Surgery on a δ-Neck

12B.9 Surgery described surgery in outline: at a singular time, find a strong $\delta$-neck deep in each horn, cut along its central sphere, discard the horn, glue in a cap, and continue. This chapter does it precisely. There are three things to get right.

1. The metric. The glued metric must be smooth, must be close to the standard initial metric of 12B.8 The Standard Solution near the cap, so that the flow afterwards is close to the standard solution, and above all must still satisfy the pinching estimate of 11A.10 Hamilton–Ivey Pinching, which is what makes blow-up limits nonnegatively curved. Gluing two metrics with a partition of unity does not preserve curvature conditions. Hamilton's solution, which Perelman adopts, is to bend the neck with a conformal factor $e^{-2f}$ that makes the curvature more positive exactly where the gluing happens.
2. The object. A flow with surgery is not a family of metrics on one manifold. Morgan and Tian make it a single geometric object, a space-time with a time function and a metric on its time slices, so that parabolic neighbourhoods, $\mathcal L$-geodesics and limits make sense across surgery times.
3. The bookkeeping. Surgery must happen at controlled scales, with controlled closeness, and leave the topology changed only in known ways. Morgan and Tian write this as seven assumptions and a choice of three parameter functions.

The proof that the a priori assumptions survive surgery is the subject of 12B.11 Noncollapsing After Surgery (noncollapsing) and 12B.12 Canonical Neighbourhoods After Surgery (canonical neighbourhoods); 12B.13 Ricci Flow with Surgery for All Time puts the parameters together.

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

• describe the surgery construction on a $\delta$-neck, with the profile $f(s) = C_0\delta e^{-q/s}$ and the interpolation to the standard cap;
• compute the change of curvature under the conformal factor and show that the smallest eigenvalue and the scalar curvature increase, so that pinching is preserved;
• state Morgan and Tian's Theorem 13.2 and explain each conclusion's later use;
• define a Ricci flow with surgery as a space-time, with exposed and continuing regions, and say what noncollapsing and canonical neighbourhoods mean there;
• state Morgan and Tian's Assumptions (1)–(7), their surgery parameters, and the topological consequence (Proposition 15.3).

10.1The construction

Fix a $\delta$-neck $(N, g)$ centred at $x_0$, with its parametrisation $\rho : S^2\times(-\delta^{-1}, \delta^{-1}) \to N$, so that $R(x_0)\rho^*g$ is $\delta$-close in $C^{[1/\delta]}$ to the cylinder $h_0 + ds^2$, $h_0$ round of scalar curvature $1$. Let $s$ be the coordinate along the neck, oriented so that the part to be discarded (toward the end of the horn) is $s > 0$. Fix the standard initial metric $g_0$ with tip $p_0$: outside the ball $B(p_0, A_0)$ it is exactly a half-cylinder, which gives a coordinate $s_1$ on $\mathbb{R}^3$ with $s_1 = A_0 + 4 - d_{g_0}(p_0, \cdot)$, equal to $4 + A_0$ at the tip and running down to $-\infty$ along the cylinder.

The manifold. Glue $\rho(S^2\times(-\delta^{-1}, 4))$ to the ball $B(p_0, A_0 + 4)$ by identifying $\rho(x, s)$ with the point of the cylindrical part of $g_0$ at the same $(x, s)$, for $0 < s < 4$. The result $S$ is the negative half of the neck capped off by a ball, and $s$ extends to a function on $S$ with values in $(-\delta^{-1}, 4 + A_0]$, the maximum at the tip.

The metric. Fix once and for all bump functions $\alpha$ (equal to $1$ near $s = 1$, to $0$ near $s = 2$) and $\beta$ (equal to $1$ near $s = 4 + A_0 - r_0$, to $0$ near the tip, where $g_0$ has constant curvature $\frac14$ on $B(p_0, r_0)$), set $\eta = \sqrt{1 - \delta}$, and fix constants $q$ and $C_0$, independent of $\delta$. Let

This is smooth, vanishes to infinite order at $s = 0$, and is tiny: at most $C_0\delta$. Define

and the new metric $\tilde g = R(x_0)^{-1}\hat g$ (Morgan–Tian, §13.1).

Read it from left to right. For $s \leq 0$ nothing changes: $\tilde g = g$ on the half of the neck that is kept. On $0 < s \leq 1$ the neck is multiplied by $e^{-2f}$, which bends it inward very slightly. On $[1, 2]$ the neck is interpolated with a slightly shrunk copy of the cylindrical part of $g_0$, both $\delta$-close to the same cylinder, so the interpolation costs only $O(\delta)$. Beyond $s = 2$ the metric is the conformally bent $g_0$, and near the tip the conformal factor is frozen to a constant so that the tip is exactly a rescaled piece of $g_0$, of constant curvature. The factor $\eta < 1$ makes the comparison map in Theorem 10.1 distance-decreasing.

