The 𝓦-Entropy

The functional $\mathcal F$ of 12A.2 Ricci Flow as a Gradient Flow has no scale in it, so it cannot see shrinking solitons or measure geometry at a given size. Perelman's remedy is to insert a scale $\tau > 0$ and a Gaussian weight. The result is the $\mathcal W$-entropy

It is invariant under scaling $(g, \tau) \mapsto (cg, c\tau)$. Along the Ricci flow, with $\tau$ decreasing at unit rate and $u = (4\pi\tau)^{-n/2}e^{-f}$ solving the conjugate heat equation, it is nondecreasing, and constant exactly on shrinking gradient solitons. On Euclidean space, its positivity is exactly Gross's Gaussian logarithmic Sobolev inequality. Its minimum over $f$, $\mu(g, \tau)$, is the quantity that Perelman uses to prove noncollapsing (12A.4 κ-Noncollapsing). This monotonicity formula is the analytic heart of the proof of the Poincaré conjecture.

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

• define $\mathcal W$ and $\mu$, and check scale invariance;
• state the coupled evolution and the monotonicity formula, and explain its proof;
• show that $\mathcal W = 0$ for the Gaussian on $\mathbb{R}^n$ and connect $\mu \leq 0$ on $\mathbb{R}^n$ to the logarithmic Sobolev inequality;
• prove that shrinking breathers are shrinking gradient solitons;
• explain Perelman's statistical analogy, and its status as a heuristic.

3.1Logarithmic Sobolev inequalities

3.2The entropy and its monotonicity

Scale invariance. If $\tilde g = cg$ and $\tilde\tau = c\tau$, then $\tilde\tau\tilde R = \tau R$, $\tilde\tau|\nabla f|^2_{\tilde g} = \tau|\nabla f|^2_g$, and $(4\pi\tilde\tau)^{-n/2}d\tilde V = (4\pi\tau)^{-n/2}dV$: so $\mathcal W(cg, f, c\tau) = \mathcal W(g, f, \tau)$, and the constraint is preserved (Exercise 3.3).

The coupled evolution (Perelman I, (3.3)):

The equation for $f$ says exactly that $u = (4\pi\tau)^{-n/2}e^{-f}$ satisfies the conjugate heat equation $\square^*u = 0$ (9B.7 The Heat Equation on a Manifold), which preserves the constraint $\int u\,dV = 1$.

Perelman calls the derivation "a routine computation", and it is, given 12A.2 Ricci Flow as a Gradient Flow. The cleanest route goes through a pointwise formula that he proves later (Proposition 9.1, 12A.6 Pseudolocality). The function $v = [\tau(2\Delta f - |\nabla f|^2 + R) + f - n]u$ has integral $\mathcal W$ (integrate $\Delta f\,u$ by parts), and it satisfies $\square^*v = -2\tau|\operatorname{Ric} + \nabla^2f - \frac{1}{2\tau}g|^2u$. Since $\partial_t\,dV = -R\,dV$ and $\int\Delta v\,dV = 0$, the time derivative of $\int v\,dV$ is $-\int\square^*v\,dV$, which is the monotonicity formula (Exercise 3.6). The pointwise formula is also how Perelman localises the argument in §§9–10.

$\mu$ and $\nu$. Let

On a closed manifold the infimum defining $\mu$ is attained by a smooth $f$; this follows from work of Oscar Rothaus (1981) on logarithmic Sobolev inequalities, and uses the Sobolev embedding (4A.10 Sobolev Embeddings and Critical Exponents) with care at the logarithmic term. By monotonicity, running the conjugate heat equation backwards from a minimiser as in 12A.2 Ricci Flow as a Gradient Flow:

and $\nu(g(t))$ is nondecreasing.

