Convergence of Manifolds

Analysis extracts convergent subsequences from bounded sequences: Bolzano–Weierstrass in $\mathbb{R}^n$, Arzelà–Ascoli for functions (2B.5 Uniform Convergence and Arzelà–Ascoli). The Ricci flow needs the same for manifolds. Blow up a flow near a singularity, rescaling time after time so that the curvature stays bounded, and you get a sequence of Riemannian manifolds. To study the singularity you need a limit of that sequence. Two notions of convergence do this. Gromov–Hausdorff convergence treats manifolds as metric spaces and needs very little to apply, but its limits can be singular and of lower dimension. Cheeger–Gromov convergence is smooth convergence up to diffeomorphism, and its compactness theorem needs bounded curvature and a lower bound on the injectivity radius, exactly the two pieces of 9B.3 Collapsing and Noncollapsing.

This chapter also states, in geometric form, the argument that the rest of the Path uses again and again: suppose an estimate fails, take a sequence of counterexamples, rescale, extract a limit, and reach a contradiction in the limit.

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

• define the Hausdorff and Gromov–Hausdorff distances, and compute simple examples;
• state Gromov's precompactness theorem and explain why packing proves it;
• define pointed smooth (Cheeger–Gromov) convergence and state the compactness theorem;
• explain why both hypotheses of the compactness theorem are needed;
• run the compactness–contradiction argument in a geometric setting.

4.1Comparing shapes

4.2Hausdorff and Gromov–Hausdorff distance

For subsets $A, B$ of a metric space $Z$, the Hausdorff distance $d_H^Z(A, B)$ is the infimum of $\varepsilon$ such that each set lies within $\varepsilon$ of the other. For compact metric spaces $X, Y$, the Gromov–Hausdorff distance is

the infimum over all metric spaces $Z$ and isometric embeddings $f : X \to Z$, $g : Y \to Z$ (Figure 4.1). It is zero exactly when $X$ and $Y$ are isometric, and it is a metric on isometry classes of compact metric spaces. A practical description uses correspondences. A relation $\mathcal R \subset X\times Y$ that relates every point of each space to some point of the other has distortion

and $d_{GH}(X, Y) = \frac12\inf_{\mathcal R}\operatorname{dis}\mathcal R$ (Burago–Burago–Ivanov, chapter 7). For example, a point and a space $X$ are at distance $\frac12\operatorname{diam}X$ (Exercise 4.4).

For noncompact spaces one uses pointed convergence: $(X_k, p_k) \to (X, p)$ if the balls $B(p_k, R)$ converge to $B(p, R)$ for every $R$ (with a small technical adjustment at the boundary). A collapsing sequence converges in this sense to something lower-dimensional: thin flat tori to a circle (Exercise 4.5), Berger spheres to $S^2(\frac12)$, the thin cylinder $S^1(\varepsilon)\times\mathbb{R}$ to a line (9B.3 Collapsing and Noncollapsing).

The proof rests on a criterion of Gromov: a class of compact metric spaces is precompact if, for each $\varepsilon > 0$, every space in the class can be covered by at most $N(\varepsilon)$ balls of radius $\varepsilon$, with $N(\varepsilon)$ independent of the space. Packing (9B.2 Volume Comparison) supplies exactly this: a maximal set of points at mutual distance at least $\varepsilon$ has the $\varepsilon$-balls around it covering, and the disjoint $\frac\varepsilon2$-balls are at most $\frac{V_{-k}(D)}{V_{-k}(\varepsilon/2)}$ in number by relative volume comparison (with $V_{-k}$ the volume of balls in the model space of curvature $-k$). The limits can be singular. Characterising them is a deep theory, begun by Cheeger and Colding in the 1990s.

4.3Smooth convergence

The diffeomorphisms are part of the definition, because a metric is only defined up to diffeomorphism (8A.6 Flows and the Lie Derivative). Smooth limits are smooth manifolds of the same dimension, so they exclude collapse.

