2. Background and Preliminaries [01XT]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

2. Background and Preliminaries

In this section we review from standard constructions and techniques, which will be used throughout the paper.

2.1. Stratification of Limit Spaces

In this subsection we recall some basic properties of pointed Gromov-Hausdorff limit spaces

\displaystyle(M^{n}_{j},d_{j},p_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}(X,d,p)\,,

(2.1)

where the ${\rm Ric}_{M^{n}_{j}}\geq-(n-1)$ and the noncollapsing assumption ${\rm Vol}(B_{1}(p_{j}))\geq{\rm v}>0$ holds. In particular, we recall the stratification of a noncollapsed limit space, which was first introduced in [ChCo1], and which will play an important role in the proof of Theorem 1.1. The effective version, called the quantitative stratification, which was first introduced in [ChNa13], will be recalled in Section 7. It will play an important role in the estimates of Theorem 1.3.

Given $x\in X$ , we call a metric space $X_{x}$ a tangent cone at $x$ if there exists a sequence $r_{i}\to 0$ such that

\displaystyle(X,r_{i}^{-1}d,x)\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}X_{x}\,.

(2.2)

That tangent cones exist at every point is a consequence of Gromov’s compactness theorem; see for instance the book [P]. A point is called regular if every tangent cone is isometric to $\mathds{R}^{n}$ and otherwise singular. The set of singular points is denoted by $\mathcal{S}$ . As explained below, for noncollapsed limit spaces with a uniform lower Ricci bound, the singular set has codimension $\geq 2$ . At singular points, tangent cones may be highly nonunique with ill-defined dimension of the singular set, and even homeomorphism type, see for instance [CoNa2]. Easy examples show that the singular set need not be closed if one just assumes a uniform a lower bound ${\rm Ric}_{M^{n}_{j}}\geq-(n-1)$ . However, under the assumption of a $2$ -sided bound $|{\rm Ric}_{M^{n}_{j}}|\leq(n-1)$ , the singular set is indeed closed; see [A90], [ChCo2].

For noncollapsed limit spaces, as shown in [ChCo1], every tangent cone is a metric cone, i.e.

\displaystyle X_{x}=C(Z)\,,

(2.3)

for some compact metric space $Z$ , with ${\rm diam}(Z)\leq\pi$ . With this as our starting point, we introduce the following notion of symmetry.

Definition 2.1.

A metric space $Y$ is called $k$ -symmetric if $Y$ is isometric to $\mathds{R}^{k}\times C(Z)$ for some compact metric space $Z$ . We define the closed $k$ th-stratum by

\displaystyle\mathcal{S}^{k}(X)\equiv\{x\in X:\text{ no tangent cone at $x$ is $(k+1)$-symmetric}\}

(2.4)

Thus, in the noncollapsed case, every tangent cone is $0$ -symmetric.

The key result of [ChCo1] is the following:

\displaystyle\dim\mathcal{S}^{k}\leq k\,,

(2.5)

where dimension is in the Hausdorff sense. Thus, away from a set of dimension $k$ , every point has some tangent cone with $(k+1)$ degrees of symmetry. For an effective refinement of this theorem see [ChNa13] and Section 7.

2.2. $\epsilon$ -Regularity Theorems

A central result of this paper is the $\epsilon$ -regularity theorem, Theorem 6.1. The original $\epsilon$ -regularity theorems for Einstein manifolds were given in [A90], [T90] , [BKN89]. They state that if $M^{n}$ is an Einstein manifold, ${\rm Ric}_{M^{n}}=\lambda g$ , with $|\lambda|\leq n-1$ , and if for $B_{2}(p)\subset M^{n}$ ,

\displaystyle\fint_{B_{2}(p)}|{\rm Rm}|^{n/2}<\epsilon(n)\,,

(2.6)

then $\sup_{B_{1}(p)}|{\rm Rm}|\leq 1$ .

