Local Noncollapsing from the Reduced Volume

12A.4 κ-Noncollapsing proved that a Ricci flow on a closed manifold, over a finite time interval, is $\kappa$-noncollapsed at scales below any fixed $\rho$, by comparing two values of the entropy $\mu$: a collapsed ball forces $\mu$ to be very negative at small scale, while monotonicity bounds it from below by its value on the initial metric. That proof is global. The entropy is an integral over the whole manifold, and its monotonicity needs the flow to exist, smoothly, everywhere. A Ricci flow with surgery does not satisfy this: at each surgery time the manifold is cut and re-glued, and the entropy of the new manifold has no reason to be bounded by the old one. Perelman therefore proved a second noncollapsing theorem, in §7.3 and §8 of his first preprint, based on the reduced volume instead. It needs only local information: curvature bounds near the point in question and a lower volume bound for a ball at an earlier time at bounded distance. That is the form that can be carried across surgeries (12B.13 Ricci Flow with Surgery for All Time).

The proof has two halves, and they use the two results of 12A.7 Monotonicity of the Reduced Volume. If a ball is collapsed at scale $r$, then the reduced volume based at its centre is small at $\tau = \varepsilon r^2$: the $\mathcal L$-geodesics with small initial vectors stay inside the collapsed ball, which has little volume, and those with large initial vectors are suppressed by the Gaussian factor $e^{-|v|^2}$ that bounds the integrand. On the other hand, if the reduced distance is bounded on some region of definite volume at a much earlier time, then the reduced volume at that earlier time is bounded below. Monotonicity forbids small-then-large. The second half requires an upper bound for $\ell$, and here the point where $\ell \leq \frac n2$ is used, together with a maximum principle argument for $\bar L = 4\tau\ell$ that Perelman carries out in §8.

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

• state Perelman's definition of $\kappa$-collapsing in terms of parabolic neighbourhoods, and explain why it differs from 12A.4 κ-Noncollapsing's;
• state Morgan and Tian's general noncollapsing theorem for generalised Ricci flows, and prove it: the metric comparison and derivative bounds on the parabolic ball, the confinement of short $\mathcal L$-geodesics, the bound on the small-vector part, the Gaussian bound on the large-vector part, and the conclusion;
• prove Perelman's Lemma 8.3 on the evolution of distance under bounded Ricci curvature;
• explain Perelman's Theorem 8.2 and its maximum principle proof, including the cut-off function $\varphi$ and the role of (7.15);
• deduce universal noncollapsing for compact three-dimensional flows with normalised initial metrics (Morgan–Tian, Theorem 8.10);
• compare the entropy and reduced-volume routes and say which parts survive surgery.

8.1Collapsing in a parabolic neighbourhood

Perelman's Definition 8.1 changes the hypothesis of 12A.4 κ-Noncollapsing in one respect. A Ricci flow is $\kappa$-collapsed at $(x_0, t_0)$ on the scale $r$ if

and $\operatorname{Vol}_{t_0}B(x_0, r) < \kappa r^n$. The curvature bound is now required on the whole backward parabolic neighbourhood $P(x_0, t_0, r, -r^2) = B_{t_0}(x_0, r)\times[t_0 - r^2, t_0]$, not only at the time $t_0$. This is natural for the reduced volume, which looks backward in time from $(x_0, t_0)$; it is also slightly weaker than the entropy version (the hypothesis of collapsing is stronger, so the conclusion "not collapsed" is weaker), which Perelman notes. In the applications both are available, because the canonical neighbourhood estimates give curvature bounds on parabolic neighbourhoods anyway (12B.5 The Canonical Neighbourhood Theorem).

Morgan and Tian separate the theorem into a general statement about reduced volume, valid for generalised Ricci flows (space-times that are locally Ricci flows, which includes flows with surgery), and its applications. Throughout, the reduced volume is normalised as in 12A.5 Reduced Distance and Reduced Volume, with the factor $(4\pi\tau)^{-n/2}$, so that its value on flat space is $1$; Morgan and Tian's has value $(4\pi)^{n/2}$ there, and their constants differ from these by that factor.

The second hypothesis is the "large reduced volume at a large $\tau$" input. It implies at once

and by the localised monotonicity of 12A.7 Monotonicity of the Reduced Volume (the sets $W(\tau) = \mathcal L\exp_\tau(\tilde W)$ are swept out by minimising geodesics, so the region they fill is closed under minimising geodesics), $\tilde V(W(\tau)) \geq (4\pi\bar\tau_0)^{-n/2}Ve^{-l_0}$ for every $\tau \leq \tau_0$. The theorem follows from an upper bound at small $\tau$.

