The Asymptotic Soliton and Three-Dimensional Shrinkers

12B.1 κ-Solutions defined $\kappa$-solutions and stated the first structural fact about them: run backward in time and rescaled by the elapsed time, a $\kappa$-solution looks like a gradient shrinking soliton, and in dimension three there are only three possible such solitons, the round sphere (and its quotients), the round cylinder $S^2\times\mathbb{R}$ and its $\mathbb{Z}_2$ quotient. That chapter gave the idea of the proof in five lines. This chapter gives the proof, following Morgan and Tian's chapter 9, which spends forty pages on it. It is worth the detail, for two reasons. The asymptotic soliton is the single most important tool in the analysis of $\kappa$-solutions: universal noncollapsing, compactness and the neck-and-cap structure of 12B.3 The Structure of κ-Solutions and 12B.4 Compactness and Universal Estimates for κ-Solutions all rest on it. And the proof is the clearest example in the whole of Perelman's work of the blow-down argument: a monotone, scale-invariant quantity (here the reduced volume of 12A.7 Monotonicity of the Reduced Volume) is bounded and therefore has a limit; in the limit it is constant; constancy forces the equality case; and the equality case is a soliton (6A.13 Self-Similar Variables and Monotonicity).

The second half of the chapter classifies the solitons that can occur. Here the work is Riemannian geometry: a gradient shrinking soliton with bounded nonnegative curvature in dimension three is compact and round, or splits off a line, or is noncompact with positive curvature, and Perelman's Lemma 1.2 rules out the third case by an elegant argument with Gauss–Bonnet on the level sets of the potential.

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

• derive, from Hamilton's Harnack inequality, the bounds $|\nabla\ell|^2 + R \leq \frac{3\ell}{\tau}$ and $-\frac{2\ell}\tau \leq \partial_\tau\ell \leq \frac\ell\tau$ on a $\kappa$-solution;
• use them to control $\ell$ and the curvature near the minimum points $q(\tau)$, and produce a limit of the rescaled flows by Hamilton's compactness theorem;
• explain why the limit of the reduced distances satisfies Perelman's inequalities with equality, and why constancy of the reduced volume is the reason;
• prove Perelman's formula $\square^*v = -2\tau|\operatorname{Ric} + \nabla^2f - \frac{1}{2\tau}g|^2u$ (Proposition 9.1 of the first preprint), and deduce the soliton equation from $v = 0$;
• prove that the limit is not flat;
• state and use the splitting-at-infinity proposition and Toponogov's splitting;
• prove the classification of gradient shrinking solitons in dimensions two and three, including the identity $\nabla R = 2\operatorname{Ric}(\nabla f, \cdot)$ and Perelman's Gauss–Bonnet argument;
• deduce that every three-dimensional asymptotic soliton is a $\kappa$-solution of one of the three types.

2.1What Hamilton's Harnack inequality says about $\ell$

Let $(M, g(t))$, $-\infty < t \leq 0$, be a $\kappa$-solution, fix $p \in M$, and let $\ell(q, \tau)$ be the reduced distance from $x = (p, 0)$, with $\tau = -t$. On a general Ricci flow the inequalities for $\ell$ (12A.6 𝓛-Geodesics and the Variational Formulas) involve the unknown $K = \int_0^\tau\sigma^{3/2}H(X)\,d\sigma$, where $H$ is Hamilton's trace Harnack expression. On a flow with nonnegative curvature operator, Hamilton's Harnack inequality bounds $H$ from below, and the unknown becomes a bound.

So on a $\kappa$-solution the reduced distance controls the scalar curvature and its own gradient at the natural scale $\sqrt\tau$, and changes in $\tau$ at most like a power: integrating, $\ell(q, \tau)$ changes by at most a bounded factor between $\tau$ and $2\tau$. Perelman's (7.16) of §7.2 (12A.6 𝓛-Geodesics and the Variational Formulas) is the same statement.

