compact type ($\varepsilon$-round or $C$-component) or were in the core of a cap, then the defining inequalities, being strict, would hold for the nearby metrics at the times of $x_n$, and the $x_n$ would be good. If $x$ is the centre of a strong $\varepsilon$-neck there are two subcases.

1. The backward evolving neck extends a little further back. Then it can be shifted forward to the times of $x_n$, and recentred, giving strong necks at $x_n$: a contradiction.
2. The backward neck reaches back exactly to a surgery time, where some flow line in it ends in a surgery cap. This case is new, and is resolved only for large $b$, that is, small $\delta$. Rescale so that the surgery scale is $1$ and the surgery time is $0$. The curvature on the flow line from the cap point $z$ to the neck stays comparable to the neck's curvature $\tilde Q$, and at the cap it is between $(2D)^{-1}$ and $2D$, so $(5D)^{-1} \leq \tilde Q \leq 5D$ (17.1). Since the standard solution has $R \geq \frac{c}{1 - t}$ (12B.8 The Standard Solution), there is $\theta_1 < 1$ with $R \geq 8D$ on the standard solution after time $\theta_1$; and along the flow line $R \leq 6D$. With the evolution of surgery caps (12B.11 Noncollapsing After Surgery), this forces the elapsed time since surgery to be less than $\theta_1$ (Claim 17.3). So the neck sits, in rescaled units, at bounded distance from the tip of a cap whose evolution is close to the standard solution at a time $< \theta_1$ (Claim 17.4). The corresponding point of the standard solution cannot be the centre of a $\frac{\beta\varepsilon}{3}$-neck, because such a neck would contain the tip, which is a fixed point of the $SO(3)$ action and so has all sectional curvatures equal, impossible in a neck. By the standard solution's canonical neighbourhood theorem (12B.8 The Standard Solution, Theorem 12.32) it lies in the core of a $(C(\frac{\beta\varepsilon}{3}), \frac{\beta\varepsilon}{3})$-cap, which by closeness and openness gives a $(C, \varepsilon)$-cap around $x$, hence around $x_n$: a contradiction.

So for each $a$, after passing to large $b$, there is a first bad point. As Morgan and Tian remark (Remark 17.5), in subcase 2 the point $x$ is the centre of a strong neck, but the neighbourhoods found for the $x_n$ are caps coming from the surgery cap. The standard solution is doing its job: near a recent surgery, what looks like a neck at one scale is part of a cap at another.

12.3The blow-up sequence

Choose $b(a)$ with $\delta_{a, b(a)} \to 0$, let $x_a$ be the first bad point of the flow $(\mathcal M_a, G_a)$, and rescale by $R(x_a) \geq r_a^{-2} \to \infty$, shifting $t(x_a)$ to $0$. In the rescaled flows $(\tilde{\mathcal M}_a, \tilde G_a, \tilde x_a)$:

• $R(\tilde x_a) = 1$;
• every point at earlier times with $R \geq 1$ has a strong $(C, \varepsilon)$-canonical neighbourhood (by the choice of first bad time), and at time $0$ every point with $R > 1$ has a $(2C, 2\varepsilon)$-one (Claim 17.6: exposed points lie in caps, by conclusion 4 of the surgery theorem in 12B.10 Surgery on a δ-Neck; other points by limits along flow lines, as in 12B.7 The Singular Time: Horns and Strong Necks);
• the flows are $\kappa$-noncollapsed on scales up to $\varepsilon R(x_a)^{1/2} \to \infty$ (by 12B.11 Noncollapsing After Surgery, since $\delta_{a, b(a)} \leq \delta(r_a)$);
• they are pinched toward positive, so any limit has nonnegative curvature.

These are the hypotheses of the blow-up theorems of 12B.7 The Singular Time: Horns and Strong Necks (Theorems 11.1 and 11.8), except the existence of backward parabolic neighbourhoods, which surgeries may destroy. Bounded curvature at bounded distance (12B.6 Bounded Curvature at Bounded Distance) holds as it stands, because Morgan and Tian proved it for generalized Ricci flows: Perelman makes the same point in II, §4.2, Claim 2, noting that at the very end of its proof a cone is excluded "because the canonical neighborhoods of types other than (a) are not close to a piece of metric cone, and type (a) is ruled out by the strong maximum principle, since the $\varepsilon$-neck in question is strong".

