Ricci Flow with Surgery for All Time

12B.4 Surgery described one surgery. To run the flow with surgery forever, Perelman has to show that the hypotheses that made the first surgery possible are still true after it, and after the next, at every time. Two hypotheses carry everything: canonical neighbourhoods at high curvature (12B.3 The Canonical Neighbourhood Theorem), and noncollapsing (12A.4 κ-Noncollapsing). Both were proved for smooth flows, and both must be re-proved for flows with surgery. They cannot be re-proved with constants that stay fixed for all time. Perelman lets the parameters depend on time, decreasing from one time interval to the next, and proves both properties on each interval by induction. Once they hold, each surgery removes a definite amount of volume, so surgeries cannot accumulate, and the flow with surgery exists for all time. It may become extinct.

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

• state the a priori assumptions (pinching and canonical neighbourhoods) and Perelman's Proposition 5.1 on their justification;
• explain the order in which the parameters $\varepsilon$, $\kappa$, $r$, $\delta$, $\rho$ and $h$ are chosen, and what forces each choice;
• explain why noncollapsing is proved across surgeries with the reduced volume and not with the $\mathcal W$-entropy;
• prove that surgery times do not accumulate;
• deduce the topology of 3-manifolds with positive scalar curvature, Perelman's first application.

5.1Ruling out Zeno

5.2The a priori assumptions

A Ricci flow with surgery is a sequence of Ricci flows on manifolds $M_k$ over time intervals $[t_k^-, t_k^+)$, each becoming singular at $t_k^+ = t_{k+1}^-$, where the limit at $t_k^+$ and the initial metric on $M_{k+1}$ contain isometric compact submanifolds that are identified (II, §4.1). Perelman works with flows that satisfy:

• Pinching: there is a function $\varphi$ decreasing to zero such that $\operatorname{Rm} \geq -\varphi(R)R$.
• Canonical neighbourhoods with parameter $r$: every point with $R \geq r^{-2}$ has a neighbourhood as in 12B.2 The Structure of κ-Solutions (strong $\varepsilon$-neck, $\varepsilon$-cap, or closed positively curved component), where for a strong neck the flow must be defined on the whole backward parabolic neighbourhood. Surgeries far away during that time are allowed.

For a smooth flow, the pinching estimate of Hamilton and Ivey (11A.5 Hamilton–Ivey Pinching) and Theorem I.12.1 (12B.3 The Canonical Neighbourhood Theorem) give both on any finite time interval. For a flow with surgery, Hamilton's capping construction preserves pinching (12B.4 Surgery). The canonical neighbourhood assumption is the one that must be earned.

Normalised initial data (II, §5.1): a closed oriented 3-manifold with $|\text{sectional curvature}| \leq 1$ and every unit ball of volume at least half that of the Euclidean unit ball. Every closed 3-manifold carries such a metric after scaling. For normalised initial data the pinching takes the form $\operatorname{Rm} \geq -\varphi(R(t + 1))R$, with $\varphi(s)$ behaving like $\frac{1}{\log s}$ at infinity.

Two points of the statement deserve emphasis. The flow may become extinct in finite time: that is not excluded, and for the Poincaré conjecture it is what happens (12C.2 Finite Extinction). And components that are $\varepsilon$-close to quotients of the round sphere have been declared extinct and removed from the list of canonical neighbourhoods. Perelman summarises (II, §6.1): there are decreasing positive functions $r(t)$ and $\bar\delta(t)$, piecewise constant, such that for normalised initial data and any $0 < \delta(t) < \bar\delta(t)$, the flow with $\delta(t)$-cutoff exists on $[0, \infty)$, with canonical neighbourhoods at scale $r(t)$ and the pinching estimate.