3.3On Euclidean space: the Gaussian

On flat $\mathbb{R}^n$, take $f = \frac{|x|^2}{4\tau}$. Then $u = (4\pi\tau)^{-n/2}e^{-|x|^2/4\tau}$ is the heat kernel (6A.3 The Heat Equation on ℝⁿ), the constraint holds, and

using $\int|x|^2u\,dx = 2n\tau$. The Gaussian is a shrinking soliton (with $\operatorname{Ric} + \nabla^2f = \frac{1}{2\tau}g$, 11B.1 Ricci Solitons), so it is the equality case. For any other $f$ on $\mathbb{R}^n$, $\mathcal W(\delta, f, \tau) \geq 0$; Perelman notes that this is Gross's inequality after the substitution $f = \frac{|x|^2}{2} - 2\log\phi$ (at $\tau = \frac12$) and an integration by parts (Exercise 3.5). So $\mu(\mathbb{R}^n, \tau) = 0$, and Figure 3.1 shows the minimisation in a one-parameter family.

3.4On a closed manifold

Perelman shows that for any closed $(M, g)$, $\mu(g, \tau) < 0$ for small $\tau > 0$ and $\mu(g, \tau) \to 0$ as $\tau \to 0$. He sketches the proof and leaves the details, which are not hard, to the reader. Negativity: run the conjugate heat equation backwards from a delta function at a later time $\bar\tau$; along it $\mathcal W \to 0$ as the delta is approached, so by monotonicity $\mathcal W < 0$ earlier. The limit: rescaling the metric by $\frac{1}{2\tau}$ (which takes $\tau$ to $\frac12$), the manifold looks Euclidean at scale $\sqrt\tau$, and a minimiser with $\mathcal W \leq c < 0$ would produce a function on $\mathbb{R}^n$ violating Gross's inequality. So $\mu$ measures the failure of the Euclidean logarithmic Sobolev inequality at scale $\sqrt\tau$. Figure 3.2 shows an upper bound for $\mu$ on the round sphere.

3.5No shrinking breathers

This is the template "monotone quantity, invariant under the symmetries, equality case is a soliton", now at full strength. The same structure appears throughout the Path:

monotone quantity	flow	equality case
Bishop–Gromov volume ratio (9B.2 Volume Comparison)	(increasing radius)	the model space
Li–Yau and the heat entropy (6A.10 Entropy, Information and Diffusion)	heat equation	the Gaussian heat kernel
Huisken's Gaussian density (6A.8 Curve Shortening and the First Geometric Flows)	mean curvature flow	self-shrinkers
Hamilton's surface entropy (11A.7 Ricci Flow on Surfaces)	2D normalised flow	round sphere
$\mathcal F$ and $\lambda$ (12A.2 Ricci Flow as a Gradient Flow)	Ricci flow	steady gradient solitons
$\mathcal W$ and $\mu$ (this chapter)	Ricci flow	shrinking gradient solitons
reduced volume (12A.5 Reduced Distance and Reduced Volume)	Ricci flow, backwards	the Gaussian soliton

3.6Perelman's statistical analogy

In §5, Perelman interprets $\mathcal W$ through statistical mechanics. For a canonical ensemble at inverse temperature $\beta$, the partition function $Z = \int e^{-\beta E}d\omega(E)$ determines the average energy $\langle E\rangle = -\partial_\beta\log Z$, the entropy $S = \beta\langle E\rangle + \log Z$ and the fluctuation $\sigma = \partial_\beta^2\log Z$. Perelman lets $\tau$ play the part of the temperature, sets $\log Z = \int(-f + \frac n2)\,dm$ with $dm = u\,dV$, and computes formally that

so that $\mathcal W$ is minus an entropy and the monotonicity formula is the statement that the fluctuation is nonnegative, vanishing only on shrinking solitons. He explicitly sets aside the question of whether a "density of states" exists that would make this literal. It is a heuristic that explains the name and suggested the formula, not a physical model of anything, and the proofs do not use it. In the same section Perelman asks whether an ancient solution along which this entropy stays bounded must be a shrinking gradient soliton, a question close to the study of $\kappa$-solutions (12B.1 κ-Solutions).

