Compactness of Ricci Flows

To understand a singularity, you blow it up: rescale the flow around points of larger and larger curvature, and look for a limit. That requires a compactness theorem for sequences of flows: under what conditions does a sequence of Ricci flows have a subsequence converging to a limit flow? Hamilton proved the answer in 1995. Uniform bounds on curvature, plus a lower bound on the injectivity radius at one point at one time, are enough. The curvature bounds come from the blow-up normalisation. The injectivity radius bound did not come from anywhere in 1995: nothing prevented the rescaled flows from collapsing. That gap is the subject of 11B.5 Hamilton’s Program in 2002, and Perelman filled it (12A.4 κ-Noncollapsing).

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

• state Hamilton's compactness theorem for Ricci flows precisely;
• explain its proof architecture: Shi's estimates, Cheeger–Gromov compactness at one time, and Arzelà–Ascoli in space-time;
• give examples of convergence and of failure by collapse;
• explain why the injectivity radius hypothesis is the crux, and how a volume bound would supply it.

3.1Zooming in

3.2The theorem

A pointed Ricci flow $(M, g(t), p)$, $t \in (a, b]$ with $a < 0 \leq b$, is a complete Ricci flow with a base point. A sequence $(M_k, g_k(t), p_k)$ converges to $(M_\infty, g_\infty(t), p_\infty)$ if there are an exhaustion of $M_\infty$ by open sets $U_k \ni p_\infty$ and embeddings $\phi_k : U_k \to M_k$ with $\phi_k(p_\infty) = p_k$ such that $\phi_k^*g_k(t) \to g_\infty(t)$ smoothly on compact subsets of $M_\infty\times(a, b]$: this is Cheeger–Gromov convergence (9B.4 Convergence of Manifolds) with time added.

Local versions exist, with the curvature bound required only on parabolic neighbourhoods of the base points, of radius tending to infinity. They are what blow-up arguments use.

3.3The architecture of the proof

1. Derivative bounds. By Shi's estimates (11A.3 Short-Time Existence and Uniqueness), the bound $|\operatorname{Rm}| \leq C$ on $(a, b]$ gives $|\nabla^j\operatorname{Rm}| \leq C_j$ on $(a + \varepsilon, b]$, for every $j$, uniformly in $k$.
2. A limit at one time. At $t = 0$, the metrics $g_k(0)$ have all derivatives of curvature bounded and injectivity radius $\geq \iota$ at $p_k$. By the Cheeger–Gromov compactness theorem (9B.4 Convergence of Manifolds), a subsequence converges: there are $\phi_k$ with $\phi_k^*g_k(0) \to g_\infty(0)$ smoothly on compact sets.
3. Time derivatives. Since $\partial_t(\phi_k^*g_k) = -2\operatorname{Ric}(\phi_k^*g_k)$, and the right-hand side and all its spatial derivatives are bounded, all mixed space-time derivatives of $\phi_k^*g_k(t)$ are bounded, in coordinates fixed by $g_\infty(0)$. The metrics $g_k(t)$ are uniformly equivalent to $g_k(0)$ on the time interval, because $|\partial_tg| \leq 2|\operatorname{Ric}| \leq C$.
4. A limit flow. Arzelà–Ascoli (2B.5 Uniform Convergence and Arzelà–Ascoli) on compact subsets of $M_\infty\times(a, b]$, and a diagonal argument, give a subsequence with $\phi_k^*g_k(t) \to g_\infty(t)$ smoothly. The limit satisfies the Ricci flow equation, being a smooth limit of solutions, and it is complete because $g_\infty(0)$ is complete and the metrics at different times are uniformly equivalent.

3.4Why the injectivity radius bound is the crux

Every hypothesis but one is cheap in a blow-up argument. When you rescale a flow at points of large curvature $Q_k$, choosing the points so that $Q_k$ is roughly the maximum of the curvature nearby in space and recent time (point picking, 11B.4 Singularities), the rescaled flows have $|\operatorname{Rm}| \leq C$ on large parabolic neighbourhoods, by construction. The injectivity radius at the base point is another matter. The rescaled flows could be collapsing: their injectivity radius at $p_k$ could tend to zero with bounded curvature, as for a cigar times a line (9B.3 Collapsing and Noncollapsing) or a shrinking circle factor. Then no smooth limit of the same dimension exists.

