On the structure of locally conformally flat orbifolds and ALE metrics

Abstract.

In this paper, we prove several structure theorems for locally conformally flat, positive Yamabe orbifolds and nonnegative scalar curvature, ALE manifolds. These two kinds of spaces can be related by conformal blow-up and conformal compactification. For the orbifolds, we prove that such orbifolds admit a manifold cover. For the ALE manifolds, the homomorphism of the fundamental group for the ALE space induced by the embedding of the ALE end is always injective. Using these properties, several classifications of such ALE manifolds and orbifolds are given in low dimensions. As an application to the moduli space, we prove that the football orbifold $\mathbb{S}^{4}/\Gamma$ cannot be realized as the Gromov-Hausdorff limit. In addition, we prove the positive mass theorem of these ALE ends and give a simple proof for the optimal decay rate. Using the positive mass theorem, we can solve the orbifold Yamabe problem in the locally conformally flat case.

1. Introduction

The locally conformally flat structure is a natural generalization of the conformal structure on two-dimensional surfaces. It is well known to modern geometric analysts because of its connection to the vanishing of the Weyl curvature. Much work has been done to study such manifolds, for example, [SYconformal]. We will write LCF for short. In this paper, we always assume $n\geq 3$. We are interested in the moduli space of closed LCF manifolds with bounded curvature $L^{\frac{n}{2}}$ norm. To gain better control over the moduli space as it approaches the boundary, we assume that the manifolds in the moduli space have the Yamabe metric with a strictly positive Yamabe constant. In this way, we avoid local collapsing and ensure a non-collapsing orbifold limit. Such moduli spaces are considered in [Aku], [TV1], [TV2], [TV3]. Similar moduli spaces are studied in the Einstein case; see, for example, [And89], [BKN].

Let $\mathfrak{M}(n,\mu_{0},C_{0})$ denote the set of closed n-dimensional Riemannian manifolds that satisfy the following:

• locally conformally flat

• unit-scalar curvature: $S(g)=1$

• $g$ is the Yamabe metric with the Yamabe constant $Y(M,[g])\geq\mu_{0}>0$

• $\int_{M}|\operatorname{Ric(g)|^{\frac{n}{2}}}dvol_{g}<C_{0}$

Let $\mathfrak{M}^{\prime}(n,\mu_{0},C_{0})$ be the moduli space if we in addition assume that the manifolds are orientable.

Note that the scalar curvatures are constant 1. Thus, the volume $\operatorname{vol}(M,g)$ is uniformly bounded from above by the Yamabe constant of $\mathbb{S}^{n}$ and from below by assumption. On the other hand, from the local ball non-collapsing, we have the uniform upper bound on the diameter; see [Aku] for details.

Definition 1.1.

A topological space $M^{n}$ is called an orbifold if for every point $x\in M$, there is a neighborhood $U\ni x$ where there exist $\tilde{U}\subseteq\mathbb{R}^{n}$ and a finite group $G$ acting on $\tilde{U}$ such that there is a $G$-invariant map $\pi:\tilde{U}\to U$ that induces a diffeomorphism $U\cong\tilde{U}/G$.

An orbifold $(M,g)$ is called a Riemannian orbifold if there is a $G$-invariant $C^{\infty}$ metric $\tilde{g}$ in $\tilde{U}$ such that away from singularities, $\pi^{*}g=\tilde{g}$.

In this paper, we consider only the case where the singularities are isolated. This means that the action of $G$ contains only one fixed point in $\tilde{U}$. We say that such an orbifold is oriented if $M\setminus\{x_{1},\dots,x_{k}\}$ is an oriented manifold. In this way, the orientation on $M\setminus\{x_{1},\dots,x_{k}\}$ induces the orientation on $\mathbb{S}^{n-1}/\Gamma$, and thus the action of every orbifold group $\Gamma\subset{\rm{SO}}(n)$ is orientation-preserving with respect to this orientation. Thus, there is a well-defined choice of orientation on $M$.

For the moduli space $\mathfrak{M}(n,\mu_{0},C_{0})$, as shown in [TV2], it can be compactified by adding compact LCF multifolds of positive orbifold Yamabe invariant with finitely many singularities. A Riemannian multifold is obtained by identifying finitely many points from

$$\tilde{M}=\coprod_{i=1}^{N}M_{i},$$

where $M_{i}$ is a Riemannian orbifold with positive orbifold Yamabe invariant. Note that orbifold points with different tangent cones can be identified together. We call a multifold singularity irreducible if it is the orbifold point in the usual sense, i.e., no other orbifold points are identified with this orbifold point.

Such non-trivial multifold points correspond to the points where curvature blows up in a sequence approaching the boundary of the moduli space. To understand the curvature blow-up behavior, a rescaling argument is needed. An important result is shown in [TV3], which demonstrates that such a limit always exists with Euclidean volume growth and, in particular, is an LCF, scalar-flat, asymptotically locally Euclidean (ALE) manifold:

Definition 1.2.

Let $(M,g)$ be a complete, non-compact Riemannian manifold. An end of $(M,g)$ is an unbounded component of the complement of some compact subset $K\subseteq M$.

An end $E\subseteq M-K$ is called asymptotically locally Euclidean (ALE) of order $\tau>0$ if there is a diffeomorphism:

$$\phi:E\to(\mathbb{R}^{n}\setminus B(0,r))/\Gamma$$

for some finite subgroup $\Gamma$ acting freely on $\mathbb{R}^{n}-B(0,r)$. Under such identification:

$$g_{ij}=\delta_{ij}+O(r^{-\tau})$$

$$\partial^{m}g_{ij}=O(r^{-\tau-|m|})$$

for all partial derivatives of order $|m|$ with multi-index $m$ as $r\to\infty$. We call an end asymptotically Euclidean (AE) if the group $\Gamma$ is trivial.

We call $(M,g)$ an ALE manifold if it contains finitely many ALE ends $\{E_{i}\}$ of order $\{k_{i}\}$, respectively. We call $(M,g)$ an AE manifold if all the groups in the ends are trivial. If $(M,g)$ is orientable, then the orientation will induce an orientation on $\mathbb{S}^{n-1}/\Gamma$, and $\Gamma\subset{\rm{SO}}(n)$ with respect to this orientation.

In this paper, our aim is to study such irreducible orbifolds and ALE manifolds. Let $(M,g)$ be a complete, non-compact, LCF, nonnegative scalar curvature, ALE manifold. It is well known that $(M,g)$ can be conformally compactified to be an LCF orbifold $M^{\prime}$ with positive orbifold Yamabe invariant; see [TV2]. Conversely, given an orbifold $(M^{\prime},g^{\prime})$ with at most isolated singularities and positive orbifold Yamabe invariant, we can associate such an orbifold with an ALE manifold by conformal blow-up using the conformal Green’s function at these singularities. We will review these facts and give short proofs in Section 3.4. Thus, all results in this paper can be equivalently stated for orbifolds or for ALE metrics.

1.1. Orbifold results

Any orbifold $M$ has an orbifold universal cover $\tilde{M}_{orb}$ in the sense that any orbifold is covered by the universal orbifold cover as a topological space, and $\pi_{1}^{orb}$ is defined as the deck transformation group of $\pi:\tilde{M}\rightarrow M$. The precise definition of $\pi_{1}^{orb}(M)$ can be found in [orbifold]. In general, $\pi_{1}^{orb}(M)$ is not isomorphic to the topological fundamental group $\pi_{1}(M)$.

