κ-Solutions

Book 12A ended with a guarantee. Blow up a three-dimensional Ricci flow at a singularity and you get a limit, and the limit is ancient, $\kappa$-noncollapsed at all scales and has nonnegative curvature. Perelman gave such limits a name: $\kappa$-solutions. They are the possible shapes of a singularity, and Book 12B is about knowing them well enough to cut them out. This chapter defines $\kappa$-solutions, lists the known examples, and proves the first structural fact about them. Run backwards in time and rescaled, every $\kappa$-solution looks like a shrinking soliton, its asymptotic soliton, and in dimension three there are only three of those.

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

• state the definition of a $\kappa$-solution and explain why blow-up limits of three-dimensional flows are $\kappa$-solutions;
• list the examples: round spheres and their quotients, the round cylinder and its $\mathbb{Z}_2$ quotient, the Bryant soliton, and Perelman's compact example;
• explain why the cigar times a line, flat space and thin cylinders are not $\kappa$-solutions;
• describe how the reduced volume produces the asymptotic soliton, and list the asymptotic solitons in dimension three;
• sort $\kappa$-solutions by their asymptotic solitons.

1.1A short list of shapes

1.2The definition

In dimension three, nonnegative curvature operator is the same as nonnegative sectional curvature, which is how Perelman states it in his second preprint. He calls these "ancient $\kappa$-solutions"; the expositions shorten this to $\kappa$-solutions. Two consequences are used constantly. By Hamilton's Harnack inequality for ancient solutions with nonnegative curvature operator (11B.2 Ancient Solutions and the Harnack Inequality), $\partial_tR \geq 0$: scalar curvature only increases with time, so the curvature at a time $t_0$ bounds it at all earlier times. And curvature controls everything else, since $0 \leq |\operatorname{Rm}| \leq C(n)R$ when the curvature operator is nonnegative.

Where they come from. Take a Ricci flow on a closed 3-manifold, a sequence of points $(p_k, t_k)$ with $Q_k = |\operatorname{Rm}|(p_k, t_k) \to \infty$, and the curvature bound of 12A.4 κ-Noncollapsing's corollary. The rescaled flows converge to a limit that is complete and ancient, with bounded curvature and $|\operatorname{Rm}| = 1$ at the base point, so it is not flat. It is $\kappa$-noncollapsed at all scales by the corollary, and its curvature is nonnegative by the Hamilton–Ivey estimate (11A.5 Hamilton–Ivey Pinching): at a point of curvature $Q_k$, negative sectional curvature is at most a small multiple of $Q_k$, and the multiple tends to zero as $Q_k \to \infty$. The limit is a $\kappa$-solution.

1.3Examples

$\kappa$-solution	type	compact?	asymptotic soliton
round $S^3$, and $S^3/\Gamma$	shrinking soliton	yes	itself
round cylinder $S^2\times\mathbb{R}$	shrinking soliton	no	itself
$(S^2\times\mathbb{R})/\mathbb{Z}_2$, containing $\mathbb{RP}^2$	shrinking soliton	no	itself
Bryant soliton (11B.1 Ricci Solitons)	steady soliton	no	round cylinder
Perelman's compact example on $S^3$	not a soliton	yes	round cylinder

The shrinking round sphere of radius $r(t)$ with $r(t)^2 = -4t$ is ancient, with $R = \frac{6}{r^2} \to 0$ as $t \to -\infty$; the round cylinder, with $r(t)^2 = -2t$ for its $S^2$ factor, likewise. In $(S^2\times\mathbb{R})/\mathbb{Z}_2$ the group acts by the antipodal map on $S^2$ and by $s \mapsto -s$ on the line; the quotient is a twisted line bundle over $\mathbb{RP}^2$ with one end. The Bryant soliton is a steady soliton, so it is defined for all time, and at distance $s$ from its tip its curvature is about $\frac1s$ and its cross-sections have radius about $\sqrt s$: balls of radius $r \lesssim \sqrt s$ with controlled curvature are nearly Euclidean, and it is $\kappa$-noncollapsed (Exercise 1.5). Figure 1.1 draws the profiles.

