Collapsing and Noncollapsing

A manifold can have bounded curvature and still be very thin. A flat torus can be made as thin as you like, and so can a Berger sphere (9A.5 Computing Curvature): its curvature stays bounded while the circles of its Hopf fibration shrink to points. Such sequences collapse: their volume goes to zero, their injectivity radius goes to zero, and from far away they look lower-dimensional. Collapse is the enemy of every compactness argument, because a limit of collapsing manifolds is not a manifold of the same dimension.

This chapter makes that precise. The key estimate, due to Cheeger and to Cheeger, Gromov and Taylor, says that collapse is the only obstruction. A curvature bound together with a lower bound on volume gives a lower bound on the injectivity radius. Hamilton's compactness theorem for Ricci flows needs an injectivity radius bound, and Perelman's noncollapsing theorem supplies a volume bound. This estimate is the bridge between them. The chapter ends with Perelman's definition of $\kappa$-noncollapsing, which 12A.4 κ-Noncollapsing turns into a theorem.

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

• state Klingenberg's lemma and explain the two ways the injectivity radius can be small;
• state the Cheeger–Gromov–Taylor estimate in its scale-invariant form, and sketch why it holds;
• describe collapsing sequences: thin flat tori, Berger spheres, nilmanifolds, and the cigar seen from far away;
• define $\kappa$-noncollapsing at a scale, and check it or its failure in examples;
• explain why noncollapsing is exactly what a blow-up argument needs.

3.1From far away, a hose is a line

3.2Two ways to have small injectivity radius

Recall from 9A.3 Geodesics and the Exponential Map that $\operatorname{inj}(p)$ is the distance from $p$ to its cut locus, and that a geodesic stops minimising either at a conjugate point or where it meets another geodesic of the same length.

The first term comes from conjugate points: when $K \leq K_0$, Jacobi fields cannot vanish before time $\frac{\pi}{\sqrt{K_0}}$ (Rauch comparison, the opposite inequality to 9A.7 Jacobi Fields and Curvature versus Topology). The second comes from loops: if two minimising geodesics from $p$ meet at a point $q$ at distance $\operatorname{inj}(p)$ and $q$ is not conjugate, they join to a geodesic loop of length $2\operatorname{inj}(p)$ through $p$, and at a point where $\operatorname{inj}$ is smallest this loop is a closed geodesic (Petersen gives the argument). So with bounded curvature, the injectivity radius is small only if there is a short closed geodesic: a small loop, like the short circles of a thin torus.

3.3Volume controls the injectivity radius

Both hypotheses and the conclusion are scale-invariant: replacing $g$ by $\lambda^2g$ multiplies $r$, $|\operatorname{Rm}|^{-1/2}$ and the injectivity radius by $\lambda$, and leaves $\frac{\operatorname{Vol}B}{r^n}$ unchanged (9A.1 Riemannian Metrics and Model Spaces). So it is enough to prove it for $r = 1$. Jeff Cheeger proved the compact version in his 1970 finiteness theorem; the local form is due to Cheeger, Gromov and Taylor (1982).

The theorem is used in exactly this form. Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows) needs, besides curvature bounds, a lower bound on $\operatorname{inj}(p_k)$ at the base points of a sequence of Ricci flows. In 1995 that bound was the missing ingredient: no general way was known to prevent collapse at the scale of a singularity. Perelman's $\kappa$-noncollapsing theorem gives the volume bound, and Cheeger–Gromov–Taylor converts it to an injectivity radius bound.

3.4Collapsing examples

Thin flat tori. $T_\varepsilon = \mathbb{R}^2/(\mathbb{Z}\times\varepsilon\mathbb{Z})$ has curvature $0$, injectivity radius $\frac\varepsilon2$ (one half of the shortest closed geodesic, 9A.3 Geodesics and the Exponential Map) and area $\varepsilon$. As $\varepsilon \to 0$ it converges to the unit circle (Figure 3.1).