An important example is as follows:

Definition 1.3.

A football orbifold is $\mathbb{S}^{n}/\Gamma$, where $\Gamma\subseteq{\rm{O}}(n)\subset{\rm{O}}(n+1)$, having two fixed antipodal points. The orbifold metric is given by the quotient of the standard spherical metric.

In this case, $\pi_{1}^{orb}(M)\simeq\Gamma$, while $\pi_{1}(M)=\{e\}$ since it is simply connected.

We call an orbifold good if it is obtained from a manifold quotient by some group with fixed points. We say such an orbifold has a manifold cover, although it is not a genuine covering space. Football orbifolds are good orbifolds. Not all orbifolds are good; a famous example is the “Teardrop” orbifold, which is constructed by identifying the boundary of two-dimensional disc $B_{1}\subseteq\mathbb{R}^{2}$ with the boundary of $B_{1}/\mathbb{Z}_{k}$. A Teardrop orbifold cannot be covered by a manifold. The “bad” Teardrop example is $2$-dimensional, LCF, and admits a metric of positive curvature. But in higher dimensions we have the following.

Theorem 1.4.

If $(M,g)$ is a compact LCF orbifold with positive scalar curvature and $\dim(M)\geq 3$, then $(M,g)$ is a good orbifold.

Theorem 1.4 is proved in Section 3.4.

We next recall some background in conformal geometry. For an LCF manifold with non-negative scalar curvature, $(M,g)$, Schoen-Yau [SYconformal] proved there is a developing map, $\phi:(\tilde{M},\tilde{g})\to(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$, which is a conformal embedding, where $\pi:\tilde{M}\to M$ is the universal covering of $M$, $\tilde{g}=\pi^{*}g$, and $g_{\mathbb{S}^{n}}$ is the round metric. This is generalized to good orbifolds in the following manner. For an LCF good Riemannian orbifold with positive scalar curvature and finitely many singular points, the orbifold universal covering $\tilde{M}_{orb}$ is a manifold by the universal property. Furthermore, $\pi^{*}g$ is a smooth metric, so the developing map is defined. We write $M=\Omega/G$, where $\Omega\subseteq\mathbb{S}^{n}$, and $G\subseteq C(n)\leq\rm{O}(n+1,1)$, the conformal group, which acts properly and discontinuously on $\Omega$. Denote $\Lambda:=\mathbb{S}^{n}\setminus\Omega$. When $M$ is compact, $\Lambda$ coincides with the limiting set of $G$. By Selberg’s lemma, there is a finite index torsion-free subgroup $H\subseteq G$. Thus, $M$ has a finite manifold cover. Note the Hausdorff dimension $dim_{\mathcal{H}}(\Lambda_{G})=dim_{\mathcal{H}}(\Lambda_{H})$.

Now we define the conformal connected sum. Let $(M_{1},g_{1})$, $(M_{2},g_{2})$ be two LCF orbifolds. Fix an orientation on $\mathbb{S}^{n}$. Let $A\subseteq\mathbb{S}^{n}$ be diffeomorphic to $[0,1]\times\mathbb{S}^{n-1}$, where $\mathbb{S}^{n}\setminus A$ has two components, $D_{1}$, $D_{2}$, and each $D_{i}$ is diffeomorphic to $B_{\mathbb{R}^{n}}(1)$.

Definition 1.5 ([Iz1]).

A conformal connected sum of $(M_{1},g_{1})$, $(M_{2},g_{2})$ is defined by:

$$(M_{1}\setminus\psi_{1}(E_{1}),g_{1})\bigcup_{\psi_{1}\circ\psi_{1}^{-1}|_{\psi_{1}(A)}}(M_{2}\setminus\psi_{2}(E_{2}),g_{2})$$

where $E_{i}=A\cup D_{i}$ and $\psi_{i}:E_{i}\to M_{i}$ is the orientation-preserving conformal map. We denote it as $(M_{1},g_{1})\#_{C}(M_{2},g_{2})$.

We can use the above definition to get a strong restriction on certain compact LCF orbifolds:

Theorem 1.6.

Let $(M,g)$ be a compact LCF orbifold with $\dim_{\mathcal{H}}(\Lambda)<1$, then $(M,g)$ is finitely covered by $\mathbb{S}^{n}$ or a conformal connected sum

$$\#_{k}\mathbb{S}^{1}\times\mathbb{S}^{n-1},$$

for some $k>0$.

If $n=3,4$, then $(M,g)$ is a conformal connected sum

$$M=\#_{i}(\mathbb{S}^{n-1}\times\mathbb{R}/G_{i})\#_{j}(\mathbb{S}^{n}/\Gamma_{j}),$$

where each $G_{i}$ is a discrete cocompact subgroup of the isometry group of the standard product metric of $\mathbb{S}^{n-1}\times\mathbb{R}$ with at most isolated singularities; each $\Gamma_{i}$ is a finite subgroup of ${\rm{O}}(n+1)$ with at most isolated singularities.

In any dimension: If $(M,g)$ is orientable, then if $n$ is odd, $(M,g)$ is a manifold; if $n$ is even, the number of non-trivial orbifold points is even, and they occur in orientation-reversing conjugate pairs.

Theorem 1.6 will be proved in Section 4.2.

Remark 1.7.

Theorem 1.6 is an extension of Theorem 6.1 in Izeki [Ilimit] to the orbifold case. Here we make use of the structure of the orbifolds proved in Theorem 1.4.

Remark 1.8.

If $n=3$ or $4$, then non-negative scalar curvature implies $\dim_{\mathcal{H}}(\Lambda)<1$. Thus, the above theorem directly applies to the LCF orbifold with non-negative scalar curvature.

Remark 1.9.

In odd dimensions, by Theorem 1.4, the only non-trivial orbifold points are $\mathbb{Z}_{2}$-singularities, which are obtained by taking the quotient of an orientation-reversing $\mathbb{Z}_{2}$ map. In even dimensions, we will need the algebraic result about the orbits of fixed points proved in the Appendix.

If $n=4$, Chen-Tang-Zhu [ChenZhu] show that $(M,g)$ is diffeomorphic to a connected sum of football orbifolds. The key point is that, in this situation, we have a conformal connected sum. As a consequence of Theorem 1.6, we have the following corollary:

Corollary 1.10.

Any compact 4-orbifold $(M,g)$ with at most isolated singularities and with positive isotropic curvature has a finite manifold cover that is diffeomorphic to $\#_{k}(\mathbb{S}^{1}\times\mathbb{S}^{3})$.

Remark 1.11.

The corollary a priori seems to have nothing to do with LCF metrics. The connection is pointed out by [ChenZhu], namely: a compact four-orbifold with at most isolated singularities admits positive isotropic curvature if and only if it admits an LCF metric with positive scalar curvature (Corollary 2 in [ChenZhu]). Since the conclusion is topological, we work on its LCF metric.

