The Proof of Pseudolocality

12A.9 Pseudolocality stated pseudolocality and sketched its proof in six steps. This chapter carries out the proof, following Perelman's §10.1 and Kleiner and Lott's sections 31–34, with every estimate written down. It is the most intricate argument of the first preprint, and also the one that assembles everything Book 12A has built: the point-picking technique of 2B.3 Compactness's contradiction–compactness template, the distance-distortion Lemma 8.3 of 12A.8 Local Noncollapsing from the Reduced Volume, the conjugate heat kernel and Perelman's Harnack inequality $v \leq 0$ of 12A.9 Pseudolocality, blow-up limits and the equality case of the monotonicity formula, which forces a shrinking soliton (12A.3 The 𝓦-Entropy), a cut-off function whose heat-operator image is controlled by the distance lemma, and finally the Gaussian logarithmic Sobolev inequality of 6A.12 Entropy, Information and Diffusion, transplanted to an almost-Euclidean ball by symmetrisation.

The logic, in one sentence: if curvature appeared in the controlled region, the conjugate heat kernel from the bad point would carry a definite amount of "negative entropy" back to time $0$, where the almost-Euclidean ball has no room for it. The negative entropy is the integral of $v$; the "no room" is the log-Sobolev inequality, which says that on Euclidean space (and nearly so on an almost-Euclidean ball) the entropy $\mathcal W$ of any probability density is nonnegative.

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

• set up the proof by contradiction, with the normalisations $r_0 = 1$, $\alpha < \frac{1}{100n}$, the set $M_\alpha$ and the minimal choice of $\varepsilon$;
• prove Claim 1, the point selection, and Claim 2, which turns its conclusion into a curvature bound on a genuine parabolic neighbourhood using Lemma 8.3(b);
• construct the fundamental solution of the conjugate heat equation on a noncompact manifold by exhaustion, and prove that it passes to limits;
• prove Claim 3, $\int_Bv \leq -\beta$, in both the noncollapsed case (where the limit would be a flat shrinking soliton) and the collapsed case (where $\int_Bv \to -\infty$);
• construct the cut-off function $h$, compute $\square h$, and carry the bounds $\int hu \geq 1 - A^{-2}$ and $-\int hv \geq \beta(1 - A^{-2})$ back to time $0$;
• rewrite $-\int hv$ at time $0$ as an entropy for $\tilde u = hu$, rescale, and contradict the sharpened logarithmic Sobolev inequality via symmetrisation;
• state the corollaries 10.2–10.4 and say where the theorem is used.

10.1The statement and the normalisations

By scaling, assume $r_0 = 1$. Decreasing $\alpha$ only strengthens the conclusion, so assume $\alpha < \frac{1}{100n}$. Write $d(x, t) = d_t(x, x_0)$ and

the set of space-time points where the curvature is large on the parabolic scale. The conclusion fails at a point $(x, t)$ exactly when $|\operatorname{Rm}| > \alpha t^{-1} + \varepsilon^{-2}$ there; such a point lies in $M_\alpha$.

10.2Claim 1: point selection

This is the point-picking argument of 11B.4 Blow-Ups, Blow-Downs and Splitting at Infinity and of Hamilton's work on singularities, adapted to a region where only part of space-time is controlled: one keeps moving to a point of at least four times the curvature, within a distance that is small on the scale of the current curvature, and the geometric series keeps the sequence inside the region of interest.

10.3Claim 2: a parabolic neighbourhood with bounded curvature

The region described by (10.2) is not a product in space-time: the condition $d(x, t) \leq \cdots$ uses the time-$t$ distance. To take limits one needs a curvature bound on a product region, a backward parabolic neighbourhood, and for that one must control how fast distances change.

