κ-Noncollapsing

In 2002 the first gap in Hamilton's program (11B.5 Hamilton’s Program in 2002) was noncollapsing: nothing prevented a region of high curvature from being very thin, like a long tube of tiny circumference. That had two consequences. Without a lower bound on volume, Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows) could not produce blow-up limits. And the cigar times a line, a steady soliton that is collapsed at large scales, could not be ruled out as a singularity model (11B.4 Singularities). Section 4 of Perelman's first preprint settles both questions in one page. A thin ball supports a test function with very negative $\mathcal W$-entropy, while monotonicity (12A.3 The 𝓦-Entropy) bounds the entropy below by its value at time zero. This chapter proves the theorem in full and derives its consequences.

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

• state Perelman's no local collapsing theorem with its dependence on the initial metric, the time and the scale;
• construct the test function on a ball with bounded curvature and prove that $\mu(g, r^2) \leq \log\frac{\operatorname{Vol}B(p, r)}{r^n} + C(n)$;
• combine this bound with monotonicity to prove the theorem;
• explain why blow-up limits exist and are $\kappa$-noncollapsed at all scales, and why this excludes the cigar.

4.1A tube cannot hide

4.2The definition

Recall from 9B.3 Collapsing and Noncollapsing: $(M^n, g)$ is $\kappa$-noncollapsed at scales below $\rho$ if every ball $B(p, r)$ with $r < \rho$ on which $|\operatorname{Rm}| \leq r^{-2}$ has $\operatorname{Vol}B(p, r) \geq \kappa r^n$. A Ricci flow is $\kappa$-noncollapsed at scales below $\rho$ if every time slice is. This is Perelman's Definition 4.2. He notes two properties that are immediate: a limit of metrics that are $\kappa$-noncollapsed at scales below $\rho$ has the same property, and $\lambda^2g$ is $\kappa$-noncollapsed at scales below $\lambda\rho$ whenever $g$ is $\kappa$-noncollapsed at scales below $\rho$. On a ball where $|\operatorname{Rm}| \leq r^{-2}$, noncollapsing gives an injectivity radius bound $\operatorname{inj}(p) \geq c(n, \kappa)r$ (Cheeger–Gromov–Taylor, 9B.3 Collapsing and Noncollapsing).

4.3The test function

It helps to write $\mathcal W$ in terms of $\psi = \sqrt u$, where $u = (4\pi\tau)^{-n/2}e^{-f}$ and $\int\psi^2\,dV = 1$. Since $\nabla f = -2\psi^{-1}\nabla\psi$ and $f = -\log\psi^2 - \frac n2\log(4\pi\tau)$,

(Exercise 4.4). This form makes sense for any nonnegative Lipschitz $\psi$ with $\int\psi^2 = 1$, including ones that vanish on open sets, and the infimum $\mu(g, \tau)$ is unchanged if we allow them (approximate $\psi$ by smooth positive functions). Perelman's own test function is positive everywhere but "very close to $0$" outside the ball, which amounts to the same thing.

The bound is sharp in its dependence on the volume: the entropy goes to $-\infty$ like the logarithm of the volume ratio, and no faster. Figure 4.1 shows the test function on a thin tube, and Figure 4.2 computes $\mathcal W$ for it on a flat product.

4.4The theorem

Perelman's proof is the same argument run as a contradiction: a sequence of collapsing balls would force $\mu(g(0), t_k + r_k^2) \to -\infty$ with $t_k + r_k^2$ bounded. He states the conclusion at scale $\rho = \sqrt T$, which is the natural choice. The proof has the shape of the Figure 4.3 chain, and uses no compactness at all; compactness enters in the corollary.

4.5Consequences

Blow-up limits exist. Perelman draws the corollary at once.

Hamilton's point picking (11B.4 Singularities) is designed to produce sequences with the curvature bound required in the corollary, so every blow-up limit of the kind studied in Book 11B exists and is noncollapsed.

The cigar is excluded. The cigar $\Sigma$ times a line is not $\kappa$-noncollapsed at all scales for any $\kappa > 0$: far from the tip, a ball of radius $r$ has $|\operatorname{Rm}| \leq r^{-2}$ and volume at most $8\pi r^2$, which is much less than $\kappa r^3$ when $r$ is large (9B.3 Collapsing and Noncollapsing). So $\Sigma\times\mathbb{R}$ is never a blow-up limit of a Ricci flow on a closed 3-manifold, and neither is any other solution that is collapsed at large scales. This was Hamilton's cigar problem.