10.2Why a conformal factor

Under $\hat h = e^{-2f}h$ the curvature changes by (Morgan–Tian, (13.1))

where $f_{ij}$ is the Hessian. In the neck, $f$ depends only on $s$, with $f' = \frac{q}{s^2}f$ and $f'' = \big(\frac{q^2}{s^4} - \frac{2q}{s^3}\big)f$. Choose $q$ so large that $q \gg (4 + A_0)^2$, which makes $\frac{q^2}{s^4} \gg \frac{q}{s^3}$ on $(0, 4 + A_0]$, and so that $\frac{q^2}{s^4}e^{-q/s} \ll 1$ there; then require $\delta \ll C_0^{-1}$. In the frame of the neck, with index $0$ for the $s$-direction and $a, b \in \{1, 2\}$ for the sphere, the result is (Corollary 13.5)

in the basis $e_1\wedge e_2$ (the sphere plane), $e_0\wedge e_1$, $e_0\wedge e_2$ (the mixed planes). The mixed planes, which on the cylinder have curvature $0$ and where negativity could hide, gain $\frac{q^2}{s^4}f$, the dominant term, from the Hessian $f''$; the sphere plane, of curvature about $\frac12$, loses only $\frac{q^2}{s^4}f^2$, which is quadratically small. For the scalar curvature, $\hat R = e^{2f}(R + 4\Delta f - 2|\nabla f|^2)$, and the same comparison gives $\hat R \geq R$ (Corollary 13.6).

Diagonalising carefully (Lemma 13.7 to Corollary 13.11), the eigenvalues $\lambda \geq \mu \geq \nu$ of the curvature of $h$, with $\lambda \approx \frac12$ and $\mu, \nu = O(\delta)$, become

so the smallest eigenvalue increases (Corollary 13.12). Finally fix $C_0 = 2Ke^q$, where $-K\delta$ bounds the eigenvalues of any $\delta$-neck of scale one from below. Then on $s \in [1, 4 + A_0]$ the gain $\frac{q^2}{2s^4}f \geq \frac{q^2}{2}e^{-q}C_0\delta > K\delta$ beats any negativity, and $\mu', \nu' > 0$ (Lemma 13.13): the bent region is positively curved.

This is the computation that Perelman summarises in one sentence: "in the region near the middle of the neck, where $f$ is small, the dominating terms in the formulas for the change of curvature are just positive constant multiples of $f''$, so the pinching improves". The choice $f = C_0\delta e^{-q/s}$ is designed to make $f''$ dominate $f'$ and $f$ dominate $f^2$, and the constant $C_0\delta$ is small enough not to spoil closeness to the standard cap and large enough to beat the $O(\delta)$ errors of the neck.

10.3The surgery theorem

What each conclusion is for. Conclusion 1 lets the Hamilton–Ivey estimate continue after surgery, so blow-up limits after surgery still have nonnegative curvature (12B.6 Bounded Curvature at Bounded Distance, 12B.12 Canonical Neighbourhoods After Surgery). Conclusion 2 says the new cap is a region of positive curvature, which 12B.12 Canonical Neighbourhoods After Surgery uses to recognise it as a cap. Conclusion 3 says surgery does not increase distances, which gives the monotonicity of the minimal-surface quantities in the extinction argument of 12C.3 Finite Extinction and keeps $\mathcal L$-lengths controlled in 12B.11 Noncollapsing After Surgery. Conclusion 4, with uniqueness and stability of the standard solution (12B.8 The Standard Solution), makes the flow after surgery near the cap close to the standard solution for a definite time: this is Perelman's Lemma 4.5 and the main input to 12B.11 Noncollapsing After Surgery.

The surgery cap occupies, in $\tilde g$, a region between the balls of radii $R(x_0)^{-1/2}(A_0 + 3)$ and $R(x_0)^{-1/2}(A_0 + 5)$ about the tip (Lemma 13.4), and balls about the tip of radius $D + 5 + A_0$ in the rescaled metric have boundary inside the kept part of the neck, at depth between $\frac D2$ and $2D + 2$ (Lemma 13.15).

