Pseudolocality

Book 12A ends with two results from §§9–10 of Perelman's first preprint, and the second is the deepest. The first is a differential Harnack inequality for the conjugate heat kernel on a Ricci flow background. It is a pointwise version of the monotonicity of $\mathcal W$, and works locally, where integrals over the whole manifold are unavailable. The second is pseudolocality: a region that is almost Euclidean at some scale stays tame for a definite time at a smaller scale, whatever happens elsewhere. Perelman's own gloss is that regions where the curvature is huge cannot, at once, have much effect on regions that are nearly Euclidean. The flow is a heat equation, and heat equations propagate information at infinite speed; pseudolocality says that for the Ricci flow, in a precise sense, they do not propagate damage that fast.

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

• state Perelman's Proposition 9.1 and the inequality $v \leq 0$ for the conjugate heat kernel, and explain how it is proved;
• derive Perelman's Harnack inequality along curves and the comparison $f \leq \ell$;
• state the pseudolocality theorem with every quantifier in place, and its corollaries;
• outline the proof of pseudolocality and say where each earlier result is used;
• say where pseudolocality is used in the rest of the proof.

6.1Far-away events

6.2Perelman's Harnack inequality

Let $g(t)$, $0 \leq t \leq T$, be a Ricci flow, set $\tau = T - t$, and let $u = (4\pi\tau)^{-n/2}e^{-f}$ solve the conjugate heat equation $\square^*u = -\partial_tu - \Delta u + Ru = 0$ (9B.7 The Heat Equation on a Manifold).

Perelman's proof is "routine computation", and it is a long one; the expositions write it out. Integrated over a closed manifold it gives the monotonicity of $\mathcal W$ (12A.3 The 𝓦-Entropy), since $\int v\,dV = \mathcal W$. Its advantage is that it holds pointwise, so it can be combined with cut-off functions.

The inequality $v \leq 0$ is the Ricci flow analogue of the Li–Yau inequality (6A.10 Entropy, Information and Diffusion). Both are one-sided bounds on a second-order expression in $\log u$ for a fundamental solution, with equality for the Gaussian. Li–Yau integrated theirs along paths to compare values of $u$ at different points and times, and Perelman does the same.

The first part is an algebraic consequence of $v \leq 0$ (Exercise 6.7). In backward time it says $\frac{d}{d\tau}\big(2\sqrt\tau f\big) \leq \sqrt\tau(R + |\dot\gamma|^2)$, so integrating along a curve gives $2\sqrt{\bar\tau}f \leq \mathcal L(\gamma)$, which is why the second part holds. Perelman proves it directly: $(4\pi\tau)^{-n/2}e^{-\ell}$ is a subsolution of the conjugate heat equation (12A.5 Reduced Distance and Reduced Volume), and the conjugate heat kernel lies above it. So $e^{-\ell}$ is a lower barrier for the heat kernel, the precise form of the least-action picture of 12A.5 Reduced Distance and Reduced Volume. Perelman adds a striking remark (9.6): among all evolutions $\partial_tg = A(t)$ of a metric, the Ricci flow is characterised by how the fundamental solutions of the conjugate heat equation behave near their starting point.

6.3Pseudolocality

Read the quantifiers carefully. The constants $\delta$ and $\varepsilon$ depend only on $\alpha$ (and the dimension), not on the flow, the manifold or $r_0$. The hypotheses concern only the ball $B(x_0, r_0)$ at time $0$: no curvature bound is assumed there at all, only a lower bound on $R$ and almost-Euclidean isoperimetry. The conclusion concerns a smaller space-time region, of size $\varepsilon r_0$ in space and $(\varepsilon r_0)^2$ in time (Figure 6.1). The statement is scale-invariant, so it is enough to prove it for $r_0 = 1$ (Exercise 6.8). Perelman states it for a smooth solution. In §7 he assumes that the manifold is closed or that the metrics are complete with bounded curvature, and some such standing assumption is needed for the maximum-principle arguments; the expositions make it explicit.

