Reduced Distance and Reduced Volume

The $\mathcal W$-entropy of 12A.3 The 𝓦-Entropy is an integral over a whole manifold, so it controls global things. For the flow with surgery, Perelman needed a monotone quantity attached to a single point of space-time, which sees only the part of the flow that can influence that point. He built it from a new notion of length for curves in space-time, the $\mathcal L$-length. Its minimisers give a reduced distance $\ell$, and the Gaussian built from $\ell$ integrates to the reduced volume $\tilde V(\tau)$. The reduced volume is monotone along every Ricci flow, with no curvature assumption, and equals $1$ on flat space. It is the Ricci flow's version of the Bishop–Gromov volume ratio, and Perelman found it by applying Bishop–Gromov to a Ricci-flat manifold of very high dimension.

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

• define the $\mathcal L$-length, $\mathcal L$-geodesics, the reduced distance and the reduced volume, and compute them on flat space;
• state the differential inequalities for $\ell$ and use them to prove that $\tilde V$ is nonincreasing in $\tau$;
• show that $\ell \leq \frac n2$ somewhere at every $\tau$;
• read the reduced volume against Bishop–Gromov, row by row, and against Huisken's monotonicity formula;
• describe Perelman's heuristic derivation in $N \to \infty$ dimensions, and the uses of $\tilde V$.

5.1Least action and the heat kernel

5.2$\mathcal L$-length

Work in backward time. Fix a Ricci flow and a base time $t_0$, set $\tau = t_0 - t$, so that $\partial_\tau g = 2\operatorname{Ric}$, and assume the manifold is closed, or that each $g(\tau)$ is complete with uniformly bounded curvature. Fix a base point $p$.

The weight $\sqrt\tau$ is what makes the theory scale correctly. With $s = \sqrt\tau$ the curve has finite energy near $\tau = 0$, and the change of variable turns $\int\sqrt\tau|\dot\gamma|^2\,d\tau$ into $\frac12\int|\gamma'(s)|^2\,ds$. Perelman's first variation (his (7.1)) gives the $\mathcal L$-geodesic equation, with $X = \dot\gamma$:

Minimisers exist and are $\mathcal L$-geodesics, and $\sqrt\tau X(\tau)$ has a limit $v \in T_pM$ as $\tau \to 0$. The $\mathcal L$-exponential map sends $v$ to $\gamma_v(\bar\tau)$. Where minimisers are not unique, or $L$ is not smooth, the inequalities below hold in the barrier sense, as for the distance function (9B.1 Laplacian Comparison).

5.3The differential inequalities

Perelman computes the first and second variations of $L$ exactly as in Riemannian geometry, with one new feature: the time derivative of the metric is $2\operatorname{Ric}$, so Ricci terms appear, and along the curve they combine into Hamilton's Harnack expressions (11B.2 Ancient Solutions and the Harnack Inequality). He collects them in a quantity $K = \int_0^{\bar\tau}\tau^{3/2}H(X)\,d\tau$, where $H$ is Hamilton's trace Harnack expression. Then (his (7.5), (7.6) and (7.10)), in terms of $\ell$:

The last is the space-time Laplacian comparison, the analogue of $\Delta r \leq \frac{n - 1}{r}$ when $\operatorname{Ric} \geq 0$ (9B.1 Laplacian Comparison). It is proved with the test fields $Y$ solving $\nabla_XY = -\operatorname{Ric}(Y, \cdot) + \frac{1}{2\tau}Y$, which play the part of the linear Jacobi fields $\frac{s}{r}E$ in Euclidean space. The unknown $K$ cancels from the right combinations, and gives (Perelman's (7.13) and (7.14))

On flat space both are equalities (Exercise 5.8). No curvature hypothesis is needed anywhere: this is the decisive difference from Bishop–Gromov.

5.4The reduced volume

Perelman's own definition omits the factor $(4\pi)^{-n/2}$; with it, $\tilde V \equiv 1$ on flat $\mathbb{R}^n$, since the integrand is the heat kernel. This normalisation is common in the expositions.