10.4Ricci flow with surgery as a space-time

A space-time (Definition 14.1) is a paracompact Hausdorff space $\mathcal M$ with a time function $t : \mathcal M \to \mathbb{R}$ whose level sets are the time slices $M_t$, locally modelled on products $U\times I$ where things are smooth, together with a vector field $\chi$ with $\chi(t) = 1$ generating the flow lines. A surgery space-time (Definition 14.2) allows three kinds of points: interior manifold points; boundary points at the initial or final time; and points of the surgery time slices where the space-time is modelled on the union of a backward half-product and a forward half-product glued along a domain. At a surgery time $t$, the slice $M_t$ is split by a closed surface $\Sigma_t$ into the continuing region $C_t$, whose points have flow lines going back in time, and the exposed region $E_t$, the new caps, whose points have flow lines only forward. A horizontal metric $G$ is a metric on $\ker dt$, and a Ricci flow with surgery is a pair $(\mathcal M, G)$ with $\mathcal L_\chi G = -2\operatorname{Ric}(G)$, one-sided at exposed points and at the initial and final times (Definition 14.8).

The construction is concrete (Lemma 14.11): given flows $(M_1, g_1(t))$ on $(a, b]$ and $(M_2, g_2(t))$ on $[b, c)$, and compact submanifolds $\Omega_1 \subset M_1$, $\Omega_2 \subset M_2$ with neighbourhoods identified by an isometry of $g_1(b)$ and $g_2(b)$ carrying $\Omega_1$ to $\Omega_2$, remove $(M_1\setminus\Omega_1)\times\{b\}$ and glue. The result is a Ricci flow with surgery; the part of $M_1$ outside $\Omega_1$ simply disappears at time $b$, and $M_2\setminus\Omega_2$ appears. A smooth Ricci flow on $M\times[0, T)$ is the special case with no surgeries, and the interior manifold points of any Ricci flow with surgery form a generalized Ricci flow in the sense of 12B.6 Bounded Curvature at Bounded Distance (Proposition 14.12), which is why the theorems of 12B.6 Bounded Curvature at Bounded Distance and 12B.7 The Singular Time: Horns and Strong Necks apply.

The notions of the previous chapters carry over with one change: parabolic neighbourhoods must exist. $P(x, t, r, -\Delta t)$ exists if $B(x, t, r)\times(t - \Delta t, t]$ embeds in $\mathcal M$ compatibly with time and $\chi$, that is, if no point of the ball was created by a surgery within time $\Delta t$ before $t$ (Definition 14.14). Then:

• noncollapsing (Definition 14.15): if $P(x, t, r, -r^2)$ exists and $|\operatorname{Rm}| \leq r^{-2}$ on it, then $\operatorname{Vol}B(x, t, r) \geq \kappa r^3$;
• the strong canonical neighbourhood assumption (Definition 14.18): every $x$ with $R(x) \geq r^{-2}$ has a strong $(C, \varepsilon)$-canonical neighbourhood, where for a strong neck the backward evolving neck, for rescaled time $1$, must exist in $\mathcal M$. A point at or near an exposed region cannot be the centre of a strong neck, because the backward neighbourhood would run into the surgery.

10.5Controlled flows with surgery

Morgan and Tian restrict to flows whose surgeries are controlled topologically and geometrically.

Normalized initial metric. $(M, g(0))$ compact with $|\operatorname{Rm}| \leq 1$ and $\operatorname{Vol}B(x, 0, 1) \geq \frac{\omega_3}{2}$ everywhere; every metric becomes normalized after scaling up. The flow then exists on $[0, 2^{-4}]$ with $|R| \leq 2$ and with a universal noncollapsing constant $\kappa_0$ at scales $\leq \varepsilon$ (Claim 15.1).

Choice of $C$ and $\varepsilon$. Fix $\varepsilon$ small enough for every structure theorem of 12B.4 Compactness and Universal Estimates for κ-Solutions, 12B.6 Bounded Curvature at Bounded Distance, 12B.7 The Singular Time: Horns and Strong Necks and the neck topology of Morgan and Tian's appendix to hold with $2\varepsilon$, and $\varepsilon \leq \frac{1}{200}$. Fix $\beta < \frac12$ from the gluing proposition below, and $C$ the larger of the constants of the $\kappa$-solution structure theorem and of the standard solution's canonical neighbourhoods (12B.8 The Standard Solution, Theorem 12.32, at $\frac{\beta\varepsilon}{3}$). These are fixed for the rest of the proof.

This is how a neck that begins at a surgery (case 2 of the standard solution's canonical neighbourhoods, a neck reaching back only to the surgery time) can still be recognised as strong when the flow before the surgery supplies the rest of the backward time. The proof is by contradiction: as $\beta \to 0$ the union converges to the round shrinking cylinder on a time interval of length at least $1$.