2.2The limit of the rescalings

By 12A.7 Monotonicity of the Reduced Volume, for each $\tau$ there is a point $q(\tau)$ where $\ell(\cdot, \tau)$ attains its minimum, with $\ell(q(\tau), \tau) \leq \frac n2$. Take $\bar\tau_k \to \infty$ and rescale:

so that the time $-\bar\tau_k$ becomes $-1$ and $q_k$ is a minimum point of the rescaled reduced distance $\ell_k$ at $\tau = 1$, with $\ell_k(q_k, 1) \leq \frac n2$ (the reduced distance is scale invariant, 12A.7 Monotonicity of the Reduced Volume).

Bounds at the base points. Integrating $\partial_\tau\ell_k \geq -\frac{2\ell_k}{\tau}$ from $\tau$ to $1$ and $\partial_\tau\ell_k \leq \frac{\ell_k}{\tau}$ from $1$ to $\tau$ gives (Morgan–Tian, Claim 9.15)

Bounds nearby. Since $|\nabla\sqrt{\ell_k}| = \frac{|\nabla\ell_k|}{2\sqrt{\ell_k}} \leq \frac12\sqrt{3/\tau}$, $\sqrt{\ell_k}$ is Lipschitz with a constant depending only on $\tau$, so (their Corollary 9.16)

bounded on every ball $B_{g_k(-\tau)}(q_k, A)$, uniformly in $k$. Then $R \leq \frac{3\ell}{\tau}$ bounds the scalar curvature there, and since $\partial_tR \geq 0$ on a $\kappa$-solution (12B.1 κ-Solutions), the bound holds at all earlier times too. Nonnegative curvature operator turns a bound on $R$ into a bound on $|\operatorname{Rm}|$. The flows are $\kappa$-noncollapsed at all scales, so Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows) gives a subsequence converging smoothly, on compact subsets, to a limit flow

complete for each $t$, with nonnegative curvature operator, $\kappa$-noncollapsed, with $\partial_tR \geq 0$ (Morgan–Tian, Corollary 9.17). It is not yet known to have bounded curvature on each time slice, which is why Morgan and Tian are careful not to call it a $\kappa$-solution until the classification below.

The limit of the reduced distances. The $\ell_k$ are uniformly locally bounded and uniformly locally Lipschitz in space-time (by the gradient bound and the bound on $\partial_\tau\ell$), so a subsequence converges locally uniformly, and weakly in $W^{1,2}_{\mathrm{loc}}$, to a locally Lipschitz function $\ell_\infty$ on $M_\infty\times(0, \infty)$ (in $\tau = -t$). It is not claimed to be the reduced distance of the limit flow from any point; it is simply the limit of the rescaled functions.

2.3Constancy of the reduced volume, and the equality case

The reduced volume of $g$ based at $x$, $\tilde V(\tau)$, is nonincreasing and positive, so it has a limit $V_\infty$ as $\tau \to \infty$. Since $g$ is not flat, $V_\infty < 1$ (12A.7 Monotonicity of the Reduced Volume: equality $\tilde V \equiv 1$ forces flat space). By scale invariance, the reduced volume of the $k$-th rescaled flow at $\tau$ is $\tilde V(\bar\tau_k\tau)$, which tends to $V_\infty$ for every fixed $\tau$ (Morgan–Tian, Corollary 9.14). With some work (their Lemma 9.25 bounds $\ell_k$ below by a multiple of the squared distance, giving the Gaussian decay needed for dominated convergence) one can pass to the limit in the integral:

The limit reduced volume is constant in $\tau$. That is the whole point of rescaling by $\bar\tau_k$: monotone quantities become constant in the limit.