As $\bar\tau \to 0$, the flow near $(p, t_0)$ looks Euclidean at the relevant scale $\sqrt{\bar\tau}$, and $\tilde V(\bar\tau) \to 1$; the expositions prove this limit carefully. So $\tilde V \leq 1$, with $1$ the value of flat space. Figure 5.2 computes it for the round shrinking 3-sphere.

5.5Bishop–Gromov in space-time

The reduced volume completes the comparison theory of 9B.1 Laplacian Comparison and 9B.2 Volume Comparison, row by row.

Riemannian geometry, $\operatorname{Ric} \geq 0$	Ricci flow, backward time $\tau$
length $\int|\dot\gamma|\,ds$	$\mathcal L$-length $\int\sqrt\tau(R + |\dot\gamma|^2)\,d\tau$
distance $d(p, q)$, with $\frac{d^2}{4}$ as "action"	reduced distance $\ell(q, \tau)$
geodesics, the exponential map	$\mathcal L$-geodesics, the $\mathcal L$-exponential map
Jacobi fields	$\mathcal L$-Jacobi fields
Laplacian comparison $\Delta r \leq \frac{n - 1}{r}$	$\Delta\ell \leq -R + \frac{n}{2\tau} - \frac{K}{2\tau^{3/2}}$
curvature hypothesis $\operatorname{Ric} \geq 0$	none: the flow equation supplies it
model: Euclidean space	model: the Gaussian soliton (flat $\mathbb{R}^n$), $\ell = \frac{|x|^2}{4\tau}$
volume ratio $\frac{\operatorname{Vol}B(p, r)}{\omega_nr^n}$, nonincreasing in $r$	reduced volume $\tilde V(\tau)$, nonincreasing in $\tau$
ratio $\to 1$ as $r \to 0$	$\tilde V \to 1$ as $\tau \to 0$
equality: Euclidean space	equality: shrinking gradient solitons
use: volume doubling, noncollapsing	use: noncollapsing that survives surgery

5.6Perelman's heuristic: Bishop–Gromov in $N$ dimensions

Perelman explains in §6 where the reduced volume comes from. Take a large integer $N$ and the manifold $\tilde M = M\times S^N\times\mathbb{R}_+$ with metric

where $g(\tau)$ is the backward Ricci flow and $g_{S^N}$ is the round metric of constant curvature $\frac{1}{2N}$. He reports that $\tilde g$ is Ricci-flat up to errors of order $N^{-1}$, and that its curvature components reproduce Hamilton's Harnack expressions. For a curve from a point $p$ at $\tau = 0$, orthogonal to the sphere factor, the $\tilde g$-length is

so shortest geodesics in $\tilde M$ minimise $\mathcal L$. The volume of the geodesic sphere of radius $\sqrt{2N\tau}$ about $p$, divided by the Euclidean value, is a constant times $N^{-n/2}$ times $\int\tau^{-n/2}e^{-\ell}$, up to $O(N^{-1})$. Bishop–Gromov for the nearly Ricci-flat $\tilde M$ then suggests that this quantity increases as $\tau$ decreases. Perelman presents this as a heuristic, and gives the rigorous proof separately in §7. The heuristic explains why no curvature hypothesis is needed: the extra dimensions make the space-time Ricci-flat.

5.7Huisken's formula and the direction of monotonicity

Huisken's monotonicity formula (6A.8 Curve Shortening and the First Geometric Flows) for mean curvature flow integrates the backward heat kernel centred at a space-time point $(x_0, t_0)$ over the surface:

nonincreasing in $t$, and constant exactly on self-shrinkers. The reduced volume has the same shape, with the Euclidean distance replaced by $\ell$. Perelman points out that the monotonicity runs the other way. In $\tau = t_0 - t$, Huisken's $\Theta$ is nondecreasing, while $\tilde V$ is nonincreasing. Both are nevertheless constant exactly on the self-similar shrinking solutions, and both are used in the same way: they are evaluated at very small and very large scales and compared, to show that a blow-up looks like a soliton.

