Ricci Flow as a Gradient Flow

Perelman's first idea is that the Ricci flow is a gradient flow. Hamilton had looked for a functional whose gradient flow is the Ricci flow and not found one, and there is a good reason: in the obvious sense there is none. Perelman found one in a less obvious sense. Add an auxiliary function $f$ to the metric, consider

and fix the measure $dm = e^{-f}\,dV$. Then the gradient flow of $\mathcal F$ is the Ricci flow, modified by a diffeomorphism. Along it, $\mathcal F$ is nondecreasing, with equality exactly on steady gradient solitons. Its minimum over $f$, $\lambda(g)$, is the first eigenvalue of the operator $-4\Delta + R$, and it too is nondecreasing along the flow. As a first application: the Ricci flow has no nontrivial breathers, solutions that return to themselves up to diffeomorphism, other than solitons.

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

• compute the first variation of $\mathcal F$ and identify its gradient for a fixed measure;
• derive the coupled flow, and relate it by a diffeomorphism to the Ricci flow coupled with the conjugate heat equation;
• prove the monotonicity formula $\frac{d}{dt}\mathcal F = 2\int|\operatorname{Ric} + \nabla^2f|^2\,dm$;
• define $\lambda(g)$, identify it as an eigenvalue, and show it is nondecreasing;
• prove that steady breathers on closed manifolds are Ricci-flat.

2.1Going downhill

2.2The first variation

Let $\delta g = v$ and $\delta f = h$, with $v = \operatorname{tr}v$. Using the variation of the scalar curvature from 11A.2 How Curvature Evolves, $\delta R = -\Delta v + \nabla^i\nabla^jv_{ij} - \langle v, \operatorname{Ric}\rangle$, and $\delta(dV) = \frac v2\,dV$, Perelman computes (Exercise 2.4)

The factor $\frac v2 - h$ vanishes exactly when the measure $dm = e^{-f}dV$ is unchanged. So if the measure is held fixed, the variation is $\int-v_{ij}(R_{ij} + \nabla_i\nabla_jf)\,dm$: the gradient of $\mathcal F$, for the $L^2(dm)$ metric on symmetric tensors, is $-(\operatorname{Ric} + \nabla^2f)$.

2.3The gradient flow and the Ricci flow

The gradient flow of $\mathcal F$ with $dm$ fixed is

the second equation being what keeps $e^{-f}dV$ fixed. Since $-2\nabla^2f = -\mathcal L_{\nabla f}g$, this differs from the Ricci flow only by the Lie derivative along $\nabla f$. Pulling back by the diffeomorphisms generated by $\nabla f$ (as in DeTurck's trick, 11A.3 Short-Time Existence and Uniqueness) turns the system into

The equation for $f$ is the conjugate heat equation for $u = e^{-f}$: $\square^*u = (-\partial_t - \Delta + R)u = 0$ (Exercise 2.5, 9B.7 The Heat Equation on a Manifold). So the gradient flow is equivalent to the Ricci flow, coupled with a solution of the conjugate heat equation running backwards in time, which keeps $\int u\,dV$ constant. Perelman remarks that different choices of $dm$ give the same flow up to diffeomorphism, as different choices of gauge.

2.4The eigenvalue $\lambda(g)$

Define

With $w = e^{-f/2}$, $\mathcal F = \int(Rw^2 + 4|\nabla w|^2)\,dV$ and the constraint is $\int w^2 = 1$, so $\lambda(g)$ is the smallest eigenvalue of $-4\Delta + R$ (Exercise 2.6). It is attained by a positive eigenfunction $w$ (by the spectral theory of 4A.7 Compact Operators and Spectra with the variational argument of 4A.6 Weak Convergence and the Direct Method), smooth by elliptic regularity (6A.5 Weak Solutions and Elliptic Regularity). For a homogeneous metric, where $R$ is constant, $\lambda = R$, attained by constants.

Figure 2.2 shows $\lambda = R$ increasing along the Berger sphere flow of 11A.6 Hamilton’s 1982 Theorem.

2.5No steady breathers

A breather is a solution with $g(t_2) = c\,\phi^*g(t_1)$ for some $t_1 < t_2$, $c > 0$ and diffeomorphism $\phi$. It is steady, shrinking or expanding as $c = 1$, $c < 1$ or $c > 1$. Solitons are breathers; the question is whether there are others.