By the Cheeger–Gromov–Taylor estimate (9B.3 Collapsing and Noncollapsing), the injectivity radius bound follows from a curvature bound and a volume bound: if $|\operatorname{Rm}| \leq r^{-2}$ on $B(p, r)$ and $\operatorname{Vol}B(p, r) \geq \kappa r^n$, then $\operatorname{inj}(p) \geq c(n, \kappa)r$. So what the blow-up argument needs is a lower bound on volume ratios, uniform at all small scales, that survives rescaling: exactly Perelman's $\kappa$-noncollapsing (12A.4 κ-Noncollapsing).

3.5History

Hamilton's compactness theorem appeared in the American Journal of Mathematics in 1995. It built on the Riemannian compactness theorems of Cheeger, Gromov, Greene–Wu and Peters (9B.4 Convergence of Manifolds) and on Shi's 1989 estimates. Peter Topping and others later gave versions with local hypotheses, and Perelman used it, in localised form, throughout his papers.

3.6Exercises

Solution

$g_k(s) = Q_k\big(1 - 4t_k - \frac{4s}{Q_k}\big)g_{S^3} = \big(Q_k(1 - 4t_k) - 4s\big)g_{S^3} = (1 - 4s)g_{S^3}$, defined for $s < \frac14$ and $s \geq -Q_kt_k$.

Solution

Products of Ricci flows are Ricci flows, and the flat circle factor does not move; the curvature is that of $S^2(\sqrt{1 - 2t})$, independent of $k$. The circle has length $\frac{2\pi}{k}$, so the injectivity radius is at most $\frac{\pi}{k} \to 0$. A smooth three-dimensional limit would have positive volume of unit balls, while $\operatorname{Vol}(g_k(0)) = \frac{8\pi^2}{k} \to 0$. In the Gromov–Hausdorff sense, the sequence converges to the shrinking $S^2$, a two-dimensional limit.

Solution

$\partial_tg = -2\operatorname{Ric}$, and spatial derivatives of $\operatorname{Ric}$ are bounded by the $C_j$; differences between $\nabla^j$ and background derivatives involve $\Gamma - \bar\Gamma$, which is controlled by integrating $\partial_t\Gamma = g^{-1}*\nabla\operatorname{Ric}$ in time. And $\partial_t\operatorname{Ric} = \Delta_L\operatorname{Ric}$ is a combination of $\nabla^2\operatorname{Rm}$ and $\operatorname{Rm}*\operatorname{Rm}$, bounded. Inductively all mixed derivatives are bounded.

Solution

For a fixed vector $v$, $\big|\frac{d}{dt}g(t)(v, v)\big| = 2|\operatorname{Ric}(v, v)| \leq 2Cg(t)(v, v)$, so $\big|\frac{d}{dt}\log g(t)(v, v)\big| \leq 2C$, and integrate. Distances at different times are then comparable up to factors $e^{\pm C|t|}$, so Cauchy sequences for one metric are Cauchy for the other; completeness transfers.

Solution

Cheeger–Gromov–Taylor (9B.3 Collapsing and Noncollapsing) gives $\operatorname{inj}(p_k) \geq c(\kappa) > 0$; Shi's local estimates give derivative bounds on a smaller neighbourhood; the local version of Hamilton's theorem gives a convergent subsequence there. Perelman's condition is scale-invariant (9B.3 Collapsing and Noncollapsing): if a rescaled flow $Qg$ has $|\operatorname{Rm}| \leq 1$ on a unit ball, then $g$ has $|\operatorname{Rm}| \leq Q$ on a ball of radius $Q^{-1/2}$, and noncollapsing for $g$ at that scale gives volume $\geq \kappa Q^{-3/2}$, which is $\geq \kappa$ after rescaling.