1.2. ALE results

Our first result for the ALE manifolds is the structure of the ALE ends. We denote: the k-th end $E_{k}$ is diffeomorphic to $\mathbb{R}^{+}\times\mathbb{S}^{n-1}/\Gamma_{k}$ for some finite subgroup $\Gamma_{k}\subset{\rm{O}}(n)$.

Theorem 1.12.

Let $(M,g)$ be a complete, non-compact, LCF, non-negative scalar curvature ALE manifold. Then the inclusion map $i_{k}:E_{k}\to M$ induces an injective homomorphism ${i_{k}}_{*}:\Gamma_{k}\to\pi_{1}(M)$.

Theorem 1.12 will be proved in Section 3.3.

Note that this is not always true for ALE manifolds without the LCF assumption. For example, in the Eguchi-Hanson metric, $M_{EH}$, the group in the end is $\Gamma\cong\mathbb{Z}_{2}$, but $M_{EH}$ is simply connected. We prove this theorem by combining the embedding theorem of [SYconformal] and the observation that the only closed totally umbilic submanifolds in $\mathbb{R}^{n}$ are $\mathbb{S}^{n-1}$.

From Remark 1.8, in dimension 3 and 4, we know that LCF orbifold with positive Yamabe satisfies $\operatorname{dim}_{\mathcal{H}}(\Lambda)<1$. Thus, Theorem 1.6 an be applied directly to give an ALE manifold classification. In particular, we have the following corollary about the ends of oriented ALE manifolds.

Corollary 1.13.

Let $M$ be an orientable, LCF, ALE manifold with non-negative scalar curvature. If $n=3$, then $M$ is AE. If $n=4$, the groups of the non-trivial ALE ends occur in orientation-reversing conjugate pairs, so there must be an even number of ends with non-trivial group. In particular, any orientable LCF ALE $4$-manifold with only one end is AE.

As an application, We have a refined structure about the boundary of the oriented moduli space.

Theorem 1.14.

Let $(M_{i},g_{i})$ be a sequence in $\mathfrak{M}^{\prime}(n,\mu_{0},C_{0})$. If $n=3$, then the Gromov-Hausdorff limit is finitely many LCF manifolds with finitely many points identified.

If $n=4$, then the Gromov-Hausdorff limit is finitely many LCF orbifolds with finitely many points identified, such that the tangent cone at any singular point has an even number of cones with non-trivial orbifold groups, which appear in pairs.

Theorem 1.14 will be proved in Section 5.1. This also relies on the result proved in the Appendix, giving certain conditions when orbifold isotropy groups must occur in pairs. For an illustration of the degeneration in Theorem 1.14, see Figure 1. In Figure 1, the limit, $(M_{\infty},g_{\infty})$, will be a multifold with exactly one multifold singularity. The rescaling limit at the point where the curvature tends to infinity, $(\hat{M}_{\infty},\hat{g}_{\infty})$, will be a 2-end ALE manifold.

Figure 1. A sequence $\{(M_{i},g_{i})\}$, where $(M_{i},g_{i})\in\mathfrak{M}^{\prime}(4,\mu_{0},C_{0})$. The limit $(M_{\infty},g_{\infty})$ is a multifold with one multifold singularity. The rescaled limit will be a 2-end ALE manifold $(\hat{M}_{\infty},\hat{g}_{\infty})$.

This has the following corollary that rules out certain multifolds being in the boundary of the oriented moduli space.

Corollary 1.15.

Let $n=3,4$, and $(M_{o},g_{o})$ be a closed, orientable, LCF multifold with finitely many isolated singularities and with only irreducible orbifold points. Then $(M_{o},g_{o})$ cannot be realized as a limit of a sequence of closed, orientable, LCF manifolds $(M_{i}^{n},g_{i})\in\mathfrak{M}^{\prime}(n,\mu_{0},C_{0})$.

In particular, we can show that the football metric $\mathbb{S}^{4}/\Gamma$ is not in the boundary of the oriented moduli space. However, for certain groups $\Gamma$, it is within the boundary of the non-orientable moduli space; see Section LABEL:nonorientable_construction for the construction.

In the Einstein case, a similar result is shown in [biquard], [ozuch2022noncollapsed], where $\mathbb{S}^{4}/\Gamma$ cannot be realized as a limit of non-collapsing Einstein manifolds with $L^{2}$ curvature bound. In the Einstein case, they identified certain local obstructions on the singularity. In our case, we only use the structure of the ALE manifolds from Theorem 1.6 when rescaling the metric near the singularity.

Note that, similar to Definition 1.2, we can define an ALE orbifold with isolated singularities, where we allow the compact region $K$ to have isolated singularities. We can prove a positive mass theorem for such ALE orbifolds.

Theorem 1.16.

Let $(M,g)$ be an LCF, nonnegative scalar curvature, ALE orbifold with $\tau>\frac{n-2}{2}$, then the ADM mass is non-negative for any end of $(M,g)$. The mass is zero at an end if and only if $(M,g)\cong(\mathbb{R}^{n},g_{\mathbb{R}^{n}})/\Gamma$, where $\Gamma\subseteq{\rm{O}}(n)$ is a finite subgroup that acts isometrically and fixes the origin. Furthermore, if $(M,g)$ is scalar-flat, then $(M,g)$ is ALE of order $n-2$, which is optimal.

The optimal decay will be proved in Section 3.4 (See Corollary 3.10 and Remark 3.11). The positive mass part will be proved in Section LABEL:PMT_proof.

This is significant since the positive mass theorem for ALE manifolds with non-negative scalar curvature is not always true. See the counterexamples in [Lebrun]. The optimal decay rate for obstruction-flat scalar-flat ALE metrics was previously proved by [AcheViaclovsky], but our proof in the LCF case is much easier by utilizing the conformal Green’s function.

For application, we consider the orbifold Yamabe problem. Given a Riemannian orbifold $(M,g)$, one wants to find a constant scalar metric in the conformal class $[g]$. Unlike the manifold Yamabe problem, this is not always solvable. Counterexamples are constructed in [viaclovsky2010monopole]. One core missing ingredient is the positive mass theorem for ALE manifolds. With the help of Theorem 1.16, we can solve the orbifold Yamabe problem in the special case.

Corollary 1.17.

If $(M^{n},g)$ is an LCF compact orbifold with positive scalar curvature, then there exists a solution $\tilde{g}=u^{\frac{4}{n-2}}g$ to the orbifold Yamabe problem on $(M,g)$.

This will be proved in Section LABEL:PMT_proof.

Acknowledgments: We thank Jeff Viaclovsky for patient guidance and helpful discussions. The author is supported by NSF Research Grant DMS-2105478.

2. Preliminaries

2.1. Yamabe invariant

Let us recall the Yamabe metric and Yamabe invariant:

Definition 2.1 (Yamabe constant).

Let $(M,g)$ be a compact Riemannian manifold. The Yamabe constant associated with its conformal class $[g]$ is

$$Y(M,[g])=\inf_{\tilde{g}\in[g]}\operatorname{Vol}(\tilde{g})^{-\frac{n-2}{n}}\int_{M}S({\tilde{g})}dvol_{\tilde{g}}.$$

The Yamabe metric is the metric in $[g]$ that achieves the above infimum.