12.4When surgeries get in the way

The key lemma handles every case in which a surgery interferes with the backward parabolic neighbourhood at bounded rescaled distance and time.

In every case $x_a$ has a canonical neighbourhood, contradicting its choice. So in the blow-up sequence, surgeries cannot interfere along bounded curvature at bounded distance.

12.5Finishing the blow-up argument

Curvature along short backward flow lines (Claim 17.9). For each $A$ there are $D(A)$ and $\delta(A) > 0$ with $|\operatorname{Rm}| \leq D(A)$ along backward flow lines from $B(\tilde x_a, 0, A)$ for time $\delta(A)$: bounded curvature at bounded distance bounds $R$ on the ball, the derivative bound $|\partial_tR| < CR^2$ at canonical neighbourhoods carries it a short time back (as in Lemma 11.2 of 12B.7 The Singular Time: Horns and Strong Necks), and pinching bounds $|\operatorname{Rm}|$.

Either parabolic neighbourhoods exist, or $x_a$ is good (Claim 17.10). If for some $A$ no uniform backward time works, there are flow lines from the $A$-ball that hit surgery caps at times $-t_a \to 0$, with curvature bounded by $D(A)$ along them, and Lemma 12.3 gives $x_a$ a canonical neighbourhood. Since $x_a$ is bad, for each $A$ the parabolic neighbourhood $P(\tilde x_a, 0, A, -t(A))$ exists with bounded curvature, and the time-$0$ slices converge to a complete nonnegatively curved limit, with bounded curvature (high-curvature points have $(2C, 2\varepsilon)$-canonical neighbourhoods, and a complete nonnegatively curved three-manifold has no $2\varepsilon$-necks of arbitrarily small scale, Morgan–Tian Proposition 2.19).

A uniform backward time (Claim 17.11). With the curvature bound $Q$ of the limit, Lemma 11.2 gives $\Delta t$ such that flow lines from points of curvature at most $2Q$ have curvature at most $16Q$ for time $\Delta t$; if surgeries interfered within $\Delta t$, Lemma 12.3 would apply. So $t(A) \geq t' > 0$ independently of $A$.

The limit is ancient. Let $T' \in (0, \infty]$ be the supremum of such uniform backward times. By the long-time blow-up theorem of 12B.7 The Singular Time: Horns and Strong Necks (Theorem 11.8) there is a limit flow on $(-T', 0]$ with bounded nonnegative curvature. If $T' < \infty$, then just beyond $T'$ flow lines from a bounded ball hit surgery caps, with curvature bounded along them by Lemma 11.2, and Lemma 12.3 applies again. So $T' = \infty$, the limit is ancient, $\kappa$-noncollapsed on all scales, non-flat: a $\kappa$-solution. By the structure theory (12B.4 Compactness and Universal Estimates for κ-Solutions, Corollary 9.95) $x_a$ has a strong canonical neighbourhood for large $a$. Contradiction. $\square$

12.6Choosing the parameters

Morgan and Tian's induction step (§17.2) reads: given sequences up to index $i$,

1. $\kappa_{i+1}$ is the $\kappa$ of the noncollapsing proposition of 12B.11 Noncollapsing After Surgery, which depends only on the sequences up to $i$;
2. $r_{i+1}$ and $\delta_{i+1}$ are given by Proposition 12.1, with $\delta_{i+1} \leq \delta(r_{i+1})$ so that noncollapsing holds;
3. the new sequences are $\{r_0, \dots, r_{i+1}\}$, $\{\kappa_0, \dots, \kappa_{i+1}\}$, and $\{\delta_0, \dots, \delta_{i-1}, \delta_{i+1}, \delta_{i+1}\}$: $\delta_i$ is replaced by $\delta_{i+1}$, because both propositions need $\delta$ small on $[T_{i-1}, T_{i+1}]$, a step back into the past.

With these, a flow with surgery satisfying everything up to $T < T_{i+1}$ is continued: run the Ricci flow to the next singular time, perform surgery with parameters $\delta(T)$ and $r_{i+1}$ (12B.10 Surgery on a δ-Neck), and check by the two propositions that the inductive properties still hold. Only finitely many surgeries happen before $T_{i+1}$ (Lemma 17.12): each surgery along a sphere removes a definite volume, of order $\delta^{-1}h^3$, with $h \geq h(T_{i+1}) > 0$, while the volume grows at most like $e^{6t}$, so the number of sphere surgeries is bounded in terms of $\operatorname{Vol}(M_0)$, $T_{i+1}$, $r_{i+1}$ and $\delta(T_{i+1})$; and the number of components removed is at most the initial number plus the number of sphere surgeries. 12B.13 Ricci Flow with Surgery for All Time puts all of this together as Perelman's Proposition 5.1 and Morgan and Tian's Theorem 15.9: a Ricci flow with surgery exists for all time from any normalized initial metric.