8.2The small-scale upper bound

Write $\operatorname{Vol}B(x, T, r) = \varepsilon^nr^n$, so that $\varepsilon$ measures collapse, and set $\tau_1 = \varepsilon r^2$. The claim to prove is:

Granting this, either $\varepsilon > \varepsilon_0$ or $(4\pi\bar\tau_0)^{-n/2}Ve^{-l_0} < 3(4\pi)^{-n/2}\varepsilon^{n/2}$, that is $\varepsilon^{n/2} > \frac13\bar\tau_0^{-n/2}Ve^{-l_0}$. In either case $\varepsilon$ is bounded below, and $\kappa = \min\{\varepsilon_0^n, (\frac13\bar\tau_0^{-n/2}Ve^{-l_0})^2\}$ works. To prove the proposition, split the initial vectors at the scale $\frac18\varepsilon^{-1/2}$:

and bound the two contributions to $\tilde V(W(\tau_1))$ separately by $2(4\pi)^{-n/2}\varepsilon^{n/2}$ and $(4\pi)^{-n/2}\varepsilon^{n/2}$.

Control on the parabolic ball. Rescale so that $r = 1$: then $|\operatorname{Rm}| \leq 1$ on $B(x, T, 1)\times[T - 1, T]$. By Shi's estimates (11A.5 Shi's Derivative Estimates), $|\nabla R| \leq C_1$ on $B(x, T, \frac12)\times[T - \frac12, T]$, for a universal $C_1$; scaling back, $|\nabla R| \leq C_1r^{-3}$. Since $|\operatorname{Ric}| \leq (n - 1)r^{-2}$ and $\partial_tg = -2\operatorname{Ric}$, integrating in time gives

so on $[T - \tau_1, T]$ the metrics agree with $g(T)$ up to a factor $1 \pm C\varepsilon$, and volumes of subsets of $B(x, T, r)$ agree up to a factor between $0.9$ and $1.1$ for $\varepsilon$ small (Morgan–Tian, Claims 8.5 and 8.8).

Short geodesics stay in the ball. The bounds on $\operatorname{Ric}$ and $\nabla R$ along a geodesic control how fast $\sqrt\tau X$ can change: from the geodesic equation in the variable $s$, $\big|\frac{d}{ds}A\big| \leq C(s^2|\nabla R| + s|\operatorname{Ric}||A|)$, and a Gronwall argument gives (Morgan–Tian, Lemma 6.60 and Claim 8.6)

as long as the geodesic stays in $B(x, T, r/2)$ and $\tau \leq \tau_1$. For $|v| \leq \frac18\varepsilon^{-1/2}$ and $\varepsilon$ small, this gives $\sqrt\tau|X| \leq \frac14\varepsilon^{-1/2}$, and then, until the first exit time $\tau'$,

So there is no exit: geodesics with small initial vectors stay in $B(x, T, r/2)$ up to time $\tau_1$ (Morgan–Tian, Claim 8.7). This is where the scale $\varepsilon^{-1/2}$ comes from: in time $\varepsilon r^2$, an $\mathcal L$-geodesic with initial vector $v$ travels a distance of about $2\sqrt{\varepsilon r^2}|v|$, which is less than $\frac r2$ when $|v| < \frac14\varepsilon^{-1/2}$.

The small-vector part. Along such geodesics $R \geq -n(n - 1)r^{-2}$, so $L \geq -\frac{2n(n-1)}{3r^2}\tau_1^{3/2}$ and $\ell(q, \tau_1) \geq -\frac{n(n - 1)}{3}\varepsilon$. The images lie in $B(x, T, r/2) \subseteq B(x, T, r)$, whose volume at time $T - \tau_1$ is at most $1.1\,\varepsilon^nr^n$. Hence

for $\varepsilon \leq \frac{1}{4n(n - 1)}$. The collapsed ball has too little volume to carry much reduced volume.

The large-vector part. Here no geometry is needed, only the monotonicity of the integrand: by 12A.7 Monotonicity of the Reduced Volume, on the injectivity set, $(4\pi\tau)^{-n/2}e^{-\tilde\ell}J \leq (4\pi)^{-n/2}2^ne^{-|v|^2} = \pi^{-n/2}e^{-|v|^2}$. So

(Exercise 8.6: compare the ball of radius $a$ with the inscribed cube of side $\frac{2a}{\sqrt n}$, and use $1 - (1 - \delta)^{n/2} \leq \frac n2\delta$). For $\varepsilon$ small this is less than $(4\pi)^{-n/2}\varepsilon^{n/2}$, because $e^{-1/(64n\varepsilon)}$ decays faster than any power. Adding the two parts proves Proposition 8.2, hence Theorem 8.1.

