Blow-Ups, Blow-Downs and Splitting at Infinity

The compactness theorem of 11B.3 Compactness of Ricci Flows produces limits of rescaled flows. This chapter is about what those limits look like, in the two directions that Perelman's analysis of singularities uses.

Blowing up, rescaling by a factor that tends to infinity around points of growing curvature, zooms in on a singularity. In dimension three the limits always have nonnegative curvature, by Hamilton–Ivey pinching; this is what makes the theory of $\kappa$-solutions, which assumes nonnegative curvature, applicable to singularities at all.

Blowing down, rescaling by a factor that tends to zero, or equivalently moving the base point off to infinity while rescaling by the distance, looks at a noncompact manifold from far away. For nonnegatively curved manifolds the result is a cone, the asymptotic or Tits cone of 9B.5 Splitting and Soul Theorems, and when the rescaling is at points far out compared with the curvature scale, the limit contains a line and splits off a factor $\mathbb{R}$. This splitting at infinity is how Perelman shows that the far parts of $\kappa$-solutions look like necks, and it is one of the main uses of the splitting theorem.

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

• define pointed Gromov–Hausdorff convergence of noncompact length spaces, with $\delta$-nets, and give the standard examples;
• define the Tits cone of a nonnegatively curved manifold, and state that blow-downs converge to it;
• define blow-up limits of sequences of flows, and prove that in dimension three they have nonnegative curvature;
• prove that if a nonnegatively curved manifold is rescaled at points going to infinity, faster than their distance from a fixed point, any smooth limit splits off a line;
• define the asymptotic scalar curvature ratio and state Hamilton's dimension reduction, and explain how these produce necks.

4.1Gromov–Hausdorff limits of noncompact spaces

9B.4 Convergence of Manifolds defined the Gromov–Hausdorff distance between compact metric spaces. For noncompact spaces, which is the situation of every blow-up and blow-down, one uses base points and balls.

A $\delta$-net in $(X, x)$ is a subset $L \ni x$, uniformly discrete, whose $\delta$-neighbourhood is all of $X$; a maximal subset whose $\frac\delta2$-balls are disjoint is one. Convergence can be tested on nets: $(X_k, x_k) \to (X, x)$, for uniformly bounded spaces, exactly when every finite net of $X$ is approximated, up to bi-Lipschitz distortion tending to $1$, by nets of the $X_k$ (Morgan and Tian's Lemma 5.22, citing Gromov). Limits that are complete and locally compact are unique up to isometry of based spaces. For length spaces, in which distances are realised by curves, there is a precompactness criterion of the kind proved in 9B.4 Convergence of Manifolds: a sequence has a convergent subsequence if, for every $\delta$ and $R$, the number of disjoint $\delta$-balls in $B(x_k, R)$ is bounded independently of $k$ (Morgan and Tian's Theorem 5.28). By Bishop–Gromov this holds whenever the Ricci curvature is bounded below uniformly, so every sequence with $\operatorname{Ric} \geq -(n - 1)\Lambda$ has Gromov–Hausdorff convergent subsequences, possibly to singular or lower-dimensional limits.

Examples: round cylinders $S^{n-1}(\rho_k)\times\mathbb{R}$ with $\rho_k \to 0$ converge, from any base points, to the line $\mathbb{R}$; the thin tori of 9B.3 Collapsing and Noncollapsing converge to a circle; and rescalings $\lambda_k^{-2}g$ of the Euclidean metric, with any $\lambda_k$, are all isometric to it. Gromov–Hausdorff limits forget smoothness; geometric (Cheeger–Gromov) limits, which need curvature and injectivity radius bounds (9B.4 Convergence of Manifolds, 11B.3 Compactness of Ricci Flows), keep it. In what follows both kinds appear, and the art is to use the weaker one where the stronger is not available.

4.2Blowing down: the Tits cone

Let $(M, g)$ be complete, noncompact, with nonnegative sectional curvature, and $p \in M$. For two rays $\gamma$, $\mu$ from $p$ (minimising unit-speed geodesics $[0, \infty) \to M$, 9B.5 Splitting and Soul Theorems), Toponogov's theorem (9B.5 Splitting and Soul Theorems) shows that the comparison angle at $p$ of the triangle $p, \gamma(r), \mu(r)$, the Euclidean angle with the same three side lengths, is a nonincreasing function of $r$. Its limit $\theta_\infty(\gamma, \mu)$ is the angle at infinity between the rays.