Ancient solutions to study. Blow-up limits in three dimensions are ancient, $\kappa$-noncollapsed at all scales, and (by Hamilton–Ivey, 11A.5 Hamilton–Ivey Pinching) have nonnegative curvature operator. Book 12B calls such solutions $\kappa$-solutions and classifies them (12B.1 κ-Solutions, 12B.2 The Structure of κ-Solutions).

4.6History

Hamilton's compactness theorem (1995) needed an injectivity radius bound, which Hamilton could obtain in some situations but not in general. He set out the cigar problem in his 1995 survey. Perelman's §4, in his first preprint (November 2002), proved the theorem in a few lines once the monotonicity of $\mathcal W$ was available. The localised version in §8 followed in the same preprint. The details of the test function were written out in the expositions by Kleiner and Lott, Morgan and Tian, and Cao and Zhu.

4.7Exercises

Solution

$f = -\log\psi^2 - \frac n2\log(4\pi\tau)$, so $\nabla f = -2\frac{\nabla\psi}{\psi}$ and $|\nabla f|^2\psi^2 = 4|\nabla\psi|^2$. Then $\mathcal W = \int[\tau(|\nabla f|^2 + R) + f - n]\psi^2 = \int[4\tau|\nabla\psi|^2 + \tau R\psi^2 - \psi^2\log\psi^2]\,dV - \frac n2\log(4\pi\tau) - n$, using $\int\psi^2 = 1$.

Solution

$V_{-1}(1) = \pi(\sinh2 - 2) \approx \pi\times1.627$ and $V_{-1}(\frac12) = \pi(\sinh1 - 1) \approx \pi\times0.175$, so $D(3) \approx 9.29$. It is slightly larger than $8$, because negative curvature makes larger balls relatively larger.

Solution

A point at distance at most $1$ from $p$ has its $T^{n-1}$ coordinate within $1$ of $p$'s, and the side $4$ means that this set is a Euclidean ball of radius $1$ in $T^{n-1}$; every circle coordinate is within $\frac L2$ of $p$'s. So the ball lies in a set of volume $L\omega_{n-1}$. Then $\mu(g, 1) \leq \log(L\omega_{n-1}) + C(n) \to -\infty$. There is no contradiction, because the theorem's $\kappa$ depends on the initial metric. For each fixed $L$ the static flow is $\kappa$-noncollapsed at scales below $\rho$ with $\kappa$ depending on $L$; what fails is a bound uniform in $L$.

Solution

Let $B(x, r)$ be a ball in the limit with $|\operatorname{Rm}| \leq r^{-2}$, and let $r' < r$. On $B(x, r')$, $|\operatorname{Rm}| \leq r^{-2} < r'^{-2}$, so for large $k$ the corresponding balls $B(x_k, r')$ in $g_k$ have $|\operatorname{Rm}| \leq r'^{-2}$, and $r' < \rho_k$. Hence $\operatorname{Vol}B(x_k, r') \geq \kappa r'^n$, and passing to the limit, $\operatorname{Vol}B(x, r') \geq \kappa r'^n$. Letting $r' \to r$ gives $\operatorname{Vol}B(x, r) \geq \kappa r^n$.

Solution

(a) For $r \geq d$ the ball is the whole torus. (b) Every ball has $|\operatorname{Rm}| = 0 \leq r^{-2}$, so every ball counts. If $r \leq i$, the ball is Euclidean and $\omega_nr^n \geq \kappa r^n$. If $i < r < \rho$, then $\operatorname{Vol}B(p, r) \geq \operatorname{Vol}B(p, i) = \omega_ni^n \geq \omega_n\big(\frac i\rho\big)^nr^n$. (c) The theorem allows $\kappa$ to depend on $\rho$, and here $\kappa \to 0$ as $\rho \to \infty$. Blow-ups rescale by $Q_k \to \infty$, which turns the scale $\rho$ into $\sqrt{Q_k}\rho \to \infty$ while keeping $\kappa$ fixed, which is why limits are noncollapsed at all scales. A flat torus is never a blow-up limit, since blow-up limits have a point where $|\operatorname{Rm}| = 1$.