The Yamabe invariant of $M$ is the maximum of the Yamabe constant among all conformal classes $[g]$:

$$Y(M)=\sup_{[g]}Y(M,[g]).$$

Note that $Y(M,[g])$ is scaling invariant. From the famous resolution of the Yamabe problem by [Ya60, Tr68, Au76, Sch84], such a Yamabe minimizer is always realized by a smooth metric with constant scalar curvature. Note that the Yamabe metric with a strictly positive Yamabe invariant implies the universal upper bound for the Sobolev constant, hence no local collapsing.

We also need a generalization of the Yamabe constant and Yamabe invariant on orbifolds. Denote the orbifold Yamabe constant of an orbifold $(M,g)$:

$$Y_{orb}(M,[g])=\inf_{\tilde{g}\in[g]}\operatorname{Vol}(\tilde{g})^{-\frac{n-2}{n}}\int_{M}S({\tilde{g}})dvol_{\tilde{g}},$$

and the orbifold Yamabe invariant:

$$Y_{orb}(M)=\sup_{[g]}Y_{orb}(M,[g]).$$

Parallel to the manifold Yamabe problem, an analog of Aubin’s existence theorem is as follows:

Theorem 2.2 ([AB04],[akutagawa2012computations]).

Let $(M,g)$ be a Riemannian orbifold with isolated singularities $\{p_{1},...,p_{k}\}$, with orbifold groups $G_{i}\leq{\rm{O}}(n)$, for $i=1,...,k$. Then:

$$Y_{orb}(M,[g])\leq Y(\mathbb{S}^{n})\min_{i}|G_{i}|^{-\frac{2}{n}}.$$

Furthermore, if this inequality is strict, then there exists a smooth conformal metric $\tilde{g}=u^{\frac{4}{n-2}}g$, which minimizes the Yamabe functional, and thus has constant scalar curvature.

To fully solve the orbifold Yamabe problem, we need to show the strict inequality. If one wants to use Schoen’s test function from [Sch84], a positive mass theorem for ALE manifolds is needed. However, a positive mass theorem for ALE manifolds is not always true. See the counterexamples of [Lebrun] for negative mass ALE metrics. This makes the orbifold Yamabe problem more subtle than in the manifold case. In fact, an example of the non-existence of the constant scalar metric in certain conformal classes is given in [viaclovsky2010monopole]. See also [ju2023conformally] for recent developments.

In this paper, we prove the orbifold Yamabe problem for LCF orbifolds with a positive orbifold Yamabe constant. See Corollary 1.17.

2.2. Locally conformally flat manifolds and Kleinian structure

We are interested in the locally conformally flat (LCF) manifolds:

Definition 2.3 (The LCF manifolds).

A Riemannian manifold $(M^{n},g)$ is called locally conformally flat (LCF) if for every point $p\in M$, there exists a neighborhood $U$, a chart $\phi:\mathbb{R}^{n}\to U$ such that $\phi^{*}g(x)=\lambda(x)g_{\mathbb{R}^{n}}$, for some smooth function $\lambda:\mathbb{R}^{n}\to\mathbb{R}^{+}$.

In order to understand the structure of such manifolds, we recall Schoen-Yau’s embedding theorem from [SYconformal], which is later verified by [ChodoshLi], [LUS].

Theorem 2.4 (Schoen-Yau).

Let $(M^{n},g)$ be an $n$-dimensional ($n\geq 3$) complete LCF manifold with scalar curvature $S(g)\geq 0$, then for the universal cover $(\tilde{M},g)$ there exists a conformal map $\phi:\tilde{M}\to\mathbb{S}^{n}$ which is an embedding.

Now, we introduce several concepts of Kleinian manifolds:

The conformal embedding map $\phi$ induces an injective homomorphism, which is called the holonomy representation, from the deck transformation of $\tilde{M}$, $\operatorname{Deck}(\tilde{M})\cong\pi_{1}(M)$, to the conformal group of $\mathbb{S}^{n}$, $C(n)$:

$$\rho:\pi_{1}(M)\to C(n).$$

Denote $\Omega\subseteq\mathbb{S}^{n}$ as the domain of discontinuity, which is the set in which $\rho(\pi_{1}(M))$ acts on properly discontinuously, and $\Lambda$ as the limit set of $\rho(\pi_{1}(M))$, which is the complement of $\Omega$ and is the minimal invariant closed subset of $\rho(\pi_{1}(M))$. A manifold $(M,g)$ is called Kleinian if $(M,g)$ is conformally equivalent to some $\Omega/\Gamma$.

In addition, we introduce the Liouville theorem [conformalbook]:

Theorem 2.5 (Liouville).

Let $U$, $V$ be open connected subsets of $\mathbb{S}^{n}$, $n\geq 3$, $f:U\to V$ be a conformal map. Then $f$ can be uniquely extended to a conformal map $f:\mathbb{S}^{n}\to\mathbb{S}^{n}$.

From Theorem 2.4 and Theorem 2.5, we have that an LCF manifold with non-negative scalar curvature is Kleinian.

We also need to characterize the different elements in $C(n)$. Note that the group $C(n)$ has an extension to the action on the hyperbolic space $\mathbb{H}^{n+1}$ as isometries. In particular, $C(n)\cong\operatorname{Isom}(\mathbb{H}^{n+1})\leq{\rm{O}}(n+1,1)$, [conformalbook]. If we consider the ball model, $(\mathbb{D}^{n+1},g_{\mathbb{H}^{n+1}})$, then by the Brouwer fixed point theorem, every element in $C(n)$ acts on $\mathbb{H}^{n+1}$ and will have fixed points, so we distinguish different conjugate classes by distinguishing their fixed points.

Definition 2.6.

[conformalbook] Let $g\in C(n)\cong\operatorname{Isom}(\mathbb{H}^{n+1})$:

• $g$ is called elliptic if $g$ has fixed points in $\mathbb{D}^{n+1}$;

• $g$ is called parabolic if $g$ has a single fixed point in $\partial\mathbb{H}^{n+1}\cong\mathbb{S}^{n}$;

• $g$ is called hyperbolic if $g$ has two distinct fixed points in $\partial\mathbb{H}^{n+1}\cong\mathbb{S}^{n}$.

Remark 2.7.

Note that if $g$ is of finite order, then $g$ is elliptic. Also, note that if $g$ is of infinite order, then the action $g$ on $\Omega$ is fixed point free.

From [conformalbook], if $g\in C(n)$ is elliptic, then there exists $\gamma\in C(n)$ such that $\gamma^{-1}g\gamma\in{\rm{O}}(n+1)$. Later, we will use this to study the local isotropy subgroups.

A key property to identify $C(n)$ with a subgroup of ${\rm{O}}(n+1)$ is the famous Selberg Lemma.

Theorem 2.8 (Selberg Lemma).

A finitely generated subgroup of a linear group over a field of characteristic 0 has a finite-index subgroup which is torsion-free.

In particular, we can use the fundamental result (Corollary 4.8, [wehrfritz2012infinite]), there is a normal, finite-index, torsion-free subgroup. We should point it out that the construction of such normal subgroup is elementary: let $G$ be a finitely generated subgroup of linear group, and let $H\leq G$ be the finite-index subgroup produced by Theorem 2.8. Then, consider $N=\bigcap_{g\in G}gHg^{-1}$. Since $H$ is of finite-index, then the intersection is over a finite collection, which means $N$ is also of finte-index. Clearly, $N$ is normal and torsion-free. Thus, if $(M,g)$ is Kleinian, there exists a finite cover whose fundamental group is torsion-free.