Now recall that, for the $\ell_k$, Perelman's inequality (7.13) holds in the sense of distributions:

(Morgan–Tian, Theorem 7.13; 12A.7 Monotonicity of the Reduced Volume). Passing to the limit needs one delicate point: $|\nabla\ell_k|^2$ converges to $|\nabla\ell_\infty|^2$ only weakly in general, and weak convergence of gradients does not give convergence of their squares. Morgan and Tian prove the convergence (their Lemma 9.21) using the other inequality, (7.14), which bounds $\Delta\ell_k$ from above and so prevents the gradients from oscillating. The result is a distributional inequality $\mathcal D_\infty \geq 0$ for $\ell_\infty$. On the other hand, the $\tau$-derivative of the constant $\tilde V_\infty$ is exactly the integral of $\mathcal D_\infty$ over $M_\infty$: the $\Delta$ term integrates to zero, and $\frac{d}{d\tau}\big(\tau^{-n/2}e^{-\ell}dV\big) = \big(-\frac{n}{2\tau} - \partial_\tau\ell + R\big)\tau^{-n/2}e^{-\ell}dV$ (Exercise 2.11). So

and a nonnegative distribution with total integral zero vanishes (Morgan–Tian argue with a cut-off $\varphi$: $0 = \mathcal D(1) = \mathcal D(\varphi) + \mathcal D(1 - \varphi)$ with both terms nonnegative). Hence $\ell_\infty$ is a weak solution of

a semilinear heat equation; by parabolic regularity (6A.7 Parabolic Regularity, 6B.11 Linear Parabolic Equations in Full) $\ell_\infty$ is smooth, and the companion inequality (7.14) then holds with equality too (their Proposition 9.20):

2.4Perelman's formula for $\square^*v$

The two equalities for $\ell_\infty$ say, in Perelman's language, that $u = (4\pi\tau)^{-n/2}e^{-\ell_\infty}$ solves the conjugate heat equation and that the associated function $v$ vanishes. The soliton equation then follows from Perelman's Proposition 9.1, whose "routine computation" Morgan and Tian write out as their Lemma 9.33. Here it is in full.

Integrating the formula over a closed manifold gives the monotonicity of $\mathcal W$ (12A.3 The 𝓦-Entropy), since $\int v\,dV = \mathcal W$ and $\frac{d}{dt}\int v = -\int\square^*v$ when the volume form evolves; for the conjugate heat kernel it gives $v \leq 0$ (12A.9 Pseudolocality). Here it is used in the opposite direction. For $f = \ell_\infty$ the first equality of the previous section is exactly the condition that $u$ solves the conjugate heat equation (in the variable $\tau$), and the second says $v = 0$. Then $\square^*v = 0$, so

the limit is a gradient shrinking soliton, with potential $\ell_\infty$ (Morgan–Tian, Proposition 9.32).

It is not flat. If $g_\infty(t)$ were flat for some $t$, the soliton equation would read $\nabla^2\ell_\infty = \frac{1}{2\tau}g$ on a flat manifold; lifting to the universal cover $\mathbb{R}^n$, $\tilde\ell_\infty - \frac{|x|^2}{4\tau}$ would be affine, which is not invariant under any nontrivial group of translations, so the manifold is $\mathbb{R}^n$ itself and $\ell_\infty = \frac{|x - a|^2}{4\tau} + b$. Morgan and Tian then identify the reduced volume of such a limit with that of flat Euclidean space, which is $1$ in our normalisation (their Theorem 7.26 and Proposition 7.27: on a complete flow of bounded curvature, reduced volume equal to the Euclidean value happens only for flat $\mathbb{R}^n$), contradicting $V_\infty < 1$ (their Claim 9.36). This proves:

2.5Splitting at infinity

The classification of the possible limits uses one more tool, which also reappears in 12B.4 Compactness and Universal Estimates for κ-Solutions: a noncompact manifold of nonnegative curvature on which the scalar curvature decays more slowly than the inverse square of the distance looks, at points far out and at the scale of the curvature, like a product with a line.

2.6The classification of shrinkers in dimensions two and three

The soliton flow. Bounded curvature bounds $\nabla^2f$, so $|\nabla f|$ grows at most linearly, the flow $\Phi_s$ of $\nabla f$ exists for all $s$, and $G(t) = |t|\Phi^*_{-\log|t|}g$ satisfies $\partial_tG = -2\operatorname{Ric}(G)$ (Exercise 2.10): a soliton generates its own Ricci flow by scaling and diffeomorphism.