5.3The parameters

The parameters are chosen in a fixed order, and each depends only on those before it (Figure 5.1).

parameter	role	chosen	depends on
$\varepsilon$	closeness of necks and caps to their models	first, once	nothing (small enough for §§1–2)
$\kappa_j$	noncollapsing on the $j$-th interval	Lemma 5.2	$\varepsilon$, earlier $r_i, \kappa_i, \bar\delta_i$
$r_j$	canonical neighbourhood scale	Proposition 5.1 (induction)	$\varepsilon$, $\kappa_j$, earlier choices
$\bar\delta_j$	how fine the cutoff must be	Lemmas 4.5, 5.2, 5.3	$r_j$, $\kappa_j$, earlier choices
$\delta(t) < \bar\delta(t)$	the cutoff actually used	free	—
$\rho = \delta r$	low-curvature region $\Omega_\rho$ kept at surgery	definition	$\delta$, $r$
$h$	radius of the necks cut	Lemma 4.3	$\delta$, $\rho$, $\varphi$

The essential feature is the separation of scales. The canonical neighbourhood scale $r$ stays bounded below on each finite time interval, while the surgery radius $h < \delta\rho = \delta^2r$ can be made as small as one likes by shrinking $\delta$. So surgery happens far inside the region already known to consist of necks, where the geometry is extremely close to a round cylinder.

5.4Noncollapsing across surgeries

The proof of 12A.4 κ-Noncollapsing used the $\mathcal W$-entropy, integrated over the whole manifold, and a conjugate heat flow run backwards from the time in question to time $0$. Through a surgery the manifold itself changes, and there is no conjugate heat flow that passes through it with the monotonicity intact. The reduced volume (12A.5 Reduced Distance and Reduced Volume) is better suited, because it is built from curves, and a curve can be required to avoid the surgeries. Perelman measures $\mathcal L$-length only for admissible curves, those that stay in the region unaffected by surgery.

The argument of I.7.3 then goes through, provided curves on the boundary of the admissible set, the barely admissible curves, have large reduced length, so that the missing curves would not have contributed much reduced volume anyway. That is where the standard solution enters. Lemma 4.5 says that near a fresh surgery cap, for a definite time, the flow is close to the standard solution; Corollary 4.6 deduces that any curve that runs into the cap region from far away, or stays near it for that time, has $\int(R + |\dot\gamma|^2)\,dt$ larger than any prescribed $l$, once $\delta$ is small enough. Lemma 5.3 turns this into the needed lower bound on the $\mathcal L$-length of curves ending near a surgery. Lemma 5.2 concludes that the flow is $\kappa$-noncollapsed at scales below $\varepsilon$ on the next time interval, with $\kappa$ depending on the earlier constants but not on $\delta$, as long as $\delta$ is below a threshold.

5.5Canonical neighbourhoods across surgeries

The proof of Proposition 5.1 (II, §5.4) repeats the proof of the canonical neighbourhood theorem (12B.3 The Canonical Neighbourhood Theorem) with one new possibility. Take the first time $\bar t$ at which the canonical neighbourhood assumption fails, at a point $\bar x$; rescale by $R(\bar x, \bar t)$; and try to take a limit backwards in time as before. The limit exists at time $\bar t$, by the curvature estimate at bounded distance (II, Claim 2 of §4.2) and noncollapsing (Lemma 5.2). Going backwards, the new obstruction is a surgery nearby in space and time. If that happens, the surgery's radius $h$ is comparable to the curvature scale at $(\bar x, \bar t)$, and then Lemma 4.5 and the properties of the standard solution (Claim 5 of II, §2) show that $(\bar x, \bar t)$ has a canonical neighbourhood after all, because it lies in a region close to the standard solution, which has them: a contradiction. Otherwise the limit extends backwards indefinitely, to a $\kappa$-solution, and the argument of 12B.3 The Canonical Neighbourhood Theorem concludes.

5.6Surgeries do not accumulate

Figure 5.2 shows what this permits: the volume rises between surgeries, at a bounded rate, and drops at each surgery by a definite amount, so the drops cannot crowd together.

5.7A first application: positive scalar curvature

If the initial metric has $R \geq a > 0$, the maximum principle gives $R \geq \frac{a}{1 - \frac23at}$, which blows up at $t = \frac{3}{2a}$, and surgery preserves the bound. So the flow with surgery becomes extinct by time $\frac{3}{2a}$ (II, §6.1; Exercise 5.8). Every component that ever existed was either discarded or became extinct, and all of these pieces are on the list of 12B.4 Surgery: spherical space forms $S^3/\Gamma$ and $S^2\times S^1$ (with $\mathbb{RP}^3\#\mathbb{RP}^3$ among the connected sums). Reading the bookkeeping backwards:

Perelman notes that the topology of such manifolds had been described by Schoen and Yau and by Gromov and Lawson more than twenty years earlier, up to quotients of homotopy spheres, and that conversely every such connected sum carries a metric of positive scalar curvature. If the scalar curvature is only nonnegative, it becomes positive at once unless the metric is flat, so the flat manifolds join the list. The same extinction mechanism, under a weaker hypothesis than positive scalar curvature, is the route to the Poincaré conjecture (12C.2 Finite Extinction).