Berger spheres. The Berger spheres $S^3_\varepsilon$ of 9A.5 Computing Curvature, with the Hopf circles of length $2\pi\varepsilon$, have sectional curvatures between $\varepsilon^2$ and $4 - 3\varepsilon^2$, bounded for $\varepsilon \leq 1$. As $\varepsilon \to 0$ their volume $2\pi^2\varepsilon$ goes to zero and they converge to the base of the Hopf fibration, the round sphere $S^2(\frac12)$ of radius $\frac12$ (because the Hopf map is a Riemannian submersion from the unit $S^3$ onto it). A three-dimensional manifold collapses to a two-dimensional one with bounded curvature.

Nilmanifolds. Mikhail Gromov proved in 1978 that a compact manifold admitting metrics with diameter at most $1$ and curvature arbitrarily close to zero must be finitely covered by a nilmanifold, a quotient of a nilpotent Lie group such as the Heisenberg group (9A.5 Computing Curvature); Ernst Ruh refined the statement in 1982. Conversely nilmanifolds carry such "almost flat" metrics: a sequence of them collapses to a point with curvature tending to zero. Collapse with bounded curvature is governed by this kind of algebraic structure, which Cheeger, Fukaya and Gromov analysed in the 1980s.

The cigar from far away. Hamilton's cigar soliton is the surface $ds^2 + \tanh^2s\,d\theta^2$ (11B.1 Ricci Solitons): a rounded cap at $s = 0$ opening into a cylinder of circumference $2\pi$. Its curvature $K = 2\operatorname{sech}^2s$ decays exponentially. Seen at scale $r \gg 1$ it looks like a half-line (Figure 3.2). On the cylindrical end, balls of radius $r$ have area about $4\pi r$, so $\frac{\operatorname{Area}B}{r^2} \to 0$: the cigar is collapsed at large scales. Whether such a collapsed region could appear as a blow-up limit of a Ricci flow was the "cigar problem" in Hamilton's program; Perelman's noncollapsing theorem rules it out (12A.4 κ-Noncollapsing).

3.5$\kappa$-noncollapsing

The definition is scale-invariant (Exercise 3.5). It only constrains balls whose curvature is controlled at their own scale. Those are the balls on which Cheeger–Gromov–Taylor applies, and for them it gives $\operatorname{inj}(x) \geq c(n, \kappa)r$. Round spheres, Euclidean space and the round cylinder $S^2\times\mathbb{R}$ are $\kappa$-noncollapsed at all scales for some $\kappa > 0$ (Exercise 3.6). The thin cylinder $S^1(\varepsilon)\times\mathbb{R}$ and the cigar are not, for any fixed $\kappa$, at scales much larger than their circles.

12A.4 κ-Noncollapsing proves Perelman's theorem: a Ricci flow on a closed manifold over a finite time interval is $\kappa$-noncollapsed at scales below a fixed $\rho$, with $\kappa$ depending only on the initial metric and the time interval. It follows that every blow-up limit is $\kappa$-noncollapsed at all scales, which excludes the cigar, thin cylinders and every other collapsed model.

3.6History

Wilhelm Klingenberg proved his lemma around 1959–61, in work on the sphere theorem. Jeff Cheeger's finiteness theorem (1970) contained the first injectivity radius estimate from volume; Cheeger, Gromov and Taylor's local estimate appeared in 1982. Gromov's almost flat manifolds theorem appeared in 1978. Hamilton introduced the cigar soliton in his study of surfaces (1988), and the cigar problem is posed in his 1995 survey of singularity formation. Perelman's noncollapsing theorem was the main new result of his first 2002 paper.