Perelman's compact example (II, §1.4). Start with a round cylinder $S^2\times[0, L]$ of radius $1$, capped by two hemispherical caps, a metric on $S^3$. The curvature is nonnegative, and by Hamilton's work (11A.6 Hamilton’s 1982 Theorem) the flow shrinks this metric to a round point after a time comparable to $1$. Translate time so that the singular time is $0$, and choose the scale so that at $t = -1$ the ratio of largest to smallest sectional curvature is $1 + \varepsilon$. Perelman shows that the solutions are $\kappa$-noncollapsed with $\kappa$ independent of $L$, using reduced volume, and that the starting time tends to $-\infty$ as $L \to \infty$, using Hamilton's Harnack inequality. A subsequence converges as $L \to \infty$ to a $\kappa$-solution on $S^3$ that looks, in the far past, like a long thin cylinder with two caps. It is not a soliton. This example also answered the question Perelman had asked at the end of his first preprint's §5 (12A.3 The 𝓦-Entropy), whether bounded entropy along an ancient solution forces a shrinking soliton: it does not.

Non-examples. Flat $\mathbb{R}^3$ is excluded by definition. $S^1\times\mathbb{R}^2$ and other flat quotients are flat and also collapsed. The cigar times a line is ancient, complete, has nonnegative curvature and is not flat, but it is collapsed at large scales (9B.3 Collapsing and Noncollapsing), so it is not a $\kappa$-solution for any $\kappa$. This is precisely how noncollapsing removed the cigar from the list (12A.4 κ-Noncollapsing). In dimension two, Perelman shows that the only oriented $\kappa$-solution is the round sphere (I, Corollary 11.3), using Hamilton's classification of two-dimensional shrinkers and the fact that nearly round spheres become rounder.

1.4The asymptotic soliton

The reduced volume (12A.5 Reduced Distance and Reduced Volume) gives a way to look at a $\kappa$-solution from very far in the past. Fix a point $(p, t_0)$, set $\tau = t_0 - t$, and for each $\tau$ choose $q(\tau)$ with $\ell(q(\tau), \tau) \leq \frac n2$ (12A.5 Reduced Distance and Reduced Volume).

In dimension three, the possible asymptotic solitons are known. If the soliton's curvature is not strictly positive, Hamilton's strong maximum principle splits it locally, and it is the round cylinder or its $\mathbb{Z}_2$ quotient. If its curvature is positive and it is compact, it is a quotient of the round $S^3$ by Hamilton's 1982 theorem (11A.6 Hamilton’s 1982 Theorem). The remaining case, noncompact with bounded positive curvature, Perelman rules out (II, Lemma 1.2). Far out along such a soliton, $f$ has no critical points and $R$ increases along its gradient towards a limit, which must be $1$ because blow-downs there are cylinders; the level sets of $f$ are then convex surfaces whose area increases towards $8\pi$, the area of the round 2-sphere of scalar curvature $1$. But Perelman computes that their intrinsic scalar curvature is less than $1$, so their Gauss curvature is less than $\frac12$, and Gauss–Bonnet ($\int K\,dA = 4\pi$) forces their area to exceed $8\pi$: a contradiction. Figure 1.2 shows the mechanism by which a noncompact $\kappa$-solution acquires the cylinder as its asymptotic soliton.

1.5Sorting $\kappa$-solutions

Perelman then classifies $\kappa$-solutions in dimension three by their asymptotic solitons (II, §1.3):

• asymptotic soliton a round quotient $S^3/\Gamma$: the solution is itself a round quotient, because curvature pinching only improves forward in time (Hamilton);
• asymptotic soliton $(S^2\times\mathbb{R})/\mathbb{Z}_2$: the solution has a double cover whose asymptotic soliton is the round cylinder;
• asymptotic soliton the round cylinder: the solution may be noncompact (the cylinder itself, the Bryant soliton, and so on), or compact, as in Perelman's example.

