Singularities

On a closed manifold, a Ricci flow that stops at a finite time $T$ does so because curvature blows up (11A.3 Short-Time Existence and Uniqueness). This chapter studies how. Singularities are classified by the rate of blow-up (Type I or Type II), and studied by blowing up: rescaling the flow at points of large curvature and taking a limit with the compactness theorem of 11B.3 Compactness of Ricci Flows. The limits are ancient solutions (11B.2 Ancient Solutions and the Harnack Inequality), in three dimensions with nonnegative curvature (11A.5 Hamilton–Ivey Pinching). The fundamental example is the neckpinch: a 3-sphere shaped like a dumbbell develops a thin neck that pinches off in finite time, while the rest of the manifold stays smooth. Near the pinch, the flow looks like a shrinking round cylinder. Since the manifold does not become singular everywhere at once, a flow that is to continue needs a way past the pinch: surgery.

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

• define Type I and Type II singularities and describe the blow-up procedure with point picking;
• describe the round shrinking sphere, the neckpinch and the degenerate neckpinch, and their blow-up limits;
• derive the PDE of rotationally symmetric Ricci flow, and bound the neck radius by the cylinder's;
• explain the cigar problem and how Perelman resolved it;
• explain why the three-dimensional flow needs surgery.

4.1Pinching in other flows

4.2Types of singularities and the blow-up procedure

Let $g(t)$ be a Ricci flow on a closed manifold with maximal time $T < \infty$. Write $K(t) = \max_M|\operatorname{Rm}(\cdot, t)|$, which satisfies $K(t) \geq \frac{c}{T - t}$ (11A.3 Short-Time Existence and Uniqueness). The singularity is

• Type I if $K(t)(T - t)$ stays bounded: curvature blows up at the rate of the shrinking sphere;
• Type II if $\sup_tK(t)(T - t) = \infty$: curvature blows up faster somewhere.

Blowing up. Choose points $(x_k, t_k)$ with $Q_k = |\operatorname{Rm}(x_k, t_k)| \to \infty$ and rescale:

For a limit to exist, the rescaled flows need curvature bounds on large parabolic neighbourhoods of $(x_k, 0)$. Point picking arranges this: for Type I, choose $Q_k$ comparable to $K(t_k)$; for Type II, Hamilton chose $(x_k, t_k)$ to nearly maximise $|\operatorname{Rm}|(x, t)(T_k - t)$ over $M\times[0, T_k]$ for $T_k \to T$, which forces $|\operatorname{Rm}| \leq (1 + \varepsilon)Q_k$ on growing neighbourhoods and makes the limit eternal. Given an injectivity radius bound at $x_k$, the compactness theorem (11B.3 Compactness of Ricci Flows) gives a limit, an ancient (Type I) or eternal (Type II) solution. In dimension three, by Hamilton–Ivey (11A.5 Hamilton–Ivey Pinching), its curvature is nonnegative.

4.3Examples

The round sphere. $S^3$ or any $S^3/\Gamma$ with constant curvature shrinks to a point; every blow-up limit is the shrinking round sphere (or quotient). This is Type I, and it is what Hamilton's theorem (11A.6 Hamilton’s 1982 Theorem) says happens whenever $\operatorname{Ric} > 0$ initially.

The neckpinch. Consider rotationally symmetric metrics $g = \varphi(x, t)^2dx^2 + \psi(x, t)^2g_{S^n}$ on $(-1, 1)\times S^n$, closed off at the ends to make $S^{n+1}$. With $\partial_s = \frac1\varphi\partial_x$ (arc length) and the warped-product curvatures of 9A.5 Computing Curvature, the Ricci flow becomes (Exercise 4.1)

Sigurd Angenent and Dan Knopf (2004) proved that if the initial metric has a sufficiently thin neck between two large bulbs (and satisfies some technical conditions), then the neck pinches at a finite time $T$, the bulbs stay large, and near the neck

the neck shrinks exactly like the round cylinder $S^n\times\mathbb{R}$, a Type I singularity whose blow-up limit is the shrinking cylinder. Miles Simon (2000) had shown earlier that such necks pinch. Figure 4.1 shows a computed example, and Figure 4.2 its blow-up.

The degenerate neckpinch. If the neck is placed asymmetrically so that one bulb is barely larger than the neck, the bulb and the neck can pinch simultaneously. The singularity is then at the tip of a shrinking "cap" and is Type II. Hui-Ling Gu and Xi-Ping Zhu (2008) proved that Type II singularities form for suitable rotationally symmetric metrics on $S^{n+1}$, and Angenent, Isenberg and Knopf (2015) analysed the degenerate neckpinch in detail: its blow-up at the tip, with Hamilton's point picking, is the Bryant soliton (11B.1 Ricci Solitons).