The structure is worth remembering: the reduced volume of a collapsed region is small at the scale of the collapse, because the Gaussian weight confines the integrand to $\mathcal L$-geodesics that cannot leave the region in that time. Perelman's §7.3 says this in one paragraph; the constants above are Morgan and Tian's.

8.3Bounding the reduced distance from above

To apply Theorem 8.1 one needs hypothesis 2: a region of definite volume at an earlier time on which $\ell$ is bounded. Perelman's §7.3, for a flow on a closed manifold over $[0, T]$ with base point at time $t_k$, gets it from $\min\ell \leq \frac n2$: at $\tau = t_k - \frac T2$ there is a point $q_k$ with $\ell \leq \frac n2$, and concatenating the minimiser to $q_k$ with short paths from $q_k$ through the region $[0, \frac T2]$, where all geometric quantities are bounded by those of the initial metric, gives a uniform bound for $\ell$ on a ball of definite volume at $t = 0$. Morgan and Tian's Theorem 8.10 makes this precise in dimension three:

Perelman's Theorem 8.2 is the local version, with no assumption on the whole initial manifold, only on one ball.

The difference from Theorem 8.3 is that nothing is assumed outside the parabolic cylinder over $B(x_0, r_0)$: the point $x$ may be far from $x_0$ (distance up to $Ar_0$), and the curvature between them is uncontrolled. The reduced distance from $(x, r_0^2)$ must be bounded on the ball around $x_0$ at an earlier time without any curvature control along the way, and that is what the maximum principle argument does.

8.4Perelman's maximum principle argument

Scale so that $r_0 = 1$ and $d_1(x, x_0) = A$. The base point is $(x, 1)$ and $\tau = 1 - t$. The reduced volume at $\tau = 1$ (time $0$) is bounded below if $\ell$ is bounded on a ball around $x_0$ at time $0$; by the curvature bounds on the cylinder over $B(x_0, 1)$, it suffices to bound $\ell$ at time $\frac12$ at some point within distance $\frac1{10}$ of $x_0$, and then concatenate. The tool for that is a lemma on distances.

The lemma says that bounded Ricci curvature near the endpoints is enough to control how fast distances can shrink and how negative $\partial_td - \Delta d$ can be: the uncontrolled curvature in the middle of the geodesic enters $\partial_td$ and $\Delta d$ with the same sign and cancels. It is used again in the canonical neighbourhood theorem (12B.5 The Canonical Neighbourhood Theorem).

The argument. Consider, for $\frac12 \leq t \leq 1$,

where $\bar L = 4\tau\ell$ is the function of (7.15), and $\varphi$ is a function of one variable equal to $1$ on $(-\infty, \frac1{20})$, increasing to $+\infty$ on $(\frac1{20}, \frac1{10})$, and chosen so that

The factor $\bar L + 2n + 1$ is at least $1$ for $t \geq \frac12$, because of the lower bound for $R$ (hence for $\bar L$) at the end of 12A.7 Monotonicity of the Reduced Volume. At $t = 1$, the base time, $h(x, 1) = \varphi(A - A)(0 + 2n + 1) = 2n + 1$, so $\min h(\cdot, 1) \leq 2n + 1$. At $t = \frac12$ the shift $A(2t - 1)$ vanishes, so $h(\cdot, \frac12)$ is infinite outside $B_{1/2}(x_0, \frac1{10})$, and its minimum is achieved at some $y$ with $d(y, \frac12) \leq \frac1{10}$. Running backward from $t = 1$ to $t = \frac12$, the shift $A(2t - 1)$ moves the support of $\varphi$ from around $x$ (where $d = A$) to around $x_0$, at speed $2A$ in $d$: the cut-off slides from the base point to the target ball.

