Compactness and Universal Estimates for κ-Solutions

12B.3 The Structure of κ-Solutions stated the compactness theorem for three-dimensional $\kappa$-solutions, the universal estimates and the canonical neighbourhood description, with the ideas of their proofs. This chapter proves them, following Morgan and Tian. The theorems are the bridge between 12B.2 The Asymptotic Soliton and Three-Dimensional Shrinkers, which understood a single $\kappa$-solution in the far past, and 12B.5 The Canonical Neighbourhood Theorem, which transfers the description of $\kappa$-solutions to every three-dimensional Ricci flow at high curvature. What makes them work is a small number of scale-invariant facts, each proved by contradiction and compactness:

• the asymptotic scalar curvature ratio $\limsup R(x)d(x, p)^2$ of a noncompact $\kappa$-solution is infinite, so far out the curvature is always large on the scale of the distance;
• the asymptotic volume ratio $\lim\operatorname{Vol}B(p, r)/r^3$ is zero;
• conversely, a lower bound on volume at some scale gives an upper bound on curvature at that scale;
• and therefore curvature at bounded distance is bounded, after normalising $R = 1$ at the base point, which is exactly what compactness needs.

Two points of method are worth noticing. First, every argument is of the form "suppose a scale-invariant quantity is unbounded along a sequence; rescale to normalise it; take a limit; the limit violates a rigidity theorem". The rigidity theorems are the strong maximum principle, the splitting theorem, Bishop–Gromov, the fact that no Ricci flow is a non-flat cone, and the classification of 12B.2 The Asymptotic Soliton and Three-Dimensional Shrinkers. Second, Hamilton's Harnack inequality enters through one simple consequence, $\partial_tR \geq 0$ on a $\kappa$-solution, which turns a bound at time $0$ into a bound at all earlier times and so turns spatial bounds into space-time bounds, as compactness requires.

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

• prove that a non-round three-dimensional $\kappa$-solution is $\kappa_0$-noncollapsed for a universal $\kappa_0$;
• prove that a noncompact $\kappa$-solution has infinite asymptotic scalar curvature ratio and zero asymptotic volume ratio;
• prove the volume-to-curvature estimates and the monotonicity in time of the asymptotic volume;
• prove the compactness theorem, including the bound on curvature at bounded distance through the closest point with $d^2R = 1$;
• deduce the universal bounds $|\nabla R| < \eta R^{3/2}$ and $|\partial_tR| < \eta R^2$;
• define $(C, \varepsilon)$-caps, $C$-components, $\varepsilon$-round components and strong $\varepsilon$-necks with every condition, and state and explain the seven-case description of $\kappa$-solutions (Morgan–Tian, Theorem 9.93).

4.1A universal $\kappa$

The definition of a $\kappa$-solution allows any $\kappa > 0$. For the canonical neighbourhood theorem it is essential that $\kappa$ can be fixed once and for all, apart from the round case.

4.2Curvature and volume at infinity

Let $(M, g)$ be complete, noncompact, with nonnegative curvature, and $p \in M$. Two scale-invariant numbers measure its behaviour at infinity:

the asymptotic scalar curvature ratio and the asymptotic volume ratio; both are independent of $p$, and the second exists by Bishop–Gromov (9B.2 Volume Comparison). On the round cylinder $S^2\times\mathbb{R}$, $\mathcal R = \infty$ (the curvature is constant) and $\mathcal V = 0$; on the Bryant soliton, $R \sim \frac{c}{d}$, so $\mathcal R = \infty$ again, and $\mathcal V = 0$ because cross-sections grow like $\sqrt d$; on Euclidean space $\mathcal R = 0$ and $\mathcal V = \omega_3$.

The asymptotic volume ratio being zero says that a noncompact $\kappa$-solution, although noncollapsed on the scale of its curvature, is much thinner than Euclidean space at large scales. It is turned around into the most useful statement of the theory:

4.3Compactness

Everything reduces to one lemma.