Case 1: compact with positive curvature. In dimension three, Hamilton's theorem (11A.11 Hamilton’s 1982 Theorem) says that the ratio of the largest to the smallest sectional curvature tends to $1$ as the flow becomes singular. That ratio is invariant under scaling and diffeomorphism, and $G(t)$ is, up to both, the same metric $g$ at every time. So the ratio is $1$ for $g$: it is round. In dimension two, Hamilton's work on surfaces gives the same conclusion (11A.12 Ricci Flow on Surfaces).

Case 2: not strictly positive curvature. By Hamilton's strong maximum principle (11A.8 The Strong Maximum Principle and Splitting), a one- or two-sheeted cover splits as a product of a surface of positive curvature with a flat one-manifold, so $n = 3$. The soliton equation in the direction $Y$ of the flat factor gives $\nabla^2f(Y, Y) = \frac12$, so $f$ is strictly convex along it, which is impossible on a circle; so the flat factor is $\mathbb{R}$. The surface factor is a two-dimensional shrinker (and a two-dimensional $\kappa$-solution), round by the two-dimensional case, and the three types (2) and (3) result.

Case 3: noncompact with positive curvature. This is Perelman's Lemma 1.2: it does not happen. The proof has four steps.

Step 1: $\nabla R = 2\operatorname{Ric}(\nabla f, \cdot)$. Trace the soliton equation: $R + \Delta f = \frac n2$, so $\nabla R = -\nabla\Delta f$. Take its divergence: $\operatorname{div}\operatorname{Ric} + \operatorname{div}\nabla^2f = 0$. By the contracted Bianchi identity $\operatorname{div}\operatorname{Ric} = \frac12\nabla R$, and by the Bochner commutation $\operatorname{div}\nabla^2f = \nabla\Delta f + \operatorname{Ric}(\nabla f, \cdot)$. So $\frac12\nabla R - \nabla R + \operatorname{Ric}(\nabla f, \cdot) = 0$ (Exercise 2.9). In particular $R$ increases along the flow lines of $\nabla f$, since $\langle\nabla R, \nabla f\rangle = 2\operatorname{Ric}(\nabla f, \nabla f) \geq 0$.

Step 2: $f$ grows quadratically and $\nabla f$ points outward. Along a unit-speed minimal geodesic $\gamma$ from a fixed point $p$, of length $\bar s$, the second variation of energy with the fields equal to $sE_i$ on $[0, 1]$, $E_i$ on $[1, \bar s - 1]$ and $(\bar s - s)E_i$ on $[\bar s - 1, \bar s]$ ($E_i$ parallel and orthonormal to $\gamma'$) gives $\int_0^{\bar s}\operatorname{Ric}(\gamma', \gamma')\,ds \leq C$, independent of $\gamma$ (Morgan–Tian, Claim 9.47). By the soliton equation, $\frac{d^2}{ds^2}f(\gamma(s)) = \nabla^2f(\gamma', \gamma') = \frac12 - \operatorname{Ric}(\gamma', \gamma')$, so $\frac{d}{ds}f(\gamma(s)) \geq \frac s2 - C'$; and similarly, using $|\operatorname{Ric}(X, \cdot)|^2 \leq R\operatorname{Ric}(X, X)$ (Claim 9.48) and Cauchy–Schwarz, the components of $\nabla f$ orthogonal to $\gamma'$ are $O(\sqrt s)$. So far out, $\nabla f$ is nearly parallel to the radial direction, $|\nabla f| \to \infty$, $f$ is proper, and $f$ has no critical points outside a compact set.