At a minimum point of $h$, $\nabla h = 0$ gives $\varphi\nabla\bar L = -(\bar L + 2n + 1)\nabla\varphi$, and substituting into the formula for $\square h = \partial_th - \Delta h$ (Perelman's (8.2)–(8.4)) leaves

Where $\varphi' \neq 0$, $d \geq \frac1{20}$, and Lemma 8.3(a) with $r_0 = \frac1{20}$, $K = 1$ (curvature at most $1$ in $B(x_0, \frac1{20})$) gives $\partial_td - \Delta d \geq -100(n - 1)$; (8.1) then bounds the bracket below by $-C(A)$. And in forward time, (7.15) reads $\partial_t\bar L - \Delta\bar L \geq -2n$ (since $\partial_\tau = -\partial_t$). So at a minimum point,

and the minimum of $h$ cannot decrease too fast as $t$ decreases: $\min h(\cdot, \frac12) \leq e^{(2n + C(A))/2}(2n + 1)$. At the minimising point $y$, $\varphi \geq 1$, so $\bar L(y, \frac12) \leq C'(A)$, and $\ell(y, \frac12) \leq C''(A)$ with $d(y, \frac12) \leq \frac1{10}$. From $(y, \frac12)$, short paths through the controlled cylinder over $B(x_0, 1)$ reach every point of $B_0(x_0, \frac12)$ at time $0$ with bounded $\mathcal L$-length, which has volume at least a constant depending on $A$ (by the volume hypothesis and Bishop–Gromov). So $\tilde V$ at $\tau = 1$ is bounded below by a constant depending on $A$, and Theorem 8.1 (or Perelman's §7.3 argument) gives the theorem.

8.5The two routes compared

	entropy route (12A.4 κ-Noncollapsing)	reduced-volume route (this chapter)
monotone quantity	$\mu(g, \tau)$, an integral over the whole manifold	$\tilde V(\tau)$, based at one space-time point
hypothesis on the collapsed ball	$|\operatorname{Rm}| \leq r^{-2}$ at time $t_0$	$|\operatorname{Rm}| \leq r^{-2}$ on the parabolic neighbourhood
input from the past	$\mu(g(0), \cdot)$ bounded below: the whole initial metric	$\ell$ bounded on a ball of definite volume at an earlier time
tool for the input	monotonicity of $\mu$	$\min\ell \leq \frac n2$, Lemma 8.3, maximum principle for $\bar L$
survives surgery?	no: surgery changes the manifold globally	yes, if the relevant $\mathcal L$-geodesics avoid surgery regions
used for	blow-up limits of smooth flows	Ricci flow with surgery (12B.13 Ricci Flow with Surgery for All Time); $\kappa$-solutions (12B.1 κ-Solutions)

In the flow with surgery, Perelman proves noncollapsing at each stage by induction on the surgery intervals: assuming the flow up to the $i$-th surgery is noncollapsed and has canonical neighbourhoods, he measures $\mathcal L$-length only for admissible curves, those that stay in the region unaffected by surgery, and observes that the argument of this chapter goes through if every barely admissible curve (one on the boundary of the admissible set) has reduced length at least $\frac32 + \kappa'$; since that is not automatic, he adds an estimate showing that curves which come near a surgery region pick up a large $\mathcal L$-length there, where the curvature is huge (Perelman II, Lemmas 5.2 and 5.3; Morgan–Tian, chapter 16). 12B.13 Ricci Flow with Surgery for All Time explains that induction.

8.6History

Perelman's §7.3 and §8 appeared in his first preprint (2002); Theorem 8.2 was the version he needed for surgery, and Lemma 8.3 refers to Hamilton's "Formation of singularities" (1995), §17, for part (b). Kleiner and Lott (2003–08), Morgan and Tian (2007) and Cao and Zhu (2006) wrote out the proofs; Morgan and Tian's Theorem 8.1 isolates the reduced-volume argument in a form valid for generalised Ricci flows. The use of the second variation of length with linearly growing fields to bound $\Delta d$ goes back to the Laplacian comparison theorem of Riemannian geometry.

8.7Exercises

Solution

(a) The cube $[-a/\sqrt n, a/\sqrt n]^n$ lies in the ball of radius $a$, and the integral over the cube factorises. For the second inequality, $\big(\int_{-b}^be^{-z^2}\big)^2 = \int_{[-b,b]^2}e^{-|w|^2} \geq \int_{|w|\leq b}e^{-|w|^2} = \pi(1 - e^{-b^2})$. (b) With $b = a/\sqrt n$: $\int_{|v|\leq a}e^{-|v|^2} \geq \pi^{n/2}(1 - e^{-a^2/n})^{n/2}$, and the total is $\pi^{n/2}$, so the tail is at most $\pi^{n/2}\big(1 - (1 - \delta)^{n/2}\big)$ with $\delta = e^{-a^2/n}$; and $1 - (1 - \delta)^m \leq m\delta$ for $0 \leq \delta \leq 1$ (Bernoulli, or the mean value theorem). (c) $a^2/n = \frac{1}{64n\varepsilon}$.