Proof of the theorem. By the lemma and $\partial_tR \geq 0$, curvature is bounded on $B(p_k, 0, r)\times(-\infty, 0]$ by $C(r)$, uniformly in $k$; with noncollapsing, Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows) gives a subsequence converging to a complete ancient flow, $\kappa$-noncollapsed, with nonnegative curvature operator, not flat ($R = 1$ at the base point), with $\partial_tR \geq 0$. That its time slices have bounded curvature is the argument at the end of 12B.2 The Asymptotic Soliton and Three-Dimensional Shrinkers: unbounded curvature would produce, by splitting at infinity, $\varepsilon$-necks of arbitrarily small scale in a positively curved complete three-manifold, which cannot exist. So the limit is a $\kappa$-solution. $\square$

Perelman's Theorem 11.7 is stated for noncompact $\kappa$-solutions; Kleiner and Lott observe (Remark 46.3) that the proof works for compact ones too, and that a limit of compact solutions may be noncompact. They also correct a statement in the first paragraph of Perelman's §11.7: there are $\kappa$-solutions on $S^3$ and $\mathbb{RP}^3$ that are not round, whose asymptotic soliton is noncompact (Perelman II, §1.4).

The universal derivative estimates. The quantities $\frac{|\nabla R|}{R^{3/2}}$ and $\frac{|\partial_tR|}{R^2}$ are scale invariant, and on a $\kappa$-solution normalised by $R(p, 0) = 1$ they are bounded at $(p, 0)$ by compactness (they are continuous under smooth convergence). So there is $\eta(\kappa)$, and by the universal $\kappa_0$ (the round case being explicit) a universal $\eta$, with

everywhere on every three-dimensional $\kappa$-solution (Morgan–Tian, Corollary 9.71). By the evolution equation, the second is equivalent to $|\Delta R + 2|\operatorname{Ric}|^2| < \eta R^2$.

4.4Canonical neighbourhoods, defined precisely

Morgan and Tian's definitions are worth having in full, because every later chapter uses them. Fix $0 < \varepsilon < \frac12$ and $C < \infty$. Closeness "in the $C^{[1/\varepsilon]}$ topology" means after the indicated rescaling, in a fixed diffeomorphism, with $[1/\varepsilon]$ derivatives.

• A $(C, \varepsilon)$-cap in a Riemannian three-manifold is an open submanifold $\mathcal C$ with an $\varepsilon$-neck $N \subset \mathcal C$ such that: $\mathcal C$ is diffeomorphic to an open ball or to a punctured $\mathbb{RP}^3$; $N$ has compact complement $Y$ in $\mathcal C$ (the core), whose boundary is a central sphere of an $\varepsilon$-neck; $R > 0$ on $\mathcal C$, $\operatorname{diam}\mathcal C < C(\sup_{\mathcal C}R)^{-1/2}$, $\sup R/\inf R < C$ and $\operatorname{Vol}\mathcal C < C(\sup R)^{-3/2}$ on $\mathcal C$; balls of radius $r_y$ about core points, where $\sup_{B(y, r_y)}R = r_y^{-2}$, have volume more than $C^{-1}r_y^3$; and $|\nabla R| < CR^{3/2}$, $|\Delta R + 2|\operatorname{Ric}|^2| < CR^2$ (Definition 9.72).
• A $C$-component is a compact component diffeomorphic to $S^3$ or $\mathbb{RP}^3$ with positive sectional curvature, $\inf K > C^{-1}\sup R$, and $C^{-1}(\sup R)^{-1/2} < \operatorname{diam} < C(\inf R)^{-1/2}$ (Definition 9.75).
• An $\varepsilon$-round component is a compact component which, after rescaling, is $\varepsilon$-close to a manifold of constant curvature $+1$ (Definition 9.76).
• A strong $\varepsilon$-neck centred at $(x, t)$ is an evolving neck: an embedding of $S^2\times(-\varepsilon^{-1}, \varepsilon^{-1})$ around $x$, extended backward for the rescaled time $1$ (that is, for actual time $R(x, t)^{-1}$), on which $R(x, t)g$ is $\varepsilon$-close to the standard shrinking cylinder $h(t') + ds^2$, $-1 < t' \leq 0$, with $h(t')$ round of scalar curvature $\frac{1}{1 - t'}$ (Definition 9.78).