Step 3: the end is asymptotic to a cylinder with $R = 1$. Since $R$ increases along $\nabla f$ and is bounded, there are points $p_k \to \infty$ with $R(p_k) \to \limsup R > 0$, and then $R(p_k)d(p, p_k)^2 \to \infty$: the splitting proposition applies, and rescaled limits at infinity are $N^2\times\mathbb{R}$ with $N$ a two-dimensional $\kappa$-solution, which by the two-dimensional classification (12B.1 κ-Solutions; Morgan–Tian, Corollary 9.50: two-dimensional $\kappa$-solutions are round, by Hamilton's surface entropy) is a round $S^2$ (not $\mathbb{RP}^2$, since $M$, being noncompact with positive curvature, is diffeomorphic to $\mathbb{R}^3$ by the soul theorem and contains no $\mathbb{RP}^2$). Since the shrinking round $S^2$ exists for all $t < 0$, its scalar curvature at $t = -1$ is at most $1$; since $R$ is bounded below by a positive constant at infinity on $M$ (it increases along $\nabla f$) and the flow is self-similar, the cross-sections must become singular exactly at $t = 0$, so their scalar curvature at $t = -1$ is exactly $1$ (Claim 9.51). Hence $R < 1$ on $M$ near infinity, and the level sets $N_u = f^{-1}(u)$ converge, in $C^1$, to the round sphere of scalar curvature $1$, of area $8\pi$.

Step 4: Gauss–Bonnet. For large $u$ the level sets are convex: for unit $Y$, $\nabla^2f(Y, Y) = \frac12 - \operatorname{Ric}(Y, Y) \geq \frac12 - \frac R2 > 0$ (in nonnegative curvature in dimension three, $\operatorname{Ric}(Y, Y) \leq \frac R2$). So their areas increase with $u$ to the limit $8\pi$, and $\operatorname{Area}(N_u) < 8\pi$. On the other hand, by the Gauss equation with the shape operator $S = \nabla^2f|_{TN}/|\nabla f|$ and an orthonormal frame $e_1, e_2, e_3 = \nabla f/|\nabla f|$,

and $\det\big(\frac12g - \operatorname{Ric}\big)|_{TN} \leq \frac14\big(1 - R + \operatorname{Ric}(e_3, e_3)\big)^2$ by the arithmetic–geometric mean inequality. Since $|\nabla f| \to \infty$, for large $u$

so the Gauss curvature of $N_u$ is less than $\frac12$. But $N_u$ is a sphere, and Gauss–Bonnet gives $4\pi = \int_{N_u}K_N < \frac12\operatorname{Area}(N_u) < 4\pi$. Contradiction.

The boundedness of curvature is not automatic and is proved by the same ingredients: if the curvature were unbounded, splitting at infinity would produce round $\varepsilon$-necks of arbitrarily small scale far out in a positively curved manifold diffeomorphic to $\mathbb{R}^3$, which is impossible (a complete positively curved three-manifold does not contain $\varepsilon$-necks of arbitrarily small scale: Morgan–Tian, Proposition 2.19, whose proof uses the soul and the fact that the central sphere of every $\varepsilon$-neck far out separates the soul from the end). Then Theorem Theorem 2.5 applies to a single time slice, and uniqueness of the flow identifies the whole limit.

2.7History

Perelman stated the existence of the asymptotic soliton as Proposition 11.2 of his first preprint (2002), with a two-paragraph proof, and ruled out noncompact positively curved three-dimensional shrinkers as Lemma 1.2 of the second (2003). The formula of Proposition 9.1 appeared in §9 of the first preprint. Hamilton had classified compact two-dimensional shrinkers (1988) and proved the strong maximum principle for the curvature operator (1986); Cheeger and Gromoll's splitting theorem dates from 1971, Toponogov's comparison from 1959. Morgan and Tian's chapter 9 (2007) and Kleiner and Lott's sections 39–51 give the complete proofs. Later work, building on Perelman's argument, weakened the curvature and noncollapsing assumptions in this classification (for example Ni and Wallach, 2008, and Cao, Chen and Zhu, 2008).