The last case is where the work lies. 12B.2 The Structure of κ-Solutions shows that any $\kappa$-solution in this case is built from necks and caps, and that the space of all of them is compact after normalising the curvature.

1.6History

Perelman introduced ancient solutions with bounded entropy in §11 of his first preprint (2002), and in §1 of the second (2003) he reviewed that section, "correcting a few inaccuracies": the first paragraph of I.11.7 had overlooked compact solutions whose asymptotic soliton is the cylinder, and the second preprint supplies the example. Hamilton had studied ancient solutions with nonnegative curvature in his 1995 survey, which Perelman cites for the earlier work. Perelman also wrote in I.11.9 that he believed the Bryant soliton to be the only noncompact $\kappa$-solution with positive curvature, up to scaling. Simon Brendle proved this in 2020 (12B.2 The Structure of κ-Solutions).

1.7Exercises

Solution

(a) $\operatorname{Ric}$ of $S^2(r)$ is $\frac{1}{r^2}g$, so $(r^2)' = -2$; $R = \frac{2}{r^2} = -\frac1t$. (b) Complete, curvature $\frac{1}{r^2}$ bounded at each time, not flat, sectional curvatures $\frac{1}{r^2}$ and $0$. (c) A ball with $|\operatorname{Rm}| \leq \rho^{-2}$ has $\rho \lesssim r$ (up to the norm's constant). For $\rho$ less than a fixed multiple of $r$ the ball is close to Euclidean, with volume at least $c\rho^3$; scale invariance makes $c$ independent of $t$. (The curvature condition caps the radius, so the large-scale thinness never enters.)

Solution

(a) Flat. (b) Not $\kappa$-noncollapsed at all scales for any $\kappa$: as $t \to -\infty$ the sphere is huge, so balls of radius $\rho \gg L$ have $|\operatorname{Rm}| \leq \rho^{-2}$ and volume about $L\rho^2 \ll \rho^3$. (c) Collapsed at large scales. (d) Not ancient; the flow must exist for all $t \leq 0$. (The full shrinking sphere, defined on $(-\infty, 0)$, is a $\kappa$-solution.)

Solution

The curvature near the point is about $\frac{c_1}{s}$, so $|\operatorname{Rm}| \leq \rho^{-2}$ forces $\rho \lesssim \sqrt{s/c_1}$. On that scale the cross-section, of radius about $c_2\sqrt s$, is comparable to $\rho$, so the ball looks like a ball in a cylinder of radius comparable to $\rho$, whose volume is at least a fixed multiple of $\rho^3$. Near the tip, curvature is bounded and the soliton is smooth, so small balls are nearly Euclidean. On the cigar times a line, balls of radius $\rho$ far out have curvature much smaller than $\rho^{-2}$ but volume about $2\pi\rho^2$, so the ratio $\operatorname{Vol}/\rho^3 \to 0$.

Solution

(a) Integrate $\partial_tR \geq 0$ from $t$ to $0$, then take the supremum over $x$. (b) No: the inequality bounds the past by the present. On the shrinking sphere below, $R$ blows up at $t = \frac14$. (c) $R = \frac{6}{r^2} = \frac{6}{1 - 4t}$, which increases with $t$, so $R(x, t) \leq 6 = R(x, 0)$ for $t \leq 0$.

Solution

(a) By Hamilton's 1982 theorem (11A.6 Hamilton’s 1982 Theorem), the normalised flow converges to a round metric, so rescaled limits at the singular time are round spheres. (b) Perelman normalises so that the singular time is $0$ and the curvature ratio is $1 + \varepsilon$ at $t = -1$. The starting time $t_0(L)$ tends to $-\infty$ as $L \to \infty$, and the limit remembers that, in the far past, the flow was a long capped cylinder. Its asymptotic soliton is the cylinder, so it is not round. (c) The two limits zoom in on different times: (a) on the end, where the shape is round, and (b) on a window that includes the long cylindrical history. A $\kappa$-solution's asymptotic soliton describes the far past, not the singular time.