The expanding case follows similarly from the scale-invariant quantity $\lambda(g)\operatorname{Vol}(g)^{2/n}$, which is nondecreasing when $\lambda \leq 0$. The shrinking case, which is the one that matters for singularities, needs a functional that knows about scale: the $\mathcal W$-entropy of 12A.3 The 𝓦-Entropy.

2.6History

Perelman introduced $\mathcal F$ in §1 of his first preprint. The functional had appeared in physics as the low-energy effective action of string theory, with $f$ the dilaton; Perelman notes the connection in the historical remark of §1, along with the Bakry–Émery Ricci tensor $\operatorname{Ric} + \nabla^2f$ of a manifold with a smooth measure, studied by Dominique Bakry and Michel Émery in the 1980s. The identification of $\lambda$ with the bottom of the spectrum of $-4\Delta + R$ is in §2, together with the no-breathers theorem for the steady and expanding cases, which had been known in special cases (Ivey 1993, Hamilton).

2.7Exercises

Solution

The terms with derivatives of $v$: $\int e^{-f}(-\Delta v + \nabla^i\nabla^jv_{ij}) = \int(-v\Delta e^{-f} + v_{ij}\nabla_i\nabla_je^{-f})$, with $\nabla_i\nabla_je^{-f} = (-\nabla_i\nabla_jf + \nabla_if\nabla_jf)e^{-f}$ and $\Delta e^{-f} = (-\Delta f + |\nabla f|^2)e^{-f}$. So they give $\int e^{-f}[v(\Delta f - |\nabla f|^2) - v_{ij}\nabla_i\nabla_jf + v_{ij}\nabla_if\nabla_jf]$. The $h$-term: $\int2\langle\nabla f, \nabla h\rangle e^{-f} = \int2h(-\Delta f + |\nabla f|^2)e^{-f}$. Adding $-v_{ij}R_{ij} - v_{ij}\nabla_if\nabla_jf$ and $(R + |\nabla f|^2)(\frac v2 - h)$, the $v_{ij}\nabla_if\nabla_jf$ terms cancel, and the rest regroup as $-v_{ij}(R_{ij} + \nabla_i\nabla_jf) + (\frac v2 - h)(2\Delta f - |\nabla f|^2 + R)$.

Solution

$\partial_tu = -(\partial_tf)u$ and $\Delta u = (-\Delta f + |\nabla f|^2)u$, so $-\partial_tu - \Delta u + Ru = (\partial_tf + \Delta f - |\nabla f|^2 + R)u$. And $\frac{d}{dt}\int u\,dV = \int(\partial_tu - Ru)\,dV = \int-\Delta u\,dV = 0$, using $\partial_t\,dV = -R\,dV$.

Solution

$\nabla w = -\frac12w\nabla f$, so $4|\nabla w|^2 = w^2|\nabla f|^2 = e^{-f}|\nabla f|^2$. The infimum of the Rayleigh quotient $\int(4|\nabla w|^2 + Rw^2)$ over $\int w^2 = 1$ is the bottom of the spectrum of $-4\Delta + R$ (4A.7 Compact Operators and Spectra). If $R$ is constant, $\int(4|\nabla w|^2 + Rw^2) \geq R$ with equality for constant $w$.

Solution

(a) Integrate by parts. (b) $\frac{d}{dt}\mathcal F \geq \frac2n\int(R + \Delta f)^2dm \geq \frac2n\big(\int(R + \Delta f)dm\big)^2 = \frac2n\mathcal F^2$. (c) If $\mathcal F(0) > 0$, comparison with $y' = \frac2ny^2$ shows $\mathcal F$ blows up before time $\frac{n}{2\mathcal F(0)}$, so $T < \frac{n}{2\mathcal F(0)}$, i.e. $\mathcal F(0) \leq \frac{n}{2T}$; if $\mathcal F(0) \leq 0$ the bound is trivial.

Solution

(i) $\lambda(g(t))$ is nondecreasing. (ii) $\lambda(\phi^*g) = \lambda(g)$ for diffeomorphisms. (iii) Constancy forces $\operatorname{Ric} + \nabla^2f = 0$. For a shrinking breather, $g(t_2) = c\phi^*g(t_1)$ with $c < 1$, and $\lambda(cg) = c^{-1}\lambda(g)$, so $\lambda$ changes under scaling and the comparison fails. A quantity invariant under scaling, which treats $(g, \tau)$ and $(cg, c\tau)$ alike, would fix it: Perelman's $\mathcal W(g, f, \tau)$, whose equality case is a shrinking soliton (12A.3 The 𝓦-Entropy).