3. ALE ends and orbifold singularities

3.1. Totally umbilic submanifolds in $\mathbb{R}^{n}$

To study the conformal embedding, we need to understand the conformal invariant geometric objects. Motivated by this, we study the conformal invariant submanifolds.

Let $N\subset(M,g)$ be a hypersurface. Consider the conformal change: $g\to\hat{g}=e^{2\psi}g$, then the second fundamental form of $N$:

$$\hat{\text{II}}(X,Y)=\text{II}(X,Y)-\hat{g}(X,Y)\nabla\psi|_{normal}$$

for every $X,Y$ smooth vector fields tangential to $N$.

Definition 3.1 (Totally umbilic submanifolds).

$N\subset(M,g)$ is called totally umbilic if $\forall x\in N$,

$${\text{II}}_{x}=\lambda(x)g|_{N},$$

for some smooth function $\lambda:N\to\mathbb{R}$.

It can be easily seen that the totally umbilic submanifolds are invariant under conformal change. Note that this invariance is pointwise, i.e., the umbilic points are invariant under conformal change.

For $(M,g)$, if there is a conformal embedding $\phi:M\to\mathbb{S}^{n}$, and $(M,g)$ is not conformally equivalent to $(\mathbb{S}^{n},g_{\mathbb{S}^{n}})$, then by the stereographic projection, we have a conformal embedding $\bar{\phi}:M\to\mathbb{R}^{n}$. Thus, all the totally umbilic submanifolds in $(M,g)$ are totally umbilic in $\mathbb{R}^{n}$, which gives a very strong constraint.

Recall the classical theorem of totally umbilic submanifolds in $\mathbb{R}^{n}$. This is an old theorem that dates back to Cartan. Since it is difficult to find the precise reference, we include the proof as well.

Theorem 3.2 (Totally umbilic submanifolds in $\mathbb{R}^{n}$).

Let $n\geq 3$. The only closed totally umbilic submanifolds in $\mathbb{R}^{n}$ are the round $\mathbb{S}^{n-1}$.

Proof.

Let $i:N\to\mathbb{R}^{n}$ be a totally umbilic submanifold. By the Gauss-Codazzi equation, we have:

$$0=\operatorname{Rm}_{N}(X,Y,Z,W)-\langle\text{II}(X,W),\text{II}(Y,Z)\rangle+\langle\text{II}(X,Z),\text{II}(Y,W)\rangle$$

By the second Bianchi identity, we have:

$$sec_{N}=\lambda^{2},$$

where $\text{II}=\lambda g|_{N}$ for some constant $\lambda$. We may assume $\lambda\geq 0$.

When $\lambda=0$, then $N$ is the flat, totally geodesic hypersurface in $\mathbb{R}^{n}$, then $N$ is a subset of a flat $\mathbb{R}^{n-1}$;

When $\lambda>0$, then $N$ has a constant positive sectional curvature. Now consider the following.

$$c:N\to\mathbb{R}^{n},x\mapsto i(x)+\frac{1}{\lambda}\vec{n},$$

where $\vec{n}$ is the inner normal vector field. Then

$$\nabla_{X}c=\nabla_{X}i+\frac{1}{\lambda}\nabla_{\nabla_{X}i}\vec{n}=\nabla_{X}i-\nabla_{X}i=0.$$

Thus, we have $c$ is a constant and $|i-c|=\frac{1}{\lambda}$. Hence, the image of $i$ is a subset of the $\frac{1}{\lambda}$-ball centered on $c$.

If we assume $N$ to be closed, then $i(N)\simeq\mathbb{S}^{n-1}$, the round sphere. ∎

Combined with Theorem 2.4, we have, if $(M,g)$ is an LCF manifold with non-negative scalar curvature, then the only closed totally umbilic submanifolds in the universal covers are the conformal round spheres.

3.2. Harmonic functions on ALE manifolds

Now, we recall the theory of harmonic functions in ALE manifolds.

Theorem 3.3 (Bounded harmonic functions on ALE manifolds).

Let $(M,g)$ be an ALE manifold. Let $u$ be a bounded function such that, outside a compact set $K$, on an ALE end it satisfies $\Delta u=0$. Then, on this end, $\lim\limits_{x\to\infty}u(x)=c$ for some constant $c$. In fact, $u$ has an expansion at the ALE end: there exists $\epsilon>0$ such that

$$u(x)=c+Ar^{2-n}+O(r^{2-n-\epsilon})$$

as $r\to\infty$.

Proof.

Here, we only sketch the proof. The proof can be found in many references; see, for example, [DK].

We first decompose $u=u_{0}+u_{1}$ at the ALE end outside the compact set, where $u_{0}$ satisfies $\Delta_{\mathbb{R}^{n}}u_{0}=0$ and $u_{1}$ is $O(r^{-\epsilon})$ for some $\epsilon$. This is done by choosing a suitable weighted Sobolev space to solve the Poisson equation and using the corresponding Schauder estimate. This is possible because of the ALE assumption and the fact that the exceptional set of the Laplacian is discrete. Now, the expansion of $u_{0}$ follows from the Green’s function expansion and the fact that the bounded global harmonic functions on $\mathbb{R}^{n}$ are constant. Finally, we can iteratively get better estimates using the fact that the exceptional weight is discrete. ∎

Remark 3.4.

The same techniques hold for equation $\Delta+cS$, where $c$ is some constant, since $S\in O(r^{-2-\tau})$, for an ALE end of order $\tau$.

Now, we can prove the theorem about the end of the LCF, ALE manifolds.

Theorem 3.5.

For $(M,g)$, LCF, ALE manifolds of non-negative scalar curvature, if there exists $\phi$, where $\phi:M\to\mathbb{S}^{n}$ is a conformal embedding, then $(M,g)$ is an asymptotically Euclidean manifold, i.e., $\Gamma=\{e\}$ for all ends.

Proof.

Without loss of generality, we may assume that there is only one ALE end of order $\tau>0$ with group $\Gamma$. Our goal is to construct totally umbilic $\mathbb{S}^{n-1}/\Gamma$ in the end when $r$ is very large, where $r$ is the distance function with respect to a point $z\in M$. Due to Theorem 3.2, we can conclude that $\Gamma=\{e\}$. In general, producing totally umbilic submanifolds directly in general manifolds is very difficult. However, in our case, we can use the ALE geometry at infinity and conformal equivalence to produce such submanifolds.

Note $\phi:M\to\mathbb{S}^{n}$ is a conformal map. Choosing a base point $o\in M$, we can do the stereographic projection based on $\phi(o)$. The resulting map: $\bar{\phi}:M\setminus\{o\}\to\mathbb{R}^{n}$ is also conformal with the pull-back metric: $\bar{\phi}^{*}g_{\mathbb{R}^{n}}=u^{\frac{4}{n-2}}g$. $u$ satisfies the equation:

$$-\Delta u+a(n)Su=0,$$

where $a(n)=\frac{n-2}{4(n-1)}$ and $S=S(g)\geq 0$ is the scalar curvature of $(M,g)$.