3.7Exercises

Solution

Curvature $0$; injectivity radius $\frac\varepsilon2$; area $\varepsilon$; diameter $\frac12\sqrt{1 + \varepsilon^2}$ (half the diagonal of the fundamental rectangle). At $r = \frac12$ the curvature hypothesis $|\operatorname{Rm}| \leq 4$ holds, but $\operatorname{Area}B(p, \frac12) \leq \varepsilon$, so no fixed $v$ works as $\varepsilon \to 0$; correspondingly $\operatorname{inj} = \frac\varepsilon2 \to 0$.

Solution

For $\tilde g = \lambda^2g$, a $\tilde g$-ball of radius $\tilde r$ is a $g$-ball of radius $r = \tilde r/\lambda$; $|\widetilde{\operatorname{Rm}}|_{\tilde g} = \lambda^{-2}|\operatorname{Rm}|_g$ (the norm of the $(1, 3)$ curvature tensor, 9A.4 Curvature and What It Means), so $|\widetilde{\operatorname{Rm}}| \leq \tilde r^{-2}$ iff $|\operatorname{Rm}| \leq r^{-2}$; and $\widetilde{\operatorname{Vol}} = \lambda^n\operatorname{Vol}$, so $\widetilde{\operatorname{Vol}}B \geq \kappa\tilde r^n$ iff $\operatorname{Vol}B \geq \kappa r^n$. The condition $\tilde r < \lambda\rho$ is $r < \rho$.

Solution

(a) $\operatorname{Vol}B(x, r) = \omega_nr^n$. (b) For $r \leq 1$, $\operatorname{Area}B(x, r) = 2\pi(1 - \cos r) \geq \frac{\pi}{2}r^2$; the inequality holds because $1 - \cos r \geq \frac{r^2}{2} - \frac{r^4}{24} \geq \frac{r^2}{2} - \frac{r^2}{24} \geq \frac{r^2}{4}$. (c) Again only $r \leq 1$ qualifies. Since $d((x, t), (y, u))^2 = d(x, y)^2 + (t - u)^2$, the ball $B((x, t), r)$ contains $B_{S^2}(x, \frac{r}{\sqrt2})\times(t - \frac{r}{\sqrt2}, t + \frac{r}{\sqrt2})$, of volume at least $\frac{\pi}{2}\cdot\frac{r^2}{2}\cdot\sqrt2r = \frac{\pi}{2\sqrt2}r^3$.

Solution

Curvature is $0$, so every ball qualifies. $\operatorname{Area}B(x, r) \leq 2\pi\varepsilon\cdot2r$. If $\operatorname{Area}B \geq \kappa r^2$ for all $r < \rho$, then $4\pi\varepsilon r \geq \kappa r^2$, so $r \leq \frac{4\pi\varepsilon}{\kappa}$ for all $r < \rho$, which fails once $\rho > \frac{4\pi\varepsilon}{\kappa}$.

Solution

The volume form scales by the factor by which one direction of an orthonormal frame is stretched: $\varepsilon$. The unit $S^3$ has volume $2\pi^2$. The Hopf map $S^3_\varepsilon \to S^2(\frac12)$ still does not increase lengths of horizontal vectors and only shrinks the vertical ones, so it is distance-nonincreasing, and surjective; hence $\operatorname{diam}S^3_\varepsilon \geq \operatorname{diam}S^2(\frac12) = \frac{\pi}{2}$.

Solution

On $B(x, r)$ the $\Sigma$-coordinate satisfies $s \geq d - r = \frac d2$, so $|\operatorname{Rm}| = K \leq 8e^{-d} \leq \frac{4}{d^2} = r^{-2}$ once $d$ is large. The ball lies in $\{|s - d| < r\}\times(-r, r)$, whose volume is at most $(2\pi\cdot2r)\cdot2r = 8\pi r^2$, since the circles have length $2\pi\tanh s < 2\pi$. Then $\frac{\operatorname{Vol}B(x, r)}{r^3} \leq \frac{8\pi}{r} \to 0$ as $d \to \infty$, so no $\kappa > 0$ works at all scales.