5.8History

Perelman introduced $\mathcal L$-length, reduced distance and reduced volume in §§6–7 of his first preprint (2002). He described the computations as natural modifications of the classical variational theory of geodesics, with Li and Yau's 1986 paper as the closest reference, and noted that Chow and Chu had found the first geometric interpretation of Hamilton's Harnack expressions, using a degenerate metric on $M\times\mathbb{R}$, to which his construction is in a sense dual. Huisken's formula dates from 1990.

5.9Exercises

Solution

(a) $d\tau = 2s\,ds$ and $\dot\gamma = \frac{\gamma'(s)}{2s}$, so $\sqrt\tau|\dot\gamma|^2d\tau = s\frac{|\gamma'|^2}{4s^2}2s\,ds = \frac12|\gamma'|^2ds$. (b) Energy is minimised by the constant-speed line, $\gamma(s) = \frac{s}{\sqrt{\bar\tau}}x$, giving $\frac12\frac{|x|^2}{\bar\tau}\sqrt{\bar\tau}$. Then $\ell = \frac{L}{2\sqrt{\bar\tau}}$. (c) $\int(4\pi\bar\tau)^{-n/2}e^{-|x|^2/4\bar\tau}dx = 1$.

Solution

$\frac{d}{d\tau}(\sqrt\tau\dot\gamma) = \sqrt\tau\big(\ddot\gamma + \frac{1}{2\tau}\dot\gamma\big) = 0$, so $\sqrt\tau\dot\gamma = v$ is constant, $\dot\gamma = v\tau^{-1/2}$, and $\gamma = 2\sqrt\tau\,v$.

Solution

$\partial_\tau\ell = -\frac{|x|^2}{4\tau^2}$, $\nabla\ell = \frac{x}{2\tau}$, $|\nabla\ell|^2 = \frac{|x|^2}{4\tau^2}$, $\Delta\ell = \frac{n}{2\tau}$. First: $-\frac{|x|^2}{4\tau^2} - \frac{n}{2\tau} + \frac{|x|^2}{4\tau^2} + \frac{n}{2\tau} = 0$. Second: $\frac n\tau - \frac{|x|^2}{4\tau^2} + \frac{|x|^2}{4\tau^2} - \frac n\tau = 0$.

Solution

$\partial_{\bar\tau}\bar L = \frac{L}{\sqrt{\bar\tau}} + 2\sqrt{\bar\tau}\partial_{\bar\tau}L = \frac{L}{\sqrt{\bar\tau}} + 4\bar\tau R - \frac{L}{\sqrt{\bar\tau}} + \frac{2K}{\sqrt{\bar\tau}} = 4\bar\tau R + \frac{2K}{\sqrt{\bar\tau}}$. And $\Delta\bar L = 2\sqrt{\bar\tau}\Delta L \leq -4\bar\tau R + 2n - \frac{2K}{\sqrt{\bar\tau}}$. Adding gives $2n$.

Solution

(a) $R$ depends only on $\tau$, so its integral is the same for every curve; and $|\dot\gamma|^2 \geq (1 + 4\tau)\dot\theta^2$, with equality along great circles. (b) By Cauchy–Schwarz, $\theta(q)^2 = \big(\int\dot\theta\big)^2 \leq \int a\dot\theta^2\int\frac1a$ with $a = \sqrt\tau(1 + 4\tau)$, with equality when $\dot\theta \propto \frac1a$. With $s = \sqrt\tau$, $I = \int_0^{\sqrt{\bar\tau}}\frac{2\,ds}{1 + 4s^2} = \arctan(2\sqrt{\bar\tau})$ and $\int\sqrt\tau R = 12\int_0^{\sqrt{\bar\tau}}\frac{s^2\,ds}{1 + 4s^2} = 3s - \frac32\arctan2s$. (c) $\ell = \frac{1}{2\sqrt{\bar\tau}}\big(\frac{\theta^2}{\arctan2\sqrt{\bar\tau}} + 3\sqrt{\bar\tau} - \frac32\arctan2\sqrt{\bar\tau}\big) \to \frac32$, since $\theta \leq \pi$. Then $\tilde V \approx (4\pi\bar\tau)^{-3/2}(4\bar\tau)^{3/2}|S^3|e^{-3/2}$, and $|S^3| = 2\pi^2$.