Since $u$ is the conformal Green’s function at point $o$, away from $o$, $u$ is bounded. Thus, applying Theorem 3.3, on the ALE end, fix a $z\in M$, and $r=d(x,z)$, we have the asymptotic on the ALE end:

$$u=Ar^{2-n}+O(r^{2-n-\epsilon}),$$

for some $\epsilon>0$. The constant term is 0 since it is a compactification conformal factor. Up to a scaling, we can assume $A=1$.

Next, we consider the annulus region

$$A_{g}(R,4R)=\{x:x\text{ in the end},R\leq\operatorname{dist}(x,z)\leq 4R\}.$$

According to the ALE assumption, when $R$ is sufficiently large, $A_{g}(R,4R)\simeq[R,4R]\times\mathbb{S}^{n-1}$ is connected to the metric $C^{1,\alpha}$ close to the flat metric.

From our conformal map, $u^{\frac{4}{n-2}}=(1+O(r^{-\epsilon}))r^{-4}$, the metric at the end $u^{\frac{4}{n-2}}g$ can be compactified by adding a point $p$. In fact, for some small $r>0$, $(B(p,r),u^{\frac{4}{n-2}}g)\subseteq(C(\mathbb{S}^{n-1}/\Gamma),g_{flat})$, the flat cone metric. Thus, the conformal annulus $(A_{g}(R,4R),u^{\frac{4}{n-2}}g)$ is embedded in the flat cone $C(\mathbb{S}^{n-1}/\Gamma)$. Notice that a priori this embedding of $A_{g}(R,4R)$ can be pretty wild, and the image may not contain any totally umbilic submanifold.

Denote $A(a,b)$ as the annulus region in the flat cone $C(\mathbb{S}^{n-1}/\Gamma)$ with respect to the distance function of the vertex.

Claim: for $R$ sufficiently large, there is a region $A(2R,3R)\subseteq(C(\mathbb{S}^{n-1}/\Gamma),g_{flat})$, which can be conformally embedded in the region $(A_{g}(R,4R),g)$.

Proof of the Claim.

: Consider the conformal metric, $\hat{g}=r^{-4}g$, with the inverted coordinate: $x^{\prime}=\frac{x}{r^{2}}$, where $r$ is the original distance function. Note, the distance to $p$, $\rho=|x^{\prime}|$, is $r^{-1}$. In particular, $A_{g}(a,b)=A_{\hat{g}}(b^{-1},a^{-1})$.

Since we consider $R$ large, it is enough to consider $((B(p,\delta),\hat{g})$, for some $\delta>0$ sufficiently small. Consider the flat conformal metric,

$$g_{0}=u^{\frac{4}{n-2}}g=(1+O(\rho^{\epsilon}))\hat{g}.$$

Note, $(B(p,\delta),g_{0})\subseteq((C(\mathbb{S}^{n-1}/\Gamma),g_{flat})$, is part of the flat cone, with the vertex $p$. Then, we can compare two distance functions: for $\delta$ small, on $((B(p,\delta),\hat{g})$, we have the function $q(x)=\frac{\operatorname{dist}_{g_{0}}(x,p)}{\operatorname{dist_{\hat{g}}(x,p)}}$, which is continuous away from $p$, and $\lim\limits_{x\to p}q(x)=1$. Thus, there exists a function $\eta$, with $\lim\limits_{t\to 0}\eta(t)=0$, such that:

$$(1-\eta(\delta))\operatorname{dist}_{\hat{g}}(x,p)\leq\operatorname{dist}_{g_{0}}(x,p)\leq(1+\eta(\delta))\operatorname{dist}_{\hat{g}}(x,p),$$

for any $x\in B(p,\delta)$. If $x\in A(2R,3R)$, then $\operatorname{dist}_{\hat{g}}(x,p)\leq(2R)^{-1}(1+\eta(\delta))\leq R^{-1}$, and similarly, $\operatorname{dist}_{\hat{g}}(x,p)\geq(4R)^{-1}$. Since all of the metrics above are conformal, if we choose $\delta$ small enough, then the claim is proved.

∎

As shown in Figure 2, there exists a conformal map, $\psi$, from the ALE end $E$, to the flat cone. This $\psi$ is obtained by composing with an additional inversion map. Thus, we can embed the region $A_{g}(R,4R)$ in the flat cone. From the Claim, it will be close to $A(R,4R)$.

On the flat metric of $A(2R,3R)\cong[2R,3R]\times\mathbb{S}^{n-1}/\Gamma$, write $([2R,3R]\times\mathbb{S}^{n-1}/\Gamma,dt^{2}+t^{2}g_{\mathbb{S}^{n-1}/\Gamma})$. Then for the slice $\{\frac{5}{2}R\}\times\mathbb{S}^{n-1}/\Gamma\subseteq([R,3R]\times\mathbb{S}^{n-1}/\Gamma,\delta)$, the second fundamental form:

$$\text{II}=\mathcal{L}_{\partial t}(dt^{2}+t^{2}g_{\mathbb{S}^{n-1}/\Gamma})=2tg_{\mathbb{S}^{n-1}/\Gamma}=\frac{2}{t}g|_{\{\frac{5}{2}R\}\times\mathbb{S}^{n-1}/\Gamma}.$$

Thus, $\{\frac{5}{2}R\}\times\mathbb{S}^{n-1}/\Gamma\cong\mathbb{S}^{n-1}/\Gamma$ is a totally umbilic submanifold. For $R$ large, $\{\frac{5}{2}R\}\times\mathbb{S}^{n-1}/\Gamma\subseteq(A(2R,3R),\delta)$, as a totally umbilic submanifold.

Since the totally umbilic submanifold is invariant under conformal mapping, there exists $N\subset(M,g)$, $(N,g|_{N})$ is $C^{1,\alpha}$ close to $(\{\frac{5}{2}R\}\times\mathbb{S}^{n-1}/\Gamma,g_{\mathbb{S}^{n-1}/\Gamma})$ and $N\cong\mathbb{S}^{n-1}/\Gamma_{i}$, which is totally umbilic. This violates Theorem 2.4 and Theorem 3.2 unless $\Gamma=\{e\}$. Thus, this end is an AE end.

We can repeat the process and conclude that $(M,g)$ contains only AE ends.

Figure 2. Conformal mapping $\psi:E\to\mathbb{R}^{n}/\Gamma$, from the ALE end to the flat cone.

∎

3.3. The local isotropy group and developbility

In addition, we can prove the following structure theorem for the universal cover of the LCF, ALE manifolds.

Proposition 3.6.

Let $(M,g)$ be an LCF, ALE manifold with non-negative scalar curvature, then its universal cover $\tilde{M}$ is diffeomorphic to $\mathbb{S}^{n}\setminus\Lambda\cup I$, where $\Lambda$ is the limit set of $\pi_{1}(M)$ under the conformal diffeomorphism, and $I$ is a discrete set of points that correspond to the AE ends.

Proof.

Now, $(M,g)$ is an ALE manifold. By the previous argument, since there is a conformal developing map $\phi:\tilde{M}\to\mathbb{S}^{n}$, the only ALE ends in $\tilde{M}$ are AE ends, possibly infinitely (countably) many of them.