10.4The contradicting sequence

Suppose the theorem fails for some $\alpha$. Then there are sequences $\varepsilon_k \to 0$, $\delta_k \to 0$ and flows satisfying the hypotheses with $\varepsilon = \varepsilon_k$, $\delta = \delta_k$, each with a bad point. Decrease each $\varepsilon_k$ as far as possible while a bad point remains (a flow on $[0, \varepsilon^2]$ restricts to $[0, \varepsilon'^2]$ for $\varepsilon' < \varepsilon$ and still satisfies the hypotheses, so this keeps a counterexample); then, by the minimality,

This is a curvature bound near $x_0$ at all small times, which will be needed for the cut-off function. Set $A = \frac{1}{100n\varepsilon} \to \infty$, so that $A\varepsilon = \frac{1}{100n}$ satisfies Claim 1's hypothesis, and construct $(\bar x, \bar t)$ and $Q$ as above.

The conjugate heat kernel. Let $u = (4\pi(\bar t - t))^{-n/2}e^{-f}$ be the fundamental solution of the conjugate heat equation $\square^*u = -\partial_tu - \Delta u + Ru = 0$ with $u \to \delta_{\bar x}$ as $t \uparrow \bar t$, and

which satisfies $v \leq 0$ by Perelman's Corollary 9.3 (12A.9 Pseudolocality). On a noncompact manifold, $u$ is constructed by exhaustion (Kleiner and Lott, §33): solve on an increasing sequence of compact domains $D_i$ with Dirichlet boundary conditions; the solutions increase with $i$ by the maximum principle; their limit is $u$, independent of the exhaustion, with $\int u\,dV \leq 1$ (equality, stochastic completeness, holds under bounded curvature, 6D.5 Diffusions on Manifolds and Curvature). Kleiner and Lott's Lemma 33.1 adds the compactness statement needed for blow-ups: if pointed flows converge smoothly to a stochastically complete limit, then the fundamental solutions converge smoothly on compact sets to the limit's fundamental solution. The proof uses the $L^1$ bound and parabolic regularity to extract a limit $U \geq u_\infty$, and then $\int(U - u_\infty) \leq 0$ by Fatou.

10.5Claim 3: a definite amount of negative $v$

Case 2 is where the dimension of collapse enters: on a collapsed space the heat kernel spreads in fewer directions than $n$, its entropy relative to the $n$-dimensional Gaussian is too low, and $v$, which measures exactly that deficit, becomes very negative. That is the heat-kernel form of the reason that $\mu$ detects collapse (12A.4 κ-Noncollapsing).

10.6The cut-off function and the passage to time $0$

The negative integral of $v$ near $(\bar x, \tilde t)$ must be carried back to time $0$, where the hypotheses live. The conjugate heat equation runs backward, and $\int hv$, for a suitable cut-off $h$, cannot change much.

The cut-off. Let $\varphi$ be a smooth function on $\mathbb{R}$, equal to $1$ on $(-\infty, 1]$, decreasing on $[1, 2]$, zero on $[2, \infty)$, with

(take $1 - 2(x - 1)^2$ on $[1, \frac32]$ and $2(x - 2)^2$ on $[\frac32, 2]$, smoothed). With $\tilde d(y, t) = d(y, t) + 200n\sqrt t$, set

At $t = 0$, $h$ vanishes outside $B(x_0, 20A\varepsilon)$, which lies in the almost-Euclidean ball (since $A\varepsilon = \frac{1}{100n}$); near $(\bar x, \bar t)$, $h = 1$, since $d(\bar x, \bar t) < (2A + 1)\varepsilon$ and $200n\sqrt{\bar t} \leq 200n\varepsilon$, so $\tilde d < 10A\varepsilon$ for large $A$.

Its heat operator. By the chain rule,

Where $\varphi' \neq 0$, $10A\varepsilon \leq \tilde d \leq 20A\varepsilon$, so $d \geq 9A\varepsilon$ for large $A$; apply Lemma 8.3(a) with $r_0 = \sqrt t \leq \varepsilon$, using the curvature bound (10.4), $|\operatorname{Rm}| \leq \alpha t^{-1} + 2\varepsilon^{-2}$ on $B(x_0, \sqrt t)$:

because $t \leq \varepsilon^2$ and $\alpha$ is small. So the first term is $\leq 0$ ($\varphi' \leq 0$ and the bracket is $\geq 0$), and $\square h \leq -\frac{\varphi''}{(10A\varepsilon)^2} \leq \frac{10\varphi}{(10A\varepsilon)^2}$. The term $200n\sqrt t$ in $\tilde d$ is there precisely to absorb the $t^{-1/2}$ from Lemma 8.3.