A point has a strong $(C, \varepsilon)$-canonical neighbourhood if it is the centre of a strong $\varepsilon$-neck, or lies in the core of a $(C, \varepsilon)$-cap, or lies in a $C$-component, or in an $\varepsilon$-round component. All these notions are scale invariant and open: a metric close enough to one with a canonical neighbourhood at $x$ has one, with slightly worse constants, at points near $x$ (Proposition 9.79). Openness is what lets canonical neighbourhoods pass from limits to the sequences that converge to them, which is how 12B.5 The Canonical Neighbourhood Theorem uses them.

4.5The structure of $\kappa$-solutions

Positive curvature or a split. By the strong maximum principle, a three-dimensional $\kappa$-solution either has positive curvature at all times, or is the shrinking cylinder $S^2\times\mathbb{R}$, or is a line bundle over $\mathbb{RP}^2$ finitely covered by the cylinder; a product with a circle is excluded because it collapses in the far past (Morgan–Tian, Proposition 9.83).

Noncompact with positive curvature. Let $p$ be a soul (9B.5 Splitting and Soul Theorems). There is a universal $D(\varepsilon)$ such that every point outside $B(p, 0, DR(p, 0)^{-1/2})$ is the centre of a strong $\varepsilon$-neck, and on that ball the sectional curvatures and the volume are universally comparable to $R(p, 0)$ and $R(p, 0)^{-3/2}$ (Proposition 9.85). The proof is a contradiction argument: far out, either $d^2R \to \infty$, and splitting plus the two-dimensional classification gives a neck, or $d^2R$ stays bounded with $R \to 0$, contradicting the lemma on bounded curvature at bounded distance after rescaling. Universality of $D$ comes from compactness. Moreover the central sphere of every $\varepsilon$-neck separates the soul from the end, so the regions between necks are products $S^2\times I$: the manifold is a $(C, \varepsilon)$-cap (around the soul) followed by a strong $\varepsilon$-tube, a $C$-capped strong $\varepsilon$-tube, diffeomorphic to $\mathbb{R}^3$.

Compact. By a similar argument (Morgan–Tian, Theorem 9.89): a compact $\kappa$-solution is round, or has diameter less than $C_1(\max R)^{-1/2}$ and is diffeomorphic to $S^3$ or $\mathbb{RP}^3$, or is a doubly $C_1$-capped strong $\varepsilon$-tube, two caps joined by a tube, again diffeomorphic to $S^3$ or $\mathbb{RP}^3$ (the other possibilities allowed by the topology of capped tubes, $\mathbb{RP}^3\#\mathbb{RP}^3$ and $S^2\times S^1$, have infinite fundamental group, which positive curvature forbids).

Case 7 is the one exception, and it is harmless. $\mathbb{RP}^2\times\mathbb{R}$ contains an embedded $\mathbb{RP}^2$ with trivial normal bundle, which cannot occur in an orientable three-manifold: a surface in an orientable three-manifold has trivial normal bundle only if it is itself orientable. Perelman assumes throughout that the manifold is oriented; Morgan and Tian assume only that it contains no embedded $\mathbb{RP}^2$ with trivial normal bundle, which every orientable manifold satisfies. In the Poincaré conjecture the manifold is simply connected and so orientable. Morgan and Tian's Corollary 9.95 then states the form in which the theorem is used: if a sequence of based flows, none of whose time slices contains an embedded $\mathbb{RP}^2$ with trivial normal bundle, converges to a $\kappa$-solution, then for large $n$ the base points have strong $(C, \varepsilon)$-canonical neighbourhoods. That is the step that 12B.5 The Canonical Neighbourhood Theorem combines with a blow-up argument.

4.6History

Perelman stated the results of this chapter in I, §§11.4–11.9 (2002), with the noncompactness assumption in 11.7 removed and the universal constants recorded in II, §1.5 (2003). The asymptotic scalar curvature ratio and its relation to cones are due to Hamilton (1995), whose "Formation of singularities" also contains the strong maximum principle argument against cones. Morgan and Tian (2007, chapter 9) and Kleiner and Lott (§§41–50) give the complete proofs. Brendle's classification (2020) later identified the noncompact positively curved case with the Bryant soliton.