One way to see that each AE end can be compactified by adding a point is to use the conformal embedding $\bar{\phi}:\tilde{M}\to\mathbb{R}^{n}$ and to use the expansion of the harmonic function on the end of the AE. Note that in this case, our group $\Gamma$ is trivial, thus, we can compactify it by adding a manifold point.

Another way to see this is to use the exhaustion of totally umbilic manifolds on the end: As before, consider the $i$-th end. For $r$ large enough, instead of $A$, we consider a family of annuli:

$$A_{k}=\{x:x\text{ in the $i$-th end },kr\leq\operatorname{dist}(x,z)\leq(k+1)r\}.$$

Repeat the above argument, $\exists N_{k}\subset A_{k}$, totally umbilic, and $N_{k}\cong\mathbb{S}^{n-1}$. Moreover, let $C_{k}\subset\tilde{M}$ be a compact subset such that $\partial C_{k}=N_{k}$. Then we have $C_{1}\subset C_{2}\subset C_{3}...\subset C_{k}\subset C_{k+1}\subset...$. Moreover, $\tilde{M}\supset\cup_{k=1}^{\infty}C_{k}$, $\{C_{k}\}$ is an exhaustion of the $i$-th end.

Now, under the conformal map $\phi:\tilde{M}\to\mathbb{S}^{n}$, $\phi(N_{k})=S_{k}\cong\mathbb{S}^{n-1}$, $S_{i}\bigcap S_{j}=\emptyset$ for $i\neq j$, $\phi(C_{k})\subset\phi(C_{k+1})$, we see that $\operatorname{rad}(S_{k})$ is decreasing. By Theorem 2.7 in [SYconformal], $\lim\limits_{k\to\infty}\operatorname{rad}(S_{k})=0$, otherwise, the Hausdorff dimension of $\mathbb{S}^{n}\setminus\phi(M)$ is $n$. Thus, $\lim\limits_{k\to\infty}S_{k}=\{x_{i}\}$. Thus, we have a one-point compactification for each end.

Let the compactified universal cover

$$\overline{M}=\phi(\tilde{M})\cup_{i}\{x_{i}\},$$

with the metric to be the spherical metric $h$. By Van Kampen’s theorem and $n\geq 3$, $\overline{M}$ is simply connected. The deck transformation can be extended by the effective action on $\overline{M}$ by maps $x_{i}$ to some $x_{j}$ with the only possible fixed point being $\{x_{i}\}$. Then $\overline{M}\cong\mathbb{S}^{n}\setminus\Lambda$, where $\Lambda$ is the limit set of deck transformations on $\tilde{M}$. This is because the local isometry group is finite.

For each $x_{i}$, if $x_{i}\in\Lambda$, it is an isolated limit point. Then $\exists\gamma\in\pi_{1}(M)=\operatorname{Deck}(\overline{M})$ and $x\in\phi(\tilde{M})$ such that $\lim\limits_{n\to\infty}\gamma^{n}(x)=x_{i}$. Then $x_{i}$ will not correspond to an AE end, which is a contradiction. Thus $x_{i}\notin\Lambda$. Thus, $\tilde{M}\cong\mathbb{S}^{n}\setminus\Lambda\cup I$, where $I$ is a discrete set of points. ∎

Now we can prove Theorem 1.12. We can show that for each $\Gamma_{k}$, there is a group homomorphism $i_{k}:\Gamma_{k}\to\pi_{1}(M)$, which is an injective map.

Proof of Theorem 1.12.

Consider

$$S_{r}=\{x:x\text{ in the $k$-th end },\operatorname{dist}(x,z)=r\}$$

to be the distance sphere. Then, for $r$ large enough, $(S_{r},j^{*}g)$ and $(\mathbb{S}^{n-1}/\Gamma_{k},g_{\mathbb{S}^{n}/\Gamma_{k}})$ are $C^{1,\alpha}$ close, and $S_{r}\cong\mathbb{S}^{n-1}/\Gamma_{k}$. ($j^{*}g$ is the pullback metric under $j:S_{r}\to M$.)

Let $(\tilde{M},p^{*}g)$ be the Riemannian cover with the covering map: $p:\tilde{M}\to M$. Then choose $z_{0}\in p^{-1}(z)$, there is a lift of $S_{r}$, $\tilde{S_{r}}$, by lifting the geodesic. Then $\tilde{S_{r}}$ is in an AE end. For $r$ large enough, $\tilde{S_{r}}\cong\mathbb{S}^{n}$. Thus, we have the map $p:\tilde{S_{r}}\to S_{r}$, which is a covering map. Since $\pi_{1}(\tilde{S_{r}})=\{e\}$, then $\tilde{S_{r}}$ is the universal cover of $S_{r}$.

Now, let

$$j_{k}:\mathbb{R}^{+}\times\mathbb{S}^{n-1}/\Gamma_{k}\to M$$

be the embedding. This induces the homomorphism

$$i_{k}:\pi_{1}(\mathbb{R}^{+}\times\mathbb{S}^{n-1}/\Gamma_{k})\cong\Gamma_{k}\to\pi_{1}(M).$$

Let $\gamma\subset\mathbb{R}^{+}\times\mathbb{S}^{n-1}/\Gamma_{k}$, a closed loop such that $[\gamma]\in\operatorname{ker}(i_{k})\lhd\pi_{1}(\mathbb{R}^{+}\times\mathbb{S}^{n-1}/\Gamma_{k})$. Note that $\gamma$ is homotopically equivalent to $\gamma\in S_{r}$, with $[\gamma]\in\pi_{1}(S_{r})\cong\Gamma_{k}$. Since $i_{k}([\gamma])$ is trivial, then $\tilde{\gamma}\subset\tilde{S_{r}}$, the lift of $\gamma$, is a closed loop, which represents the trivial deck transformation of $\pi_{1}(M)$. Since $\pi_{1}(\tilde{S_{r}})=\{e\}$, then $[\gamma]$ represents the trivial deck transformation in $\pi_{1}(S_{r})$, thus, $[\gamma]=e\in\pi_{1}(S_{r})=\Gamma_{k}$. Thus, $i_{k}$ is injective. ∎

In particular, for each $\gamma\in\Gamma_{k}<\pi_{1}(M)$, as a deck transformation, it fixes $\{x_{i}\}$. And it induces a map on a small ball $B_{\epsilon}$ centered on $x_{i}$. Then from the Liouville theorem, we immediately have:

Corollary 3.7.

For each $\Gamma_{k}<\pi_{1}(M)$, $\rho(\Gamma_{k})<C(n)$ which fixes $\{x_{k}\}$, where $C(n)$ is the conformal group of $\mathbb{S}^{n}$.

Note that the action $\rho(\pi_{1}(M))$ extends to the compactification $\overline{M}$. Moreover, this action is properly discontinuous. Thus $M^{\prime}:=\overline{M}/\rho(\pi_{1}(M))$, is a compact orbifold ([Thurston_orbifold]). In particular,

$$M^{\prime}=M\cup\{x_{1},...,x_{m}\},$$

which is the compactification of $M$. Since it is a quotient of a manifold $\bar{M}$, $M^{\prime}$ is a good orbifold, with each local group $\Gamma_{i}$ injecting into $\pi_{1}(M)$.