3.7History

Gross proved the Gaussian logarithmic Sobolev inequality in 1975, and Rothaus studied minimisers of logarithmic Sobolev functionals in 1981. Bakry and Émery (1985) connected such inequalities to curvature. Perelman introduced $\mathcal W$ in §3 of his first preprint, noting in his historical remarks that the logarithmic Sobolev inequality had appeared in geometric evolution equations in work of Klaus Ecker, and that Ivey had proved the no-breathers theorem in dimension three.

3.8Exercises

Solution

$c\tau\cdot c^{-1}(|\nabla f|^2 + R) = \tau(|\nabla f|^2 + R)$; $f - n$ is unchanged; $(4\pi c\tau)^{-n/2}e^{-f}c^{n/2}dV = (4\pi\tau)^{-n/2}e^{-f}dV$. Both the integrand and the measure are invariant.

Solution

Taking logarithms, $f = \frac{x^2}{4\sigma} + \frac12\log\frac{4\pi\sigma}{4\pi\tau}$. Then $|f'|^2 = \frac{x^2}{4\sigma^2}$, and with $\int x^2u = 2\sigma$: $\mathcal W = \tau\frac{2\sigma}{4\sigma^2} + \frac{2\sigma}{4\sigma} + \frac12\log\frac\sigma\tau - 1 = \frac12\big(\frac\tau\sigma + 1 + \log\frac\sigma\tau - 2\big)$. With $s = \frac\tau\sigma$: $s - 1 - \log s \geq 0$, with equality only at $s = 1$.

Solution

$\nabla f = x - \frac{2\nabla\phi}{\phi}$, so $\frac12|\nabla f|^2u = \big(\frac{|x|^2}{2} - 2\frac{\langle x, \nabla\phi\rangle}{\phi} + \frac{2|\nabla\phi|^2}{\phi^2}\big)\phi^2\gamma$. And $(f - n)u = \big(\frac{|x|^2}{2} - \log\phi^2 - n\big)\phi^2\gamma$. Integrating by parts against the Gaussian, $\int2\phi\langle x, \nabla\phi\rangle\gamma = \int\langle x, \nabla\phi^2\rangle\gamma = \int\phi^2(|x|^2 - n)\gamma$. Collecting: $\int\big(|x|^2 - (|x|^2 - n) - n\big)\phi^2\gamma + 2\int|\nabla\phi|^2\gamma - \int\phi^2\log\phi^2\gamma = 2\int|\nabla\phi|^2d\gamma - \int\phi^2\log\phi^2d\gamma$.

Solution

$\frac{d}{dt}\int v\,dV = \int(\partial_tv - Rv)dV = \int(-\square^*v - \Delta v)dV = -\int\square^*v\,dV$, since $\int\Delta v = 0$. And $\int v = \int[\tau(2\Delta f - |\nabla f|^2 + R) + f - n]u$, with $\int\Delta f\,u = \int|\nabla f|^2u$ (integrate by parts, $\nabla u = -u\nabla f$), so $\int v = \mathcal W$. Hence $\frac{d}{dt}\mathcal W = \int2\tau|\cdots|^2u \geq 0$.

Solution

With constant $f$, $|\nabla f|^2 = 0$, $R = n(n - 1)$, and $\int u = 1$, so $\mathcal W = \tau n(n - 1) + f - n$. For $n = 2$, $|S^2| = 4\pi$ and $f = \log\frac{4\pi}{4\pi\tau} = -\log\tau$. The derivative $2 - \frac1\tau$ vanishes at $\tau = \frac12$, where the value is $1 + \log2 - 2 = \log2 - 1 < 0$. Since $\mu$ is an infimum, $\mu(S^2, \frac12) \leq \log2 - 1 < 0$.