12.7History

Perelman's §5.4 (II, 2003) proved the canonical neighbourhood assumption for flows with cutoff in under a page, by the blow-up argument with the case of nearby surgeries handled by his Lemma 4.5, Corollary 4.7 and the standard solution's Claim 5. Kleiner and Lott (§80) and Morgan and Tian (chapter 17, 2007) wrote out the cases; Morgan and Tian's first-counterexample lemma, with its subcase of a neck reaching back into a surgery cap, and their use of the gluing of evolving necks, make explicit what Perelman's parenthetical remark indicates. Cao and Zhu gave another complete account.

12.8Exercises

Solution

Negation: for every $r \leq r_i$ and every $\delta' \leq \delta(r)$ there is a flow satisfying the hypotheses with $\delta(t) \leq \delta'$ that violates the canonical neighbourhood assumption with parameter $r$. So for each $r_a$ there are bad flows for every $\delta_{a, b} \to 0$. Lemma 17.2 (existence of a first bad time) needs, for fixed $a$, $b$ large, that is $\delta$ small relative to quantities that depend on the particular flow's first bad point (subcase 2 of the neck case). A single sequence with $\delta_a \leq \delta(r_a)$ would give no control of how small $\delta_a$ is relative to what Lemma 17.2 needs; the double sequence lets one first make $b$ large for each $a$, then diagonalise.

Solution

(a) The isotropy group $SO(3)$ acts on $T_{p_0}$ by the standard representation, which is transitive on $2$-planes; isometries preserve sectional curvature, so all planes at $p_0$ have the same curvature. (b) In the neck, at every point there are planes with curvature near $\frac12R$ and planes with curvature near $0$ (relative to the local $R$, which varies by a factor close to $1$ across the neck); at $p_0$ all are equal. For small $\varepsilon$ these are incompatible.

Solution

If $t_a \leq \frac{h_a^2}{2}$ the rescaled elapsed time is at most $\frac12$. Otherwise $h_a^2 < 2t_a \leq 2t'$, so in units of $h_a$ (multiplying curvatures by $h_a^2$) the curvature along the flow line is at most $D'h_a^2 \leq 2t'D'$, a fixed bound. On the standard solution the curvature at rescaled time $s$ is at least $\frac{c}{1 - s}$, which exceeds $4t'D'$ once $s \geq \theta_1$ for a suitable $\theta_1 < 1$. For small $\delta$ the evolving cap is close enough to the standard solution that its curvature is at least half the standard one, so the flow line cannot still be in the cap at a rescaled time $\geq \theta_1$. Less than $1$ is essential because the standard solution only describes the evolving cap on $[0, \theta]$ for $\theta < 1$: near time $1$ the standard solution becomes singular and closeness says nothing.

Solution

(a) A strong neck must extend backward for rescaled time $1$; this one extends only to the surgery time, which may be less. (b) The surgery was done on the central sphere of a strong $\delta$-neck, which existed for rescaled backward time $1$ before the surgery. Its kept half, near the initial slice of our neck, contains a strong $\frac{\beta\varepsilon}{2}$-neck centred at the corresponding point; concatenating it with our evolving neck (Proposition 15.2) gives a strong $\varepsilon$-neck. (c) Points of the cap are exposed: they have no past, so nothing can be glued there. The kept half of the old neck is in the continuing region, whose points have flow lines extending back before the surgery.

Solution

A sphere surgery increases the number of components by at most one (cutting along one sphere separates into at most two pieces); removing a component decreases it by one; smooth flow does not change it. So components at time $t$ $\leq N_0 + S(t) - D(t)$, and this is $\geq 0$, so $D(t) \leq N_0 + S(t)$. Every surgery time involves at least one sphere surgery or one component removal, so the number of surgery times is at most $S + D \leq N_0 + 2S$. Both $N_0$ (bounded by volume for normalized metrics, since each component has a unit ball of volume at least $\frac{\omega_3}{2}$) and $S$ are bounded.