The Structure of κ-Solutions

12B.1 κ-Solutions sorted $\kappa$-solutions by their behaviour in the far past. Surgery needs something else: a description of a $\kappa$-solution now, at each of its points, with constants that do not depend on which $\kappa$-solution it is. Perelman obtains it from one theorem: $\kappa$-solutions form a compact family up to scaling. Compactness turns qualitative facts into uniform ones. It gives universal bounds on the derivatives of curvature, and it shows that every point of every $\kappa$-solution lies in a region of one of four standard kinds: a neck, a cap, or one of two kinds of closed component. These are the canonical neighbourhoods, and the rest of the proof works with them.

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

• define $\varepsilon$-necks, strong $\varepsilon$-necks, $\varepsilon$-caps, tubes and horns, with every parameter explicit;
• state that noncompact $\kappa$-solutions have asymptotic volume ratio zero, and explain why that matters;
• state the compactness theorem for $\kappa$-solutions and outline its proof;
• state the universal estimates $|\nabla R| < \eta R^{3/2}$, $|\partial_tR| < \eta R^2$ and the canonical neighbourhood description of $\kappa$-solutions;
• describe the neck–cap structure of a noncompact $\kappa$-solution, and state the later complete classification.

2.1Pipes and end caps

2.2Necks and caps

Perelman fixes the vocabulary at the start of his second preprint. Throughout, $\varepsilon > 0$ is a fixed small constant, and "$\varepsilon$-close" means close in the $C^N$ topology with $N > \varepsilon^{-1}$, after the indicated rescaling.

A neck is a piece of the round cylinder of length $2\varepsilon^{-1}$ in units of its radius, so it is long and thin. A strong neck has also been a neck for a while: it is close to the shrinking cylinder over a backward time interval. Figure 2.1 draws the model. The central 2-sphere of a neck separates it into two halves (Exercise 2.5); this is the sphere along which surgery cuts. A cap is a ball, or a punctured $\mathbb{RP}^3$, finished off by necks. The $\mathbb{RP}^3$ case comes from $(S^2\times\mathbb{R})/\mathbb{Z}_2$, whose end is a neck and whose core is a neighbourhood of an $\mathbb{RP}^2$.

2.3Volume and curvature

The first structural fact is that a noncompact $\kappa$-solution is thin at infinity in an average sense. For a complete noncompact manifold with $\operatorname{Ric} \geq 0$, the ratio $\operatorname{Vol}B(p, r)/r^n$ is nonincreasing by Bishop–Gromov (9B.2 Volume Comparison), so it has a limit as $r \to \infty$, the asymptotic volume ratio $\mathcal V$.

The proof is by induction on dimension. If $\mathcal V > 0$, consider how fast the curvature decays relative to distance, $\limsup R(x)d(x)^2$. If it is infinite, blowing up at well-chosen points gives a $\kappa$-solution that splits off a line, and one passes to a lower dimension. If it is finite and positive, a blow-down gives a piece of a non-flat metric cone, which Hamilton's strong maximum principle rules out. If it is zero, the manifold is flat. Zero asymptotic volume ratio is a strong statement about noncollapsed solutions, which are thick at the scale of their curvature but must be thin compared with Euclidean space at large scales. Its use is a pair of corollaries (I, 11.5 and 11.6) which turn it around: a lower bound on volume at some scale gives an upper bound on curvature there. Precisely, if a ball $B(x_0, r_0)$ has volume at least $wr_0^n$ along a flow with nonnegative curvature operator on $[t_0, 0]$, then $R \leq Cr_0^{-2} + B(t - t_0)^{-1}$ on $B(x_0, \frac14r_0)$, with $B$ and $C$ depending only on $w$.

2.4Compactness

Perelman stated it for noncompact solutions and noted in his second preprint that this assumption was redundant, and that the limit need not have the same topology as the $M_k$.

The fact used in step 5, that a nonnegatively curved complete manifold cannot contain arbitrarily thin necks going off to infinity, recurs constantly. Perelman uses it without proof; it comes from comparison geometry in nonnegative curvature (9B.5 Splitting and Soul Theorems), and the expositions write the argument out.

2.5Universal estimates and canonical neighbourhoods

Compactness, applied to the whole class at once, gives constants that hold for every $\kappa$-solution. Perelman also shows that $\kappa$ itself can be made universal.

The estimates in part 2 are scale-invariant (Exercise 2.6), and they are what makes high curvature local: they say that $R^{-1/2}$, the natural length scale, changes at a bounded rate in space, and $R^{-1}$ at a bounded rate in time (Exercise 2.9). Part 3 is the template for the canonical neighbourhood assumption that 12B.3 The Canonical Neighbourhood Theorem proves for every three-dimensional Ricci flow at high curvature.

The neck–cap structure. For a noncompact $\kappa$-solution, Perelman shows (I, Corollary 11.8) that the set $M_\varepsilon$ of points that are not centres of $\varepsilon$-necks is compact, with diameter at most $CQ^{-1/2}$ and curvature comparable to $Q$ on it, where $Q$ is the curvature at a point of its boundary. So a noncompact $\kappa$-solution is a compact core followed by a tube of necks running out to infinity. If its curvature is positive, it is diffeomorphic to $\mathbb{R}^3$ by the soul theorem (9B.5 Splitting and Soul Theorems), and the core with its first necks is an $\varepsilon$-cap. If its curvature is not positive, the strong maximum principle splits it, and it is the round cylinder or its $\mathbb{Z}_2$ quotient. Figure 2.2 shows the picture on the Bryant soliton.