Perelman draws two further statements. From the proof: under the same hypotheses, $\operatorname{Vol}B(x, \sqrt t) \geq c(n)\sqrt t^n$ at time $t$ for $x \in B(x_0, \varepsilon r_0)$ (Corollary 10.2), a noncollapsing statement at all small scales. And a variant (Theorem 10.3): if at $t = 0$, $|\operatorname{Rm}| \leq r_0^{-2}$ on $B(x_0, r_0)$ and $\operatorname{Vol}B(x_0, r_0) \geq (1 - \delta)\omega_nr_0^n$, then $|\operatorname{Rm}| \leq (\varepsilon r_0)^{-2}$ for $0 \leq t \leq (\varepsilon r_0)^2$ and $\operatorname{dist}_t(x, x_0) < \varepsilon r_0$. Perelman leaves its proof, "a slight modification", to the reader, and asks whether the volume assumption can be dropped.

6.3.1The architecture of the proof

The proof is by contradiction and compactness, the [CC] template in its most elaborate form so far. Figure 6.2 sets out the steps; here is what each does.

1. Suppose it fails. Scale to $r_0 = 1$ and take sequences $\varepsilon, \delta \to 0$ of flows violating the conclusion.
2. Point picking (Claims 1–2). Starting from a bad point, move to points of larger curvature until reaching a point $(\bar x, \bar t)$ with $Q = |\operatorname{Rm}|(\bar x, \bar t) \geq \alpha\bar t^{-1}$ such that $|\operatorname{Rm}| \leq 4Q$ on a backward parabolic neighbourhood of radius $\frac{1}{10}AQ^{-1/2}$ in space and duration $\frac12\alpha Q^{-1}$ in time, where $A$ is a large constant. The search must end, because curvature quadruples at each step and the flow is smooth; Lemma 8.3(b), on how distances change, keeps the neighbourhood inside the region.
3. The conjugate heat kernel. Let $u$ be the conjugate heat kernel starting from a $\delta$-function at $(\bar x, \bar t)$, and $v \leq 0$ its function from Corollary 6.2.
4. $v$ is definitely negative near $(\bar x, \bar t)$ (Claim 3). At some time $\tilde t$ slightly before $\bar t$, $\int_Bv \leq -\beta < 0$ on a small ball $B$ about $\bar x$. Otherwise, rescaling by $Q$ and passing to a limit, (9.1) would make the limit a gradient shrinking soliton, which Perelman notes is incompatible with $|\operatorname{Rm}|(\bar x, \bar t) = 1$. If instead the rescaled metrics collapse, one rescales differently, towards a flat limit, and can arrange $\int_Bv \to -\infty$.
5. Carry this back to $t = 0$. With a cut-off function $h$ built from the distance to $x_0$ (Lemma 8.3(a) controls $\square h$), the formula for $\square^*v$ gives $\int -hv \geq \beta(1 - A^{-2})$ at $t = 0$, while $\int hu \geq 1 - A^{-2}$.
6. Contradict log-Sobolev. At $t = 0$, $\tilde u = hu$ is nearly a probability density, supported in the almost-Euclidean ball. Rescale by $\frac12\bar t^{-1}$, so that $\tau$ becomes $\frac12$. In the limit $\varepsilon, \delta \to 0$ the balls become large, the scalar curvature term disappears, and $\mathcal W$ of $\tilde u$ at $\tau = \frac12$ stays below a negative constant. The isoperimetric constants tend to the Euclidean one, so spherical symmetrisation moves these functions to Euclidean space without spoiling the estimate. That contradicts Gross's inequality (12A.3 The 𝓦-Entropy).

Every tool of Book 12A appears: the entropy and its link to log-Sobolev (step 6), the pointwise Harnack formula (steps 3–5), compactness and the soliton equality case (step 4), and noncollapsing in the alternative of step 4.

6.3.2Where it is used

Pseudolocality controls a region from information at an earlier time, locally, which is exactly what surgery needs. In Perelman's second preprint, Theorem 10.1 is applied to show that the standard solution, the model flow glued in at surgery, exists until its natural extinction time (II, §2), and the point-picking argument of its proof reappears in the analysis of the long-time behaviour (II, §6).