The seven assumptions (§15.1).

1. Each time slice is a compact three-manifold with no embedded $\mathbb{RP}^2$ with trivial normal bundle.
2. Singular times are discrete.
3. The initial time is $0$ and the initial metric is normalized.
4. At each singular time the exposed region $E_t$ is a finite disjoint union of $3$-balls.
5. Each component of $\Sigma_t = \partial E_t$ is the central sphere of a strong $\delta(t)$-neck in the maximal extension of the flow to time $t$ (12B.7 The Singular Time: Horns and Strong Necks), for a nonincreasing surgery control function $\delta(t)$.
6. Just before a singular time, every point of the disappearing region (what is removed at time $t$) has a strong $(C, \varepsilon)$-canonical neighbourhood.
7. Between singular times the flow is maximal: it is continued until its curvature blows up.

So if a later time slice satisfies the geometrization conjecture, so does every earlier one, and if the flow becomes extinct, $M$ is a connected sum of space forms and $S^2$-bundles over $S^1$; in particular a simply connected $M$ whose flow with surgery becomes extinct is $S^3$ (Corollary 15.4). This is the topological half of the proof of the Poincaré conjecture, and 12C.1 Reading Off the Topology returns to it.

The surgery parameters (Definitions 15.5–15.7). Time is divided into steps $[T_i, T_{i+1}]$ with $T_i = 2^it_0$, $t_0 = 2^{-5}$. On each step there are a canonical neighbourhood parameter $r_{i+1}$ (points with $R \geq r^{-2}$ have canonical neighbourhoods), a noncollapsing parameter $\kappa_{i+1}$, and an upper bound $\delta_{i+1}$ for the surgery control $\delta(t)$, all positive and nonincreasing in $i$. The surgery scale is $h(t) = h(\rho(t), \delta(t))$ with $\rho = \delta r$, the function of the deep-neck theorem of 12B.7 The Singular Time: Horns and Strong Necks, so $h \leq \delta^2r$, and surgeries are performed on strong $\delta(t)$-necks centred at points with $R = h(t)^{-2}$.

The surgery process (§15.4, Definition 15.8). At a singular time $T \in (T_i, T_{i+1}]$: form the limit $(\Omega(T), G(T))$ of 12B.7 The Singular Time: Horns and Strong Necks; set $\rho = \delta(T)r_{i+1}$ and $h = h(\rho, \delta(T))$; discard every component of $\Omega(T)$ not meeting $\Omega_\rho$; in each of the finitely many $2\varepsilon$-horns of the remaining components, take a point $y$ with $R(y) = h^{-2}$, which is the centre of a strong $\delta(T)$-neck; perform the surgery of Theorem 10.1 on that neck, removing its positive half and the end of the horn; and restart the Ricci flow from the resulting compact manifold. Everything else in $M$ (the set where the curvature blew up, the components missing $\Omega_\rho$, the ends of horns) disappears.

The two remaining properties, noncollapsing and canonical neighbourhoods after surgery, are not automatic: they hold only if $\delta$ is chosen small enough depending on $r$ and $\kappa$, and that is the delicate part of Perelman's second preprint. Morgan and Tian's Theorem 15.9 states the conclusion:

The proof is an induction on the steps $[T_i, T_{i+1}]$: 12B.11 Noncollapsing After Surgery proves noncollapsing on the next step assuming canonical neighbourhoods; 12B.12 Canonical Neighbourhoods After Surgery proves canonical neighbourhoods on the next step, for $\delta$ small enough; and 12B.13 Ricci Flow with Surgery for All Time shows that there are only finitely many surgeries on each step, because each removes a definite amount of volume ($h^3$ up to a constant), while the volume grows at most exponentially.

10.6History

The capping construction is Hamilton's, in "Four-manifolds with positive isotropic curvature" (1997, §4), where he showed that a neck can be capped preserving a pinching condition in dimension four. Perelman (II, §4.4, 2003) observed that "his argument works in our case too (and the estimates are much easier to verify)", and described it in a paragraph. Morgan and Tian (2007, chapters 13–15) carried out the estimates in full, introduced surgery space-times as the formal framework, and isolated the seven assumptions; Kleiner and Lott (§§68–73) and Cao and Zhu did the same in their own frameworks.