In [CCT02], [Ch2], [CD13], $\epsilon$ -regularity theorems were proved under the assumption of $L^{q}$ curvature bounds, $1\leq q<n/2$ , provided $B_{2}(p)$ is assumed sufficiently close to a ball in a cone which splits off an isometric factor $\mathds{R}^{n-2q}$ .

On the other hand, the regularity theory of [ChNa13] for Einstein manifolds depends on $\epsilon$ -regularity theorems which do not assume any $L^{q}$ curvature bounds. In particular, it follows from the work of [A90] that there exists $\epsilon(n)>0$ such that if $|{\rm Ric}_{M^{n}}|\leq\epsilon(n)$ and if

\displaystyle d_{GH}(B_{2}(p),B_{2}(0^{n}))<\epsilon(n)\,,

(2.7)

where $B_{2}(0^{n})\subseteq\mathbb{R}^{n}$ , then $|{\rm Rm}|\leq 1$ on $B_{1}(x)$ .

This result can be extended in several directions. In order to state the extension in full generality, we first recall the notion of the harmonic radius:

Definition 2.2.

For $x\in X$ , we define the harmonic radius $r_{h}(x)$ so that $r_{h}(x)=0$ if no neighborhood of $x$ is a Riemannian manifold. Otherwise we define $r_{h}(x)$ to be the largest $r>0$ such that there exists a mapping $\Phi:B_{r}(0^{n})\to X$ such that:

(1)

$\Phi(0)=x$ with $\Phi$ is a diffeomorphism onto its image.
(2)

$\Delta_{g}x^{\ell}=0$ , where $x^{\ell}$ are the coordinate functions and $\Delta_{g}$ is the Laplace Beltrami operator.

(3)

If $g_{ij}=\Phi^{*}g$ is the pullback metric, then

\displaystyle||g_{ij}-\delta_{ij}||_{C^{0}(B_{r}(0^{n}))}+r||\partial_{k}g_{ij}||_{C^{0}(B_{r}(0^{n}))}\leq 10^{-3}\,.

(2.8)

We call a mapping $\Phi:B_{r}(0^{n})\to X$ as above a harmonic coordinate system. Harmonic coordinates have an abundance of good properties when it comes to regularity issues; see the book [P] for a nice introduction. In particular, if the Ricci curvature is uniformly bounded then in harmonic coordinates, the metric, $g_{ij}$ has a priori $C^{1,\alpha}\cap W^{2,q}$ bounds, for all $\alpha<1$ and $q<\infty$ . If in addition, there is a bound on $|\nabla{\rm Ric}_{M^{n}}|$ , then in harmony coordinates, $g_{ij}$ has $C^{2,\alpha}$ bounds, for all $\alpha<1$ .

The primary theorem we wish to review in this subsection is the following:

Theorem 2.3 ([A90], [ChCo1]).

There exists $\epsilon(n,{\rm v})>0$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq\epsilon$ , ${\rm Vol}(B_{1}(p))>{\rm v}>0$ , and

\displaystyle d_{GH}(B_{2}(p),B_{2}(0))<\epsilon(n)\,,

(2.9)

where $0\in\mathds{R}^{n-1}\times C(Z)$ , then the harmonic radius $r_{h}(x)$ satisfies

\displaystyle r_{h}(x)\geq 1\,.

(2.10)

If $M^{n}$ is further assumed to be Einstein, then the regularity scale $r_{x}$ satisfies $r_{x}\geq 1$ .

By the results of the previous subsection, it is possible to find balls satisfying the above constraint off a subset of Hausdorff codimension $2$ . Moreover, when combined with the quantitative stratification of [ChNa13], see also Section 7, this $\epsilon$ -regularity theorem leads to a priori $L^{p}$ bounds on the curvature. The primary result of the present paper can be viewed as Theorem 6.1, which states that the conclusions of Theorem 2.3 continue to hold if $\mathds{R}^{n-1}$ is replaced by $0\in\mathds{R}^{n-3}\times C(Z)$ .