4.7Exercises

Solution

(a) $\mathcal R = 0$, $\mathcal V = \omega_3$: flat, not a $\kappa$-solution. (b) $R = \frac{2}{r^2}$, so $Rd^2 \to \infty$; $\operatorname{Vol}B(p, \rho) \approx 4\pi r^2\cdot2\rho$, so $\mathcal V = 0$. (c) $Rd^2 \approx \frac{2d}{c^2} \to \infty$; $\operatorname{Vol}B(p, \rho) \approx \int_0^\rho4\pi c^2s\,ds = 2\pi c^2\rho^2$, so $\mathcal V = 0$. (b) and (c) are consistent with Proposition 4.2 and Theorem 4.3.

Solution

A fundamental domain for $\mathbb{Z}$ acting with translation part of length $a$ along an axis is a slab of width $a$ perpendicular to the axis (for a screw motion the slab still works). The ball of radius $r$ in $L$ lifts into the part of the slab within distance $r$ of the lift of $p$, of volume at most $a\cdot\pi r^2\cdot C$. So $\operatorname{Vol}B_L(p, r)/r^3 \to 0$. Any nontrivial $\Gamma$ contains an element of infinite order (a free properly discontinuous action on $\mathbb{R}^3$ has no elements of finite order, since a finite-order isometry has a fixed point), and $L$ is covered by $\mathbb{R}^3/\langle\gamma\rangle$, whose asymptotic volume ratio is zero; volumes of balls in $L$ are at most those of the corresponding balls in the cover.

Solution

(a) $\partial_tdV = -R\,dV \leq 0$. (b) By Lemma 8.3(b) of 12A.8 Local Noncollapsing from the Reduced Volume with $r_0$ fixed (say $r_0 = K^{-1/2}$), $\frac{d}{dt}d_t(x, y) \geq -2(n - 1)(\frac23Kr_0 + r_0^{-1}) = -A$; integrating from $a$ to $b$ gives $d_b \geq d_a - A(b - a)$. The best choice $r_0 = \sqrt{3/(2K)}$ gives $A = 4(n - 1)\sqrt{2K/3}$, Morgan and Tian's constant. (c) By (b), $B(p, b, r) \subseteq B(p, a, r + A(b - a))$; by (a), $\operatorname{Vol}_bB(p, b, r) \leq \operatorname{Vol}_aB(p, a, r + A(b - a))$. Dividing by $r^n$, the right side is $\frac{\operatorname{Vol}_aB(p, a, r')}{r'^n}\big(\frac{r'}{r}\big)^n$ with $r' = r + A(b - a)$; let $r \to \infty$.

Solution

Substituting, $\log\frac{R(p, 0)}{R(q, -cQ^{-1})} \geq -\frac{\tilde C^2Q^{-1}}{2cQ^{-1}} = -\frac{\tilde C^2}{2c}$, which rearranges to the claim. A Harnack inequality compares the values of a positive solution of a parabolic equation at a later time with those at an earlier time and nearby point: heat cannot appear from nothing, so a large value in the past at $q$ would force a large value later nearby. Here $R$ plays the role of the heat, and the bound $R(p, 0) = 1$ now forbids $R$ from having been huge at $q$ shortly before, provided $q$ is close on the parabolic scale, which Claim 2 guarantees.

Solution

Under $g_\lambda$: $R_\lambda = \lambda^{-1}R$, $|\nabla R_\lambda|_{g_\lambda} = \lambda^{-1}\cdot\lambda^{-1/2}|\nabla R|_g$ (the gradient of a function scales by $\lambda^{-1/2}$ in norm), and $\partial_tR_\lambda = \lambda^{-2}\partial_tR$ (time scales by $\lambda$). So both ratios are invariant. And $|\nabla(R^{-1/2})| = \frac12R^{-3/2}|\nabla R| < \frac\eta2$.