Solution

$\big|\partial_t\log|w|^2_{g(t)}\big| = \frac{2|\operatorname{Ric}(w, w)|}{|w|^2} \leq 2(n - 1)r^{-2}$; integrating over an interval of length at most $\varepsilon r^2$ gives $|\log\frac{|w|^2_{g(t)}}{|w|^2_{g(T)}}| \leq 2(n - 1)\varepsilon$. The volume form is $\sqrt{\det g}$, and $\partial_t\log\sqrt{\det g} = -R$ with $|R| \leq n(n - 1)r^{-2}$, giving a factor at most $e^{n(n-1)\varepsilon}$.

Solution

$\gamma_v(\tau) = 2\sqrt\tau v$, so $|\gamma_v(\varepsilon r^2)| = 2\sqrt\varepsilon r|v| \leq \frac r2$ iff $|v| \leq \frac14\varepsilon^{-1/2}$. In a curved flow $\sqrt\tau|X|$ is only approximately $|v|$, with error $2\varepsilon(1 + |v|)$, and the metrics $g(t)$ and $g(T)$ differ slightly; the margin from $\frac14$ to $\frac18$ absorbs these errors for small $\varepsilon$.

Solution

(a) The ball of radius $r$ lies in $S^1\times(\text{disc of radius } r)$, of volume $2\pi\delta\cdot\pi r^2$, and $|\operatorname{Rm}| = 0 \leq r^{-2}$, so it is $\kappa$-collapsed when $2\pi^2\delta r^2 < \kappa r^3$. (b) The volume hypothesis at the earlier time fails for $r_0 \gg \delta$: $\operatorname{Vol}B(x_0, r_0) \leq 2\pi^2\delta r_0^2$, which is less than $A^{-1}r_0^3$ unless $A \geq r_0/(2\pi^2\delta)$; noncollapsing propagates forward in time but cannot be created. Likewise the normalisation $\operatorname{Vol}B(p, 0, 1) \geq \omega$ fails when $\delta$ is small. (c) The universal cover is $\mathbb{R}^3$ and $\mathcal L$-geodesics lift; $\ell(q, \tau) = \min_k\frac{|\tilde q - \tilde p_k|^2}{4\tau}$ over the lifts $\tilde p_k$ of the base point. So $\tilde V(\tau) = \int_{\text{fundamental domain}}(4\pi\tau)^{-3/2}e^{-\ell}$, which for $\tau \gg \delta^2$ is approximately $(4\pi\tau)^{-3/2}\cdot2\pi\delta\cdot4\pi\tau = \frac{2\pi\delta}{\sqrt{4\pi\tau}} \to 0$: the reduced volume detects the collapse at large scales.

Solution

$\partial_td = 0$ and $\Delta d = \frac{n - 1}{d}$, so $\partial_td - \Delta d = -\frac{n - 1}{d} \geq -\frac{n - 1}{r_0}$ for $d \geq r_0$, with equality on the sphere $d = r_0$. So it is sharp at $d = r_0$. The proof's choice of fields (linear on $[0, r_0]$, parallel beyond) is not optimal for $d > r_0$, where linear fields along the whole geodesic would give $\frac{n - 1}{d}$; the lemma uses the cruder fields because the curvature is controlled only on $B(x_0, r_0)$.

Solution

$\tilde g$ is a Ricci flow; $|\widetilde{\operatorname{Rm}}| = r_0^2|\operatorname{Rm}| \leq 1$ where $|\operatorname{Rm}| \leq r_0^{-2}$; distances scale by $r_0^{-1}$, so $\tilde d_0(x, x_0) < 1$ iff $d_0(x, x_0) < r_0$ and $\tilde d_1(x, x_0) \leq A$ iff $d_{r_0^2}(x, x_0) \leq Ar_0$; volumes scale by $r_0^{-n}$, so $\widetilde{\operatorname{Vol}}\tilde B(x_0, 1) \geq A^{-1}$. A ball $B(x, r)$ with $|\operatorname{Rm}| \leq r^{-2}$ and volume $< \kappa r^n$ corresponds to $\tilde B(x, r/r_0)$ with $|\widetilde{\operatorname{Rm}}| \leq (r/r_0)^{-2}$ and volume $< \kappa(r/r_0)^n$.