10.7Exercises

Solution

(a) $f' = \frac{q}{s^2}f$, $f'' = \big(\frac{q^2}{s^4} - \frac{2q}{s^3}\big)f$. (b) $\frac{q^2}{s^4} > \frac{4q}{s^3}$ iff $s < \frac q4$; then $\frac{2q}{s^3} < \frac12\frac{q^2}{s^4}$, so $f'' > \frac12\frac{q^2}{s^4}f$. (c) Each derivative is a polynomial in $\frac1s$ times $e^{-q/s}$, and $s^{-k}e^{-q/s} \to 0$. (d) With $u = \frac qs$, $\frac{q^2}{s^4}e^{-q/s} = q^{-2}u^4e^{-u}$, largest at $u = 4$, that is $s = \frac q4$. Requiring $q \gg (4 + A_0)^2$ puts the whole surgery region $s \leq 4 + A_0$ far to the left of this maximum, where $u \gg 1$, so the dominant-term comparisons hold uniformly; it also makes $\frac{q^2}{s^4}e^{-q/s}$ tiny there, so the conformal change is a small perturbation.

Solution

(a) For $\hat h = e^{-2f}h$ and $h$-orthonormal $X$, $Y$, the sectional curvature is $\hat K(X, Y) = e^{2f}\big(K(X, Y) + \nabla^2f(X, X) + \nabla^2f(Y, Y) + (Xf)^2 + (Yf)^2 - |\nabla f|^2\big)$. For $X = \partial_s$, $Y = e_a$ on the cylinder: $K = 0$, $\nabla^2f(\partial_s, \partial_s) = f''$, $\nabla^2f(e_a, e_a) = 0$, and $(\partial_sf)^2 = |\nabla f|^2$ cancels. So $\hat K = e^{2f}f''$, positive wherever $f'' > 0$. (b) For $X = e_1$, $Y = e_2$: $\hat K = e^{2f}(\frac12 - f'^2)$, a change that is second order in $f$ ($f'$ is of the size of $f$ times $\frac{q}{s^2}$). Shrinking the cross-sections by $e^{-f}$ as $s$ increases makes the neck taper, like the start of a cap, and a tapering surface of revolution has positive mixed curvature where its profile is concave as seen from outside.

Solution

(a) $\Phi_t'(X) = 2(\log X + \log(1 + t) - 2)$, positive iff $X(1 + t) > e^2$; $\Phi_t(X) < 0$ iff $X(1 + t) < e^3$. (b) If $\hat X(1 + t) \geq e^2$: $\hat R \geq R \geq \Phi_t(X) \geq \Phi_t(\hat X)$, using monotonicity on $[\hat X, X]$. If $\hat X(1 + t) < e^2$: $\Phi_t(\hat X) < 0$, and $\hat R \geq R \geq 0$ in a neck (where $R \approx 1$ after rescaling). (c) The condition says "$R$ is large compared with the negative curvature $X$". Increasing $X$ or decreasing $R$ could break it; the conformal bending does neither.

Solution

(a) The disappearing region near the pinch is a tube $S^2\times I$; replacing it by two balls is surgery on a non-separating sphere, which splits off $S^2\times S^1$ and discards it: topologically, $M = M_t\#(S^2\times S^1)$. (b) Cutting $S^2\times S^1$ along one fibre and capping both ends gives $S^3$: $M_t \cong S^3$, and $S^3\#(S^2\times S^1) \cong S^2\times S^1$. (c) The corollary concludes that $M$ is a connected sum of space forms and $S^2$-bundles over $S^1$; for simply connected $M$ no $S^2\times S^1$ summand can occur (it would contribute a free factor $\mathbb{Z}$ to $\pi_1$), so only $S^3$ remains.

Solution

The surgery discards at least the positive half of a $\delta$-neck of scale $h$, close to $S^2(\sqrt2h)\times(0, \delta^{-1}h)$, of volume about $8\pi h^2\cdot\delta^{-1}h = 8\pi\delta^{-1}h^3$, and glues in a cap inside a ball of radius $(A_0 + 5)h$, of volume at most $K'h^3$ for a universal $K'$. For $\delta$ small the net loss is at least $4\pi\delta^{-1}h^3 \geq h^3$. Under Ricci flow $\frac{d}{dt}\operatorname{Vol} = -\int R\,dV \leq \frac{6}{1 + 4t}\operatorname{Vol}$ by the pinching lower bound $R \geq \frac{-6}{1 + 4t}$, so $\operatorname{Vol}(t) \leq \operatorname{Vol}(0)(1 + 4t)^{3/2}$, and surgery only decreases volume. So on $[0, T]$, with $h \geq h_T > 0$, the number of surgeries is at most $\operatorname{Vol}(0)(1 + 4T)^{3/2}/(ch_T^3)$. This is the argument of 12B.13 Ricci Flow with Surgery for All Time.