The architecture of the proof. (1) The injectivity radius bound at $p_k$ spreads to every point at bounded distance, with a smaller constant. Bishop–Gromov (9B.2 Volume Comparison) moves the volume lower bound from $B(p_k, 1)$ to nearby balls, and Cheeger–Gromov–Taylor (9B.3 Collapsing and Noncollapsing) turns volume back into injectivity radius. (2) Around every point there is then a chart of uniform size in which the metric coefficients and all their derivatives are bounded: normal coordinates, or better, harmonic coordinates. (3) Arzelà–Ascoli (2B.5 Uniform Convergence and Arzelà–Ascoli), applied chart by chart, with a diagonal argument over $j$ and over $R$, gives limits of the metric coefficients. (4) The charts are glued into a limit manifold and maps $\phi_k$, with a centre-of-mass construction smoothing the transition maps. Hamilton proved the version for sequences of Ricci flows in 1995 (11B.3 Compactness of Ricci Flows), where the curvature bounds on derivatives come for free from Shi's estimates.

Both hypotheses are necessary. Without the curvature bounds, smooth surfaces can converge to a cone or develop a corner. Without the injectivity radius bound, thin tori have zero curvature and no smooth two-dimensional limit.

4.4History

Felix Hausdorff defined his distance in 1914. David Edwards introduced the distance between metric spaces in 1975, and Mikhail Gromov independently developed it, with precompactness and its first uses in geometry, in 1981. Jeff Cheeger's finiteness theorem (1970) contained the first smooth compactness argument; the smooth compactness theorem in its modern form is due to Gromov, with refinements by Stefan Peters (1987) and by Robert Greene and Hung-Hsi Wu (1988). Richard Hamilton proved the compactness theorem for Ricci flows in 1995. Cheeger and Colding's structure theory for limits with lower Ricci bounds dates from 1996–2000.

4.5Exercises

Solution

The only correspondence relates $\ast$ to every $x \in X$, with distortion $\sup_{x, x'}|0 - d(x, x')| = \operatorname{diam}X$, so $d_{GH} = \frac12\operatorname{diam}X$. The circle of length $L$ has intrinsic diameter $\frac L2$.

Solution

$d_T((x, y), (x', y'))^2 = d_S(x, x')^2 + d(y, y')^2$ with $d(y, y') \leq \frac\varepsilon2$ in $\mathbb{R}/\varepsilon\mathbb{Z}$. So $0 \leq d_T - d_S \leq \sqrt{d_S^2 + \frac{\varepsilon^2}{4}} - d_S \leq \frac\varepsilon2$.

Solution

The farthest point of the polygon from the circle is the midpoint of a side, at distance $\cos\frac{\pi}{m}$ from the centre; every point of the circle is within $1 - \cos\frac\pi m$ of the polygon too (the nearest side's midpoint direction). Since both sit in the plane isometrically, $d_{GH} \leq d_H$. With intrinsic metrics the spaces change (the polygon has perimeter $2m\sin\frac{\pi}{m}$, the circle $2\pi$), but the polygons still converge to the circle, as one checks with the correspondence matching points at the same angle.

Solution

(a) The diameter $\frac\pi k \to 0$, so the limit is a point. (b) $k^2g_k$ is the unit round sphere for every $k$: a constant sequence. (c) $k^4g_k$ is the round sphere of radius $k$; its curvature $\frac{1}{k^2} \to 0$ and its injectivity radius $\pi k \to \infty$, and the pointed limit is $\mathbb{R}^n$. Which limit you see depends on the scale you choose, and choosing it well is the art of blow-up arguments.

Solution

All derivatives of $\operatorname{Rm}$ vanish, so the curvature hypotheses hold with $C_j = 0$, but $\operatorname{inj} = \frac{1}{2k} \to 0$. In a smooth limit, $\phi_k^*g_k \to g$ on compact sets, so volumes of unit balls would converge to a positive number (the limit is a smooth surface), but $\operatorname{Area}T_{1/k} = \frac1k \to 0$.

Solution

Suppose infinitely many diffeomorphism types occur; choose $M_k$ pairwise non-diffeomorphic (failure of the uniform statement). Pick any $p_k$ (the normalisation is already uniform). The compactness theorem gives a subsequence converging smoothly to a complete $(M, g, p)$ (extraction); $\operatorname{diam}M \leq D$, so $M$ is closed. By the assumption, $M_k \cong M$ for all large $k$ in the subsequence, so two of them are diffeomorphic to each other: a contradiction.