Mass is almost conserved. Since $\square^*u = 0$, $\frac{d}{dt}\int hu = \int(\square h)u \leq \frac{10}{(10A\varepsilon)^2}\int u \leq \frac{1}{10(A\varepsilon)^2}$. Integrating from $0$ to $\bar t \leq \varepsilon^2$, and using $\int hu \to 1$ as $t \to \bar t$ (because $h = 1$ near $\bar x$),

Negative $v$ is almost conserved. Using $\square^*v = -2\tau|\cdots|^2u \leq 0$ and $v \leq 0$,

using $\varphi'' \geq -10\varphi$ and $v \leq 0$. At $\tilde t$, $h = 1$ on $B$ (for large $A$), so $\int(-hv) \geq -\int_Bv \geq \beta$. Integrating back from $\tilde t$ to $0$ (Gronwall):

10.7At time $0$: an entropy that is too negative

Now work at $t = 0$, where $\tau = \bar t$. Put $\tilde u = hu$ and $\tilde f = f - \log h$ (on the support of $h$), so that $\tilde u = (4\pi\bar t)^{-n/2}e^{-\tilde f}$. By the definition of $v$,

An integration by parts (Exercise 10.7; Kleiner–Lott (34.8)–(34.9)) gives

so

The error terms are small: $\frac{|\nabla h|^2}{h} = \frac{(\varphi')^2}{\varphi}\frac{1}{(10A\varepsilon)^2} \leq \frac{10}{(10A\varepsilon)^2}$ and $-Rh \leq 1$ (the hypothesis $R \geq -1$ at time $0$), so $\bar t\int(\ldots)u \leq \varepsilon^2\big(\frac{10}{(10A\varepsilon)^2} + 1\big) \leq A^{-2} + \varepsilon^2$; and $-h\log h \leq 1$ is supported where $h < 1$, outside $B(x_0, 10A\varepsilon)$, where the mass of $u$ is at most $cA^{-2}$ (by the same argument as for $\int hu$, with $\varphi(\tilde d/5A\varepsilon)$). Altogether

Rescale to $\tau = \frac12$. Put $\hat g = \frac{1}{2\bar t}g$ and $\hat u = (2\bar t)^{n/2}\tilde u = (2\pi)^{-n/2}e^{-\hat f}$. Then $\bar t|\nabla\tilde f|^2_g = \frac12|\nabla\hat f|^2_{\hat g}$, $\int\hat u\,d\hat V = \int\tilde u\,dV \to 1$, and (34.14) becomes, for large $k$,

Normalising, $U = \hat u/\int\hat u = (2\pi)^{-n/2}e^{-F}$, the same holds for $U$ and $F$ with $\frac12\beta$ replaced by a slightly smaller positive constant, still written $\frac12\beta$:

The support of $U$ lies in the ball $B(x_0, 20A\varepsilon)$ at time $0$, which in the metric $\hat g$ has radius $20A\varepsilon/\sqrt{2\bar t} \geq 20A/\sqrt2 \to \infty$, and on which the isoperimetric constant is within $\delta_k \to 0$ of the Euclidean one.

Logarithmic Sobolev. On $\mathbb{R}^n$, for $U = (2\pi)^{-n/2}e^{-F}$ with $\int U = 1$ and compact support, Gross's inequality in Perelman's form ($\mathcal W \geq 0$ at $\tau = \frac12$, 12A.3 The 𝓦-Entropy) says

which (34.16) would contradict outright, if the manifold were Euclidean. Peter Topping observed a sharper form, obtained by applying (34.17) to the rescaled densities $c^nU(cx)$ and optimising over $c$ (Exercise 10.8):