Note that we can also define a Kleinian orbifold (See Section 2.2 for Kleinian manifolds). Thus, $M^{\prime}$ is a Kleinian orbifold.

Proposition 3.8.

Let $M^{\prime}$ be a good Kleinian orbifold. Then there exists a compact Kleinian manifold $N$, such that:

$$p:N\to M^{\prime}$$

is a finite covering map.

Proof.

Since $M=\Omega/\Gamma$, for some open set $\Omega\subseteq\mathbb{S}^{n}$, and $\Gamma\leq C(n)$, by Theorem 2.8, there exists a normal subgroup $H\leq\Gamma$ of finite-index and torsion-free. Note that the torsion-free action in $C(n)$ is fixed-point-free (Remark 2.7). Thus, the resulting normal cover space $N\cong\Omega/H$ a compact manifold, whose fundamental group is isomorphic to $H$, and the deck transformation group is isomorphic to $\Gamma/H$. ∎

3.4. The connection with the orbifolds

Now, we link the geometry of LCF orbifolds with positive scalar curvature with ALE manifolds of nonnegative scalar curvature. In the following, we show that the conformal compactification of the nonnegtative scalar curvature, LCF, ALE manifolds is a compact LCF orbifold with positive Yamabe invariant.

Theorem 3.9 (Conformal compactification with positive Yamabe invariant).

If $(M,g)$ is a LCF, nonnegative scalar curvature ALE manifold. The compactified ALE manifold $M^{\prime}$ has positive Yamabe invariant, that is, $Y_{orb}(M^{\prime})>0$.

Proof.

Without loss of generality, we assume that $(M,g)$ has only one ALE end. First, we need a well-defined orbifold metric. Fix $z\in M$, let $r(x)=\operatorname{dist}(x,z)$ be the distance function of $z$. Consider the metric:

$$\hat{g}=r^{-4}g.$$

In the inverted coordinate, let $\rho=\frac{1}{r}$, then by the ALE assumption, on the ball $B(x_{0},\delta):=\{x:\rho(x)\leq\delta\}$, where $x_{0}$ is an orbifold point and $\delta$ is sufficiently small, we have

$$\hat{g}=\delta+O(\rho^{\tau}).$$

Note that $B(x_{0},\delta)\cong B_{\mathbb{R}^{n}}(0,1)/\Gamma$ is an orbifold with an isolated orbifold point $0$.

Second, we want a removable singularity argument. We take the universal cover $P:B_{\mathbb{R}^{n}}(0,\delta)\setminus\{0\}\to B(x_{0},\delta)\setminus\{x_{0}\}$, and the $\Gamma$-invariant, pull-back metric $(B_{\mathbb{R}^{n}}(0,\delta)\setminus\{0\},P^{*}\hat{g})$. There are many ways to prove this. We will prove the removable singularity theorem using injection of the local group into $\pi_{1}(M)$, and conformal embedding. Consider the universal cover: $\pi:\tilde{M}\to M$. Since $\Gamma$ injects into $\pi_{1}(M)$, there exists a conformal diffeomorphism $\Phi:B_{\mathbb{R}^{n}}(0,\delta)\setminus\{0\}\to B_{\mathbb{R}^{n}}(0,\delta)\setminus\{0\}$ such that $\Phi^{*}P^{*}\hat{g}=u^{\frac{4}{n-2}}g_{\mathbb{R}^{n}}$, hence it extends smoothly at the origin by a smooth function $u\in C^{\infty}(B_{\mathbb{R}^{n}}(0,\delta))$ since $u$ satisfies the conformal equation $-\Delta u+S(\hat{g})u=0$. Thus, it follows from the removability of the singularity of $u$. We denote the compactified metric $(M^{\prime},g^{\prime})$.

By lemma 3.4 of [AB04], on the orbifold $(M^{\prime},g^{\prime})$ there exists a conformal factor such that the conformal metric $g^{\prime\prime}\in[g^{\prime}]$ has the scalar curvature $S(g^{\prime\prime})$ which does not change sign. To get the ALE metric $g$ on $M$, it is equivalent to finding a conformal factor $v$ such that:

$$L^{\prime\prime}v=-\Delta_{g^{\prime\prime}}v+a(n)S(g^{\prime\prime})v=S(g)v^{\frac{n+2}{n-2}}\geq 0$$

on $M\subset M^{\prime}$. Since $v>0$, $M^{\prime}$ is compact, $v(x)\to+\infty$ as $x\to x_{\infty}$, there exists $x_{0}\in M$ such $v(x_{0})=\min_{x\in M^{\prime}}v(x)=\min_{x\in M}v(x)$. At that point, $-\Delta_{g^{\prime\prime}}v(x_{0})\leq 0$, and thus $S(g^{\prime\prime})(x_{0})\geq 0$. Since $S(g^{\prime\prime})$ does not change sign, by strong maximal principle, $S(g^{\prime\prime})>0$. Thus, the Yamabe invariant $Y_{orb}(M^{\prime})>0$.

∎

As a corollary, for every such LCF, ALE manifold, we have that such an ALE metric is equivalent to the conformal Green’s function metric on a compact LCF orbifold with positive Yamabe invariant. The conformal factors are the superposition of conformal Green’s functions by Bôcher’s theorem. Thus, we have the optimal decay rate.

Corollary 3.10.

Let $(M,g)$ be an LCF, scalar-flat, ALE manifold of order $\tau$, for any $\tau>0$, then there exist charts at infinity for every end such that:

$$g=\delta+O(r^{-n+2})$$

$$\partial^{m}g=O(r^{-n+2-|m|}).$$

i.e. they are ALE of order $n-2$. If $(M,g)$ is an LCF, nonnegative scalar curvature, ALE manifold, then there exists a conformal metric such that it is scalar-flat, ALE manifold of order $n-2$.

Proof.

Note that the ALE manifold in this case is scalar-flat. It is known that on the orbifold of positive Yamabe invariant, there exists positive conformal Green’s functions. Using Bôcher’s theorem, the conformal factors are the superposition of conformal Green’s functions. In the conformal normal coordinates, the conformal Green’s function has local expansion:

$$G=r^{2-n}+A+O(r),$$

which is without log terms, by the LCF assumption (See Lemma 6.4 of [lee1987yamabe]). The optimal decay rate follows from direct computation. ∎

Remark 3.11.

The same argument works for LCF, nonnegative scalar curvature, ALE orbifolds as well. To relate the manifolds and orbifolds with at most isolated singularities, we need the conformal blow-up at the singularities to produce ALE ends. This will be proved in Lemma LABEL:ALE_orbifold_blow-up.

This is a much easier proof of the optimal ALE order for locally conformally flat scalar-flat ALE manifolds. The optimal decay rate for obstruction-flat scalar-flat ALE manifolds is proved by [AcheViaclovsky].

Conversely, we can reverse the above picture, starting from an LCF orbifold of positive orbifold Yamabe invariant with at most isolated singularities to construct the ALE manifolds. This will prove our good orbifold theorem for all LCF orbifolds with positive scalar curvature (Theorem 1.4).

Proof of Theorem 1.4.

Let $(M,g)$ be our orbifold, with isolated singularities $\{x_{i}\}