6.4History

Li and Yau proved their differential Harnack inequality in 1986, and Hamilton used Harnack inequalities for the backward heat equation to prove monotonicity formulas for parabolic flows; Klaus Ecker obtained a local monotonicity formula for mean curvature flow with solutions of the backward heat equation. Perelman cites all three in §9.7*. Pseudolocality is §10 of the first preprint (2002). Perelman interpreted it in terms of his renormalisation-group picture: a region that looks trivial at a higher energy scale cannot suddenly become highly nontrivial at a slightly lower one.

6.5Exercises

Solution

With $R = 0$, $\square^*u = -\partial_tu - \Delta u = \partial_\tau u - \Delta u = 0$, the heat equation in $\tau$. With $f = \frac{|x|^2}{4\tau}$: $2\Delta f = \frac n\tau$, $|\nabla f|^2 = \frac{|x|^2}{4\tau^2}$, so $\tau(2\Delta f - |\nabla f|^2) + f - n = n - \frac{|x|^2}{4\tau} + \frac{|x|^2}{4\tau} - n = 0$.

Solution

$\frac{d}{dt}\int hw = \int(h_tw + hw_t - Rhw)$. And $\int(\square h)w - h\square^*w = \int(h_tw - w\Delta h + hw_t + h\Delta w - Rhw)$, where $\int(h\Delta w - w\Delta h) = 0$. With $w = u$, $\square^*u = 0$; with $w = v$, $-h\square^*v = 2\tau h|\cdots|^2u$.

Solution

$v \leq 0$ gives $\Delta f \leq \frac12(|\nabla f|^2 - R) + \frac{n - f}{2\tau}$. Then $\partial_tf \geq -\frac12(|\nabla f|^2 - R) - \frac{n - f}{2\tau} + |\nabla f|^2 - R + \frac{n}{2\tau} = \frac12|\nabla f|^2 - \frac12R + \frac{f}{2\tau}$. So $-\partial_tf \leq \frac12R - \frac12|\nabla f|^2 - \frac{f}{2\tau}$, and adding $-\langle\nabla f, \dot\gamma\rangle \leq \frac12|\nabla f|^2 + \frac12|\dot\gamma|^2$ gives $-\frac{d}{dt}f \leq \frac12(R + |\dot\gamma|^2) - \frac{f}{2\tau}$.

Solution

(a) For every $\alpha > 0$ there exist $\delta, \varepsilon > 0$, depending only on $\alpha$ and $n$, such that for every $r_0 > 0$ and every smooth Ricci flow on $[0, (\varepsilon r_0)^2]$ satisfying the two hypotheses on $B(x_0, r_0)$ at $t = 0$, the curvature bound holds for all $(x, t)$ with $0 < t \leq (\varepsilon r_0)^2$ and $\operatorname{dist}_t(x, x_0) < \varepsilon r_0$. (b) $\tilde g$ is a Ricci flow; $\tilde R = r_0^2R \geq -1$ on $\tilde B(x_0, 1)$; isoperimetric ratios are scale-invariant; and $|\widetilde{\operatorname{Rm}}|(x, s) = r_0^2|\operatorname{Rm}|(x, r_0^2s) \leq \frac\alpha s + \varepsilon^{-2}$ translates to the bound for $g$ at $t = r_0^2s$. (c) In applications the flow is unknown (it is the flow with surgery, near some point); the theorem gives control from local initial data alone only because $\varepsilon$ and $\delta$ are universal.

Solution

(a) $R = \frac{n(n - 1)}{\rho^2} > 0$. (b) Small balls on a smooth manifold are nearly Euclidean, with isoperimetric constants tending to the Euclidean one as the radius shrinks relative to the curvature scale. (c) The sectional curvature at time $t$ is $\big(\rho^2 - 2(n - 1)t\big)^{-1}$. Since $2(n - 1)t \leq 2(n - 1)\varepsilon^2r_0^2 \leq \frac{r_0^2}{2} \leq \frac{\rho^2}{2}$, it is at most $\frac2{\rho^2} \leq \frac{2}{r_0^2}$, and $\frac2{r_0^2} \leq (\varepsilon r_0)^{-2}$ because $\varepsilon^2 \leq \frac14 < \frac12$. The term $\frac\alpha t$ does no work: it matters only when curvature can be large at small times, which a smooth flow with bounded curvature never needs.