2.6The complete classification

Perelman's proof needs only the qualitative description above. He conjectured more (I, §11.9): that the Bryant soliton is the only noncompact $\kappa$-solution with positive curvature, up to scaling. Simon Brendle proved this ("Ancient solutions to the Ricci flow in dimension 3", Acta Mathematica 225, 2020): every noncompact three-dimensional $\kappa$-solution is a family of shrinking round cylinders, a quotient of one, or the Bryant soliton. Brendle, Panagiota Daskalopoulos and Nataša Šešum then showed ("Uniqueness of compact ancient solutions to three-dimensional Ricci flow", Inventiones Mathematicae 226, 2021) that a $\kappa$-noncollapsed ancient solution on $S^3$ is either a family of shrinking round spheres or Perelman's compact example of 12B.1 κ-Solutions. The list of singularity models is now complete in a form Perelman did not have, but the proof of the Poincaré conjecture never depended on it.

2.7History

The definitions of necks, horns and caps are those of Perelman's second preprint (2003); Hamilton had worked with necks in his four-dimensional surgery paper (1997), and Perelman cites it for the capping construction (12B.4 Surgery). The compactness theorem and the asymptotic volume ratio proposition are in §11 of the first preprint (2002), with the noncompactness assumption removed in the second. The universal $\kappa_0$, $\eta$ and the four kinds of neighbourhood are in II, §1.5. The complete classification is due to Brendle (2020) and Brendle–Daskalopoulos–Šešum (2021).

2.8Exercises

Solution

$N\setminus\Sigma = S^2\times(-L, 0)\ \cup\ S^2\times(0, L)$, a disjoint union of two connected open sets, since $S^2$ and intervals are connected. A diffeomorphism carries components to components, so the same holds in the neck.

Solution

(a) $\partial_tR = \frac{1}{t^2} = R^2$, so the time ratio is $1$; $\nabla R = 0$. (b) $\partial_tR = \frac{3}{2t^2} = \frac23R^2$, ratio $\frac23$; $\nabla R = 0$. (c) Under $g \mapsto \lambda^2g$, $t \mapsto \lambda^2t$: $R \mapsto \lambda^{-2}R$, $\partial_tR \mapsto \lambda^{-4}\partial_tR$, $|\nabla R| \mapsto \lambda^{-3}|\nabla R|$, so both ratios are invariant. From (a), the universal constant must satisfy $\eta > 1$.

Solution

(a) For $r \gg \rho$, $B(p, r)$ is roughly $S^2\times[-r, r]$, of volume about $4\pi\rho^2\cdot2r$, so $\operatorname{Vol}/r^3 \to 0$. (b) $\operatorname{Vol}B(p, r) \approx \int_0^r4\pi\psi(s)^2\,ds \approx 4\pi c^2\frac{r^2}{2}$, so the ratio is about $\frac{2\pi c^2}{r} \to 0$. (c) $\mathbb{R}^3$ is flat, so it is not a $\kappa$-solution.

Solution

There are $r < C_1(\varepsilon)R(x, t)^{-1/2}$ and $B$ with $B(x, t, r) \subset B \subset B(x, t, 2r)$ such that $B$ is the final time slice of a strong $\varepsilon$-neck: for some $r'$, $P(y, t, \varepsilon^{-1}r', -r'^2)$, scaled by $r'^{-2}$, is $\varepsilon$-close in $C^N$, $N > \varepsilon^{-1}$, to the evolving standard neck of length $2\varepsilon^{-1}$ and scalar curvature $(1 - t')^{-1}$ at time $t' \in [-1, 0]$. On $B$, $C_2(\varepsilon)^{-1}R(x, t) \leq R \leq C_2(\varepsilon)R(x, t)$, and $\operatorname{Vol}B > C_2(\varepsilon)^{-1}R(x, t)^{-3/2}$. $C_1$ and $C_2$ depend only on $\varepsilon$; $\kappa_0$ and $\eta$ are universal.

Solution

(a) $\nabla(R^{-1/2}) = -\frac12R^{-3/2}\nabla R$ and $\partial_t(R^{-1}) = -R^{-2}\partial_tR$. (b) Along a minimising geodesic, $R^{-1/2}(y) \geq R^{-1/2}(x) - \frac\eta2d(x, y) \geq \frac12R^{-1/2}(x)$, so $R(y) \leq 4R(x)$. (c) $R^{-1}(x, t') \geq R^{-1}(x, t) - \eta(t - t') \geq \frac12R^{-1}(x, t)$. So on a parabolic region of size comparable to $R(x, t)^{-1/2}$ in space and $R(x, t)^{-1}$ in time, the curvature stays within a fixed factor of $R(x, t)$: a point of high curvature controls its surroundings at its own scale, which is what lets the canonical neighbourhood argument rescale and take limits.