The idea: a point at distance $\lambda_ks$ from $p$ on a ray $\gamma$ becomes the point $(\gamma, s)$ after rescaling, and by Toponogov the rescaled distance between two such points converges to the cone distance, because the comparison angle converges to $\theta_\infty$. Every point far away is close, on the rescaled scale, to a point on some ray (Morgan and Tian's Proposition 5.31). Examples (9B.5 Splitting and Soul Theorems): the paraboloid blows down to a ray, $\mathbb{R}^n$ to itself, a round cylinder $S^2\times\mathbb{R}$ to a line, and the Bryant soliton, whose cross-sections grow like $\sqrt s$, to a ray: its spheres at distance $s$ have radius about $\sqrt s$, which is negligible compared with $s$.

The dimension of the Tits cone measures the asymptotic volume ratio of 9B.2 Volume Comparison: the cone is $n$-dimensional exactly when the volume of large balls grows like $r^n$. Perelman proves that a $\kappa$-solution has asymptotic volume ratio $0$ (12B.3 The Structure of κ-Solutions); the proof combines the blow-down with the no-cones proposition of 11A.8 The Strong Maximum Principle and Splitting: if the ratio were positive, a blow-down of the flow would be a smooth flow through a nonflat cone with nonnegative curvature, which is impossible.

4.3Blowing up: limits are nonnegatively curved

The point of normalising by $R(x_k)$ rather than $|\operatorname{Rm}|(x_k)$ is that $R$ is the quantity the canonical neighbourhood theorem is phrased in, and in dimension three, by pinching, the two are comparable where they are large.

Hamilton–Ivey pinching holds for every Ricci flow on a closed 3-manifold after normalising the initial metric (11A.10 Hamilton–Ivey Pinching, 11A.9 Flows on Complete Manifolds), and Perelman arranges that it survives surgery (12B.9 Surgery), so the theorem applies to every flow in the proof of the Poincaré conjecture. This is the reason three dimensions are special for the singularity analysis: in higher dimensions blow-up limits can have curvature of both signs.

4.4Splitting at infinity

The hypothesis $\lambda_nd(p, x_n)^2 \to \infty$ says that, on the scale at which we look ($\lambda_n^{-1/2}$), the point $p$ recedes to infinity. Seen from $x_n$, there is a long segment back to $p$ and, by the choice of $x_{m(n)}$, a long segment onwards, and nonnegative curvature forces them to line up. For a cylinder the conclusion is evident at every scale; for the paraboloid $z = x^2 + y^2$ at a large distance $d$ from the vertex, the circles of revolution have radius about $\sqrt d$; rescaling by their width, $\lambda = \frac1d$, makes them of radius about $1$, and $\lambda d^2 = d \to \infty$, so the limit is a flat cylinder $S^1(1)\times\mathbb{R}$, with the line along the axis: far out, at the scale of its own width, the paraboloid looks like a cylinder.

4.5The asymptotic scalar curvature ratio and dimension reduction

How the curvature decays at infinity, compared with the distance, decides which kind of rescaling sees a splitting. For a complete noncompact $(M, g)$ and $p \in M$, the asymptotic scalar curvature ratio is

independent of $p$. It is scale-invariant. Examples: $\mathcal R = 0$ for flat space and for the cigar (curvature decays exponentially); $\mathcal R = \infty$ for the cylinder $S^2\times\mathbb{R}$ (curvature constant); for the two-dimensional paraboloid, whose circles have radius about $\sqrt d$ at distance $d$, the Gauss curvature is about $\frac{1}{4d^2}$ and $\mathcal R = \frac12$, finite; for the three-dimensional Bryant soliton, whose cross-sectional spheres have radius about $\sqrt d$, $R \sim \frac cd$ and $\mathcal R = \infty$.

When $\mathcal R = \infty$, there are points $x_n \to \infty$ with $R(x_n)d(p, x_n)^2 \to \infty$, and rescaling by $\lambda_n = R(x_n)$ satisfies the hypothesis of Theorem 4.6. If the rescaled manifolds converge (which needs curvature bounds near $x_n$ at the scale $R(x_n)^{-1/2}$, arranged by point picking as in 11B.5 Singularities, and an injectivity radius bound, supplied by noncollapsing), the limit splits off a line. Hamilton called this dimension reduction (1995): an ancient solution with nonnegative curvature and $\mathcal R = \infty$ has blow-up limits at infinity of the form $N^{n-1}\times\mathbb{R}$ (statement here; Morgan and Tian, chapter 9, §3, carry it out for $\kappa$-solutions). In dimension three, $N$ is a surface, and the two-dimensional theory (a noncollapsed ancient solution on a surface with positive curvature is a round sphere, 12B.1 κ-Solutions) identifies $N\times\mathbb{R}$ as the round cylinder. That is how necks are found far out on $\kappa$-solutions.