2.8Exercises

Solution

$\mathcal L(\gamma_{\tau'}) = L(q, \tau) + \int_\tau^{\tau'}\sqrt\sigma R(q, \sigma)\,d\sigma$ (the constant path has $\dot\gamma = 0$). Differentiating $\frac{\mathcal L(\gamma_{\tau'})}{2\sqrt{\tau'}}$ at $\tau' = \tau$: $-\frac{L}{4\tau^{3/2}} + \frac{\sqrt\tau R}{2\sqrt\tau} = -\frac{\ell}{2\tau} + \frac R2$. Since $\ell(q, \tau') \leq \frac{\mathcal L(\gamma_{\tau'})}{2\sqrt{\tau'}}$ with equality at $\tau' = \tau$, the right derivative of $\ell(q, \cdot)$ at $\tau$ is at most $-\frac{\ell}{2\tau} + \frac R2 \leq -\frac{\ell}{2\tau} + \frac{3\ell}{2\tau} = \frac\ell\tau$.

Solution

(a) $\operatorname{Ric}(g(t)) = \operatorname{Ric}_h = \frac12h$ (scale invariant), and $\partial_tg = \partial_t(-t)h = -h = -2\cdot\frac12h$. (b) $\nabla^2f = \frac{1}{2|t|}ds^2$, so $\operatorname{Ric} + \nabla^2f = \frac12h + \frac{1}{2|t|}ds^2 = \frac{1}{2|t|}(|t|h + ds^2)$. (c) At $t = -1$, $R = 2K = 1$. The potential is the reduced distance in the $\mathbb{R}$ direction, $\frac{s^2}{4\tau}$, the Gaussian part of the cylinder: unbounded potentials are typical of noncompact shrinkers and are not a problem; boundedness is required only of the curvature.

Solution

Trace: $R + \Delta f = \frac n2$, so $\nabla R = -\nabla\Delta f$. Divergence of the equation: $\frac12\nabla R + \nabla\Delta f + \operatorname{Ric}(\nabla f) = 0$, so $\frac12\nabla R - \nabla R + \operatorname{Ric}(\nabla f) = 0$. For the second part, $\nabla|\nabla f|^2 = 2\nabla^2f(\nabla f, \cdot) = \nabla f - 2\operatorname{Ric}(\nabla f, \cdot)$ by the soliton equation, so $\nabla(R + |\nabla f|^2 - f) = 2\operatorname{Ric}(\nabla f) + \nabla f - 2\operatorname{Ric}(\nabla f) - \nabla f = 0$.

Solution

With $s(t) = -\log|t|$, $\frac{ds}{dt} = \frac{1}{|t|}$ for $t < 0$. Then $\partial_tG = -\Phi_s^*g + |t|\cdot\frac{1}{|t|}\Phi_s^*(2\nabla^2f) = \Phi_s^*(-g + 2\nabla^2f) = \Phi_s^*(-2\operatorname{Ric}(g)) = -2\operatorname{Ric}(\Phi_s^*g) = -2\operatorname{Ric}(G)$, using that $\operatorname{Ric}$ is invariant under scaling and natural under diffeomorphisms.

Solution

$\frac{d}{d\tau}\big((4\pi\tau)^{-n/2}e^{-\ell}dV\big) = \big(-\frac{n}{2\tau} - \partial_\tau\ell + R\big)(4\pi\tau)^{-n/2}e^{-\ell}dV$. Also $\int(|\nabla\ell|^2 - \Delta\ell)e^{-\ell} = \int\Delta(e^{-\ell}) = 0$. Adding zero gives the formula. If $\tilde V$ is constant, the integral of the nonnegative quantity (7.13) against the positive weight vanishes, so (7.13) is an equality almost everywhere, and everywhere by continuity.

Solution

$\int K\,dA < \frac12A < 4\pi$, while Gauss–Bonnet gives $\int K\,dA = 2\pi\chi(S^2) = 4\pi$. The area bound comes from convexity: convex level sets have area increasing in $u$ (each encloses the previous one, and area is monotone for nested convex surfaces), with limit $8\pi$, the area of the round cross-section of scalar curvature $1$. The curvature bound comes from the Gauss equation together with $R < 1$ near infinity and $|\nabla f| \to \infty$.