5.8History

Perelman's second preprint (March 2003) carried out the construction in §§2–5, verifying, as he says in its opening lines, most of the assertions made in §13 of the first. He noted two exceptions, a collapsing result deferred to a separate paper and a claim about the smoothness of the solution after some time that he withdrew as unjustified and unnecessary. He also wrote that he could not fix a step in Hamilton's surgery argument, and that his own approach differed, with the two scales $h$ and $r$. The parameter choices were checked in detail in the expositions by Kleiner and Lott, Morgan and Tian, and Cao and Zhu, and an alternative construction, "Ricci flow with bubbling-off", was given by Bessières, Besson, Boileau, Maillot and Porti in their book Geometrisation of 3-Manifolds (2010).

5.9Exercises

Solution

(a) After the $k$-th bounce the speed is $v_0e^k$, and a vertical flight with launch speed $v$ lasts $\frac{2v}{g}$. (b) The total is $\sqrt{2H/g} + \frac{2v_0}{g}\sum_{k\geq1}e^k = \sqrt{2H/g} + \frac{2v_0}{g}\frac{e}{1 - e} < \infty$, so infinitely many bounces happen before a finite time. (c) It would make the number of bounces in any finite time finite, so no accumulation.

Solution

The function $\phi(t) = -\frac{6}{1 + 4t}$ solves $\phi' = \frac{24}{(1 + 4t)^2} = \frac23\phi^2$ with $\phi(0) = -6$. By ODE comparison, $R_{\min}(t) \geq \phi(t)$: if $R_{\min} < \phi$ somewhere, at the first time they meet the derivative inequality is contradicted.

Solution

$\frac{d}{dt}\log\operatorname{Vol} \leq \frac{6}{1 + 4t}$, which integrates to $\log\frac{\operatorname{Vol}(t)}{\operatorname{Vol}(0)} \leq \frac32\log(1 + 4t)$. Each surgery decreases the volume and does not decrease $R_{\min}$, so on each smooth stretch the same inequality holds starting from a smaller volume. With $N$ surgeries each removing at least $h_T^3$, the volume at time $T$ is at most $\operatorname{Vol}(0)(1 + 4T)^{3/2} - Nh_T^3$, which must be positive, so $N < \operatorname{Vol}(0)(1 + 4T)^{3/2}h_T^{-3}$.

Solution

The $\mathcal W$-argument compares $\mu(g(t), r^2)$ with $\mu(g(0), t + r^2)$ through the monotonicity of $\mathcal W$, which needs a solution of the conjugate heat equation on the whole manifold over the whole interval; a surgery changes the manifold, and nothing guarantees monotonicity across it. The reduced volume is an integral over curves, and one can restrict to curves that avoid the surgeries. The monotonicity argument then works, up to the contribution of curves that would have crossed the surgery regions; if barely admissible curves have large reduced length, that contribution is negligible, and the noncollapsing estimate survives with a slightly worse constant.

Solution

(a) $\phi = \frac{a}{1 - \frac23at}$ solves $\phi' = \frac23\phi^2$, $\phi(0) = a$; compare as in Exercise 5.5. (b) A component that survived past $\frac{3}{2a}$ would have $R \geq \phi(t) \to \infty$ everywhere, impossible for a smooth metric at a regular time; surgery preserves the lower bound. So every component is discarded or extinct by then. (c) The manifold is the connected sum of the discarded and extinct pieces and of copies of $S^2\times S^1$; these pieces are spherical space forms, $S^2\times S^1$ and connected sums of them. (d) A connected sum is simply connected only if every summand is, and among $S^3/\Gamma$ and $S^2\times S^1$ only $S^3$ is; so it is $S^3$. That is the Poincaré conjecture in the special case of positive scalar curvature.