The cigar problem. In dimension three, the cigar times a line (11B.1 Ricci Solitons) is a steady soliton with nonnegative curvature: an eternal solution of exactly the kind Type II blow-ups produce. Hamilton could not rule out that it appears as a blow-up limit, and if it did, his program would stall: near such a singularity the manifold would look like a thin tube with a cap, not like a neck that can be cut. Perelman resolved the problem: the cigar times a line is not $\kappa$-noncollapsed at large scales (9B.3 Collapsing and Noncollapsing), while every blow-up limit is, by his noncollapsing theorem (12A.4 κ-Noncollapsing). So the cigar never appears.

4.4Why surgery

The neckpinch shows that a three-dimensional Ricci flow can develop a singularity on a small set (the neck's central sphere) while most of the manifold is smooth. The flow cannot be continued as a smooth flow past $T$: the metric degenerates on the neck. Hamilton proposed to stop the flow just before the neck pinches, cut along the central 2-sphere of the neck, glue in two caps (round hemispheres, suitably bent), and restart the flow. Topologically, cutting a neck undoes a connected sum or removes an $S^2\times S^1$ summand (10A.3 The Prime Decomposition). Making this work requires knowing that every high-curvature region looks like a neck or a cap (so that there is always somewhere to cut), and that surgeries do not accumulate. Hamilton did this for four-manifolds with positive isotropic curvature (1997). Perelman did it for every three-manifold (12B.3 The Canonical Neighbourhood Theorem–12B.5 Ricci Flow with Surgery for All Time).

4.5History

Hamilton classified singularity types and introduced point picking in his 1995 survey, where he also posed the cigar problem. Grayson's dumbbell neckpinch for mean curvature flow appeared in 1989. Miles Simon (2000) proved neckpinching for a class of rotationally symmetric metrics, Angenent and Knopf (2004) gave the precise asymptotics, Gu and Zhu (2008) proved the existence of Type II singularities on spheres, and Angenent, Isenberg and Knopf (2015) analysed degenerate neckpinches.

4.6Exercises

Solution

The $dx^2$ component: $\partial_t(\varphi^2) = -2\operatorname{Ric}(\partial_x, \partial_x) = -2\varphi^2\operatorname{Ric}(\partial_s, \partial_s) = 2n\varphi^2\frac{\psi_{ss}}{\psi}$, so $\varphi_t = n\varphi\frac{\psi_{ss}}{\psi}$. The sphere component: $\partial_t(\psi^2) = -2\psi^2\big(-\frac{\psi_{ss}}{\psi} + (n - 1)\frac{1 - \psi_s^2}{\psi^2}\big)$, so $\psi_t = \psi_{ss} - (n - 1)\frac{1 - \psi_s^2}{\psi}$, at fixed $x$.

Solution

At the minimum, $\psi_t = \psi_{ss} - \frac{n - 1}{\psi} \geq -\frac{n - 1}{\psi}$, so $(\psi^2)_t \geq -2(n - 1)$ there; Hamilton's trick gives the same for $\psi_{\min}^2$. Integrating from $t$ to $T$, where $\psi_{\min} \to 0$: $0 - \psi_{\min}(t)^2 \geq -2(n - 1)(T - t)$. The sectional curvature of the sphere factor at the neck, $\frac{1 - \psi_s^2}{\psi^2} = \frac{1}{\psi_{\min}^2}$, is then at least $\frac{1}{2(n - 1)(T - t)}$.

Solution

$|\operatorname{Rm}_{g_k}|(s) = \frac{1}{Q_k}|\operatorname{Rm}|\big(t_k + \frac{s}{Q_k}\big) \leq \frac{1}{Q_k}\cdot\frac{C}{T - t_k - \frac{s}{Q_k}} = \frac{C}{Q_k(T - t_k) - s} = \frac{C}{1 - s}$.

Solution

When a round sphere becomes singular, the whole manifold shrinks to a point and becomes round: its topology is identified ($S^3/\Gamma$). At a neckpinch, only the neck becomes singular; the two bulbs are large and their geometry is arbitrary, so nothing is learned about their topology, and the flow must continue to reveal it. Surgery cuts the neck's central 2-sphere and caps both sides. On a dumbbell $S^3$ the sphere separates, and the result is two 3-spheres (undoing $S^3\#S^3 = S^3$). On $S^2\times S^1$ with a neck around the circle, the sphere is non-separating and surgery leaves one $S^3$: the $S^2\times S^1$ summand has been removed.

Solution

If $g$ is $\kappa$-noncollapsed at scales below $\rho$, then $Q_kg$ is $\kappa$-noncollapsed at scales below $\sqrt{Q_k}\rho$ (9B.3 Collapsing and Noncollapsing). As $Q_k \to \infty$ these scales grow without bound, and noncollapsing at each fixed scale passes to smooth limits. So the limit is $\kappa$-noncollapsed at every scale, which $\text{cigar}\times\mathbb{R}$ is not.