Its advantage is that it involves only $\int|\nabla F|^2U$ and $\int FU$, quantities that symmetrisation handles: replacing $U$ by the radially decreasing function on $\mathbb{R}^n$ whose superlevel sets are balls with the same volumes (Pólya–Szegő, the method of Faber–Krahn in 6B.16 Eigenvalue Problems and Perelman’s λ) preserves $\int FU$ and does not increase $\int|\nabla F|^2U$ if the isoperimetric inequality on the support is Euclidean. With the almost-Euclidean constant $(1 - \delta_k)c_n$, symmetrisation gives (Kleiner–Lott, (34.19), citing Proposition 4.1 of Lei Ni's 2004 paper on the entropy formula for the linear heat equation, which attributes the argument to Perelman)

Put $x_k = 1 - \frac2n\int F_kU_k$. Combining with (34.16), $\frac12\beta \leq -\frac12(1 - \delta_k)^{2/n}n\,e^{x_k} + \frac n2(1 + x_k)$, that is,

But $\inf_x\big((1 - \delta)^{2/n}e^x - 1 - x\big) = \frac2n\log(1 - \delta) \to 0$ as $\delta \to 0$ (the infimum is at $e^x = (1 - \delta)^{-2/n}$). For large $k$ the left side is greater than $-\frac12\beta$: a contradiction. This proves the theorem.

10.8Corollaries and uses

Perelman draws three consequences.

• Corollary 10.2 (from the proof): under the same hypotheses, $\operatorname{Vol}B(x, t, \sqrt t) \geq c(n)\sqrt t^n$ for $x \in B(x_0, \varepsilon r_0)$ and $0 < t \leq (\varepsilon r_0)^2$. The almost-Euclidean ball stays noncollapsed at all smaller scales for a definite time (Kleiner–Lott, §35).
• 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 $0 \leq t \leq (\varepsilon r_0)^2$, $d_t(x, x_0) < \varepsilon r_0$. A curvature bound plus almost-Euclidean volume replaces the isoperimetric hypothesis, and the $\alpha t^{-1}$ term disappears. Perelman leaves the proof to the reader and asks whether the volume assumption is superfluous (it is not, in general, as a flat cone shows: bounded curvature away from the vertex, no control near it).
• Corollary 10.4 (from 8.2, 10.1, 10.2): under the hypotheses of 10.1, the flow is $\kappa(A)$-noncollapsed at later times on scales less than $\sqrt t$ in $B(x_0, A\sqrt t)$ (Kleiner–Lott, §36).

Where it is used. In the second preprint Perelman uses Theorem 10.1 in the proof that the standard solution exists on the whole time interval $[0, 1)$ (II, §2, Claim 4; 12B.9 Surgery): far out along the cylindrical end, balls at a time slightly before a putative singular time are nearly Euclidean at a small fixed scale, and pseudolocality bounds the curvature there up to the singular time. The point-selection technique of Claim 1 reappears in II, §6 (long-time behaviour). Morgan and Tian avoid the theorem altogether, proving the properties of the standard solution they need by other means (12B.9 Surgery). Since 2002 pseudolocality has become a standard tool in the theory, with versions for complete flows (Chau, Tam and Yu), for the Kähler–Ricci flow, and in Bamler's structure theory, where it plays the role of an $\varepsilon$-regularity theorem (6C.8 Monotonicity, ε-Regularity and Tangent Cones).

10.9History

Perelman's §10 appeared in his first preprint (2002), with the proof compressed into two pages; Kleiner and Lott's sections 30–36 (2003–08) wrote it out, and their acknowledgement records Topping's sharpening of the log-Sobolev step. Albert Chau, Luen-Fai Tam and Chengjie Yu (2011) extended pseudolocality to complete noncompact flows. The point-selection argument has roots in Hamilton's work on singularities (1995), and symmetrisation in Pólya and Szegő (1951). The interpretation in terms of the renormalisation group is Perelman's own.