2.3. Examples

In this subsection, we indicate some simple examples which play an important role in guiding the results of this paper.

Example 2.1.

(The Cone Space $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ ) The main result of this paper, Theorem 1.1, states that $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ , with $\beta<2\pi$ , is not the noncollapsed Gromov-Hausdorff limit of a sequence of manifolds with bounded Ricci curvature. However, it is clear that this space is the Gromov-Hausdorff limit of a a sequence of noncollapsed manifolds with a uniform lower Ricci curvature bound. Indeed, by rounding off $C(S^{1}_{\beta})$ we see that $\mathds{R}^{n-2}\times C(S^{1}_{\beta})$ can appear as a noncollapsed limit of manifolds with nonnegative sectional curvature.

In this example, let us just consider the two dimensional cone $C(S^{1}_{\beta})$ with $\beta<2\pi$ . Regard $S^{1}_{\beta}$ as $0\leq\theta\leq 2\pi$ , with the end points identified. Then the Laplacian on $S^{1}_{\beta}$ is $(\frac{2\pi}{\beta})^{2}\cdot\frac{\partial^{2}}{\partial\theta^{2}}$ . The eigenfunctions are of the form $e^{ik\theta}$ , where $k$ is an integer. Written in polar coordinates, a basis for the bounded harmonic functions on $C(S^{1}_{\beta})$ is $\{r^{\frac{2\pi}{\beta}|k|}\cdot e^{ik\theta}\}$ . In particular, we see from this that if $\beta<2\pi$ then $|\nabla(r^{\frac{2\pi}{\beta}|k|}\cdot e^{ik\theta})|\to 0$ as $r\to 0$ . As a consequence, every bounded harmonic function has vanishing gradient at the vertex, which is a set of positive $(n-2)$ -dimensional Hausdorff measure. By considering examples with more vertices, we can construct limit spaces where bounded harmonic functions $h$ must have vanishing gradient on bounded subsets sets of arbitrarily large, or even infinite, $(n-2)$ -dimensional Hausdorff measure. This set can even be taken to be dense.

Example 2.2.

(The Eguchi-Hanson manifold) The Eguchi-Hanson metric $g$ is a complete Ricci flat metric on the cotangent bundle of $S^{2}$ , which at infinity, becomes rapidly asymptotic to the metric cone on $\mathds{R}\mathds{P}(3)$ or equivalently to $\mathds{R}^{4}/\mathds{Z}_{2}$ , where $\mathds{Z}_{2}$ acts on $\mathds{R}^{4}$ by $x\to-x$ . When the metric $g$ is scaled down by $g\to r^{2}g$ , with $r\to 0$ , one obtains a family of Ricci flat manifolds whose Gromov-Hausdorff limit is $C(\mathds{R}\mathds{P}(3))=\mathds{R}^{4}/\mathds{Z}_{2}$ . This is the simplest example which shows that even under the assumption of Ricci flatness and noncollapsing, Gromov-Hausdorff limit spaces can contain codimension 4 singularities.

Example 2.3.

(Infinitely many topological types in dimension 4) Let $T^{3}$ denote a flat $3$ -torus. According to Anderson [A93], there is a collapsing sequence of manifolds $(M^{4}_{j},d_{j})\stackrel{{\scriptstyle d_{GH}}}{{\longrightarrow}}T^{3}$ satisfying

		$\displaystyle{\rm diam}(M^{4}_{j})\leq 1\,,$
		$\displaystyle\|{\rm Ric}_{M^{n}_{j}}\|\leq\epsilon_{j}\to 0\,,$
		$\displaystyle{\rm Vol}(M^{4}_{j})\to 0\,,$
		$\displaystyle b_{2}(M^{4}_{j})\to\infty\,,$		(2.11)

where $b_{2}(M^{4}_{j})$ denotes the second Betti number of $M^{4}_{j}$ . In particular, Theorem 1.4 , the finiteness theorem in dimension $4$ , does not extend to the case in which the lower volume bound is dropped.