When $\mathcal R$ is finite, blow-downs by the distance see a cone, and the analysis goes through Theorem 4.3 and the no-cones proposition of 11A.8 The Strong Maximum Principle and Splitting instead. Perelman shows that for a noncompact $\kappa$-solution the asymptotic volume ratio is zero and $\mathcal R = \infty$, so that the dimension reduction always applies (12B.3 The Structure of κ-Solutions).

4.6History

Gromov introduced the Gromov–Hausdorff distance and its pointed version around 1980 and proved the precompactness criterion in 1981. The angle comparison behind the Tits cone is Toponogov's (1959); the cone at infinity of nonnegatively curved manifolds was studied by Ballmann, Gromov and Schroeder, among others, in the 1980s. Hamilton's dimension reduction and the asymptotic scalar curvature ratio are from "The formation of singularities in the Ricci flow" (1995). Perelman used splitting at infinity throughout his analysis of $\kappa$-solutions (2002), and Morgan and Tian's chapter 5 states it in the form proved here.

4.7Exercises

Solution

Relate $(\theta, s)$ to $s$. For two points, $d^2 = d_{S^1}(\theta, \theta')^2 + (s - s')^2$ with $d_{S^1} \leq \pi\rho$, so $0 \leq d - |s - s'| \leq \pi\rho$. The correspondence between $B((\theta_0, 0), r)$ and $(-r, r)$ (up to the small adjustment at the boundary in the definition) has distortion at most $\pi\rho \to 0$, so the Gromov–Hausdorff distance between the balls is at most $\frac{\pi\rho}{2} + o(1)$.

Solution

Far out, the paraboloid is a surface of revolution whose circles at height $z$ have radius $\sqrt z$, and the distance from the vertex is about $z$. Two rays (meridians) from the vertex at distance $r$ are on the circle of radius about $\sqrt r$, so their distance is at most $\pi\sqrt r$. The Euclidean comparison triangle with sides $r$, $r$ and at most $\pi\sqrt r$ has angle at $p$ at most about $\frac{\pi\sqrt r}{r} \to 0$. So all angles at infinity vanish, $S_\infty$ is a point, and the Tits cone is the cone over a point: a ray.

Solution

Take $M_k = S^2(\rho_k)\times\Sigma_k$, a product of a round sphere of radius $\rho_k$ with a complete hyperbolic surface of curvature $-\frac{1}{2\rho_k^2}$, as $\rho_k \to 0$. The scalar curvature is $\frac{2}{\rho_k^2} - \frac{1}{\rho_k^2} = \frac{1}{\rho_k^2} \to \infty$. Rescaling by $R = \rho_k^{-2}$ gives, for every $k$, $S^2(1)\times\Sigma$ with $\Sigma$ of curvature $-\frac12$: a constant sequence, whose limit has sectional curvature $-\frac12$ on the planes tangent to $\Sigma$. The sequence violates Hamilton–Ivey pinching, since $|\nu|$ is comparable to $R$ at arbitrarily large $R$; it cannot arise in a Ricci flow on a closed 3-manifold with normalised initial data, for which the theorem's hypothesis holds.

Solution

For the cigar $R = 4\operatorname{sech}^2s \approx 16e^{-2s}$, so $R\,d^2 \approx 16s^2e^{-2s} \to 0$: $\mathcal R = 0$. Rescaling far out by $\lambda_n = R(x_n) \to 0$ shrinks the cylindrical end of circumference $2\pi$ to nothing: the Gromov–Hausdorff limit is a line, collapsed, and there is no smooth limit. For the Bryant soliton, $R\,d^2 \sim cs \to \infty$: $\mathcal R = \infty$. Rescaling by $R(x_n) \approx \frac{c}{s_n}$ at $x_n$ far out makes the cross-sectional spheres, of radius about $\sqrt{s_n}$, of size about $1$, and the limit is the round cylinder $S^2\times\mathbb{R}$: a neck, with the line along the axis.

Solution

$\lambda_nd(p, x_n)^2 = d(p, x_n)^2 \to \infty$. The pointed limit at the scale $1$ is the round cylinder $S^1(1)\times\mathbb{R}$ times the extra $\mathbb{R}$, that is $S^1\times\mathbb{R}^2$; it contains the line along the cylinder's axis, and it splits off two lines in fact. Rescaling by $\lambda_n \to 0$ shrinks the circle factor to nothing, so the Gromov–Hausdorff limit is $\mathbb{R}^2$, of lower dimension; no smooth limit of dimension three exists, because the injectivity radius at $x_n$ tends to zero in the rescaled metrics (collapse, 9B.3 Collapsing and Noncollapsing).