10.10Exercises

Solution

$|\operatorname{Rm}|^{-1/2}(x_j, t_j) \leq 2^{-(j-1)}\varepsilon$, so the sum is at most $A\varepsilon\sum_{j\geq1}2^{-(j-1)} = 2A\varepsilon$. With a factor $1 + \eta$ the radii shrink like $(1 + \eta)^{-(j-1)/2}$ and the sum is about $\frac{2A\varepsilon}{\eta}$, so the selected point could be far from $x_0$, outside the region where the hypotheses give control; the constant $(2A + 1)\varepsilon$ would become $(2A/\eta + 1)\varepsilon$, and the choice $A\varepsilon = \frac{1}{100n}$ would no longer keep it in the almost-Euclidean ball. Any fixed factor greater than $1$ works with different constants; $4$ makes the bookkeeping clean.

Solution

From $u = (4\pi\tau)^{-n/2}e^{-f}$: $f = -\log u - \frac n2\log(4\pi\tau) = \frac{|x|^2}{4\tau} + \frac m2\log(4\pi\tau) + \log\operatorname{vol}(C) - \frac n2\log(4\pi\tau)$. With $R = 0$: $\Delta f = \frac{m}{2\tau}$, $|\nabla f|^2 = \frac{|x|^2}{4\tau^2}$, so $\tau(2\Delta f - |\nabla f|^2) = m - \frac{|x|^2}{4\tau}$, and $\tau(\ldots) + f - n = m - n + \frac{m - n}{2}\log(4\pi\tau) + \log\operatorname{vol}(C)$, a constant in space. Since $\int u = 1$, $\int v = (m - n)(1 + \frac12\log4\pi\tau) + \log\operatorname{vol}(C) \to -\infty$ for $m < n$.

Solution

$\int-2\Delta f\,he^{-f} = \int2\langle\nabla f, \nabla(he^{-f})\rangle = \int2\langle\nabla f, \frac{\nabla h}{h} - \nabla f\rangle he^{-f}$. Adding $\int|\nabla f|^2he^{-f}$ gives $\int\langle\nabla f, 2\frac{\nabla h}{h} - \nabla f\rangle he^{-f}$. With $\nabla f = \nabla\tilde f + \frac{\nabla h}{h}$, the integrand is $\langle\nabla\tilde f + \frac{\nabla h}{h}, \frac{\nabla h}{h} - \nabla\tilde f\rangle = \frac{|\nabla h|^2}{h^2} - |\nabla\tilde f|^2$.

Solution

(a) $c^nU(cx) = (2\pi)^{-n/2}e^{-F(cx) + n\log c}$. Then $\nabla F_c(x) = c(\nabla F)(cx)$, and substituting $y = cx$, $\int|\nabla F_c|^2U_c\,dx = c^2\int|\nabla F|^2(y)U(y)\,dy$, $\int F_cU_c = \int FU - n\log c$. (b) (34.17) for $U_c$: $-\frac12c^2I - a + n\log c + n \leq 0$, with $I = \int|\nabla F|^2U$, $a = \int FU$. So $I \geq \frac{2}{c^2}(n - a + n\log c)$. The right side is maximised where $\frac{d}{dc}$ vanishes: $\frac{n}{c}c^2 - 2c(n - a + n\log c) = 0$, i.e. $n\log c = a - \frac n2$. There $n - a + n\log c = \frac n2$ and $c^2 = e^{2a/n - 1}$, so $I \geq n\,e^{1 - 2a/n}$.

Solution

Let $a = (1 - \delta)^{2/n} < 1$. The function $ae^x - 1 - x$ has derivative $ae^x - 1$, zero at $x = -\log a$, where its value is $1 - 1 + \log a = \log a = \frac2n\log(1 - \delta)$. With $x_k = 1 - \frac2n\int F_kU_k$, the two inequalities give $\frac n2(ae^{x_k} - 1 - x_k) \leq -\frac12\beta$ with $a = (1 - \delta_k)^{2/n}$. Bounding the left side below by its infimum, $\frac n2\cdot\frac2n\log(1 - \delta_k) = \log(1 - \delta_k) \leq -\frac12\beta$, which fails once $\delta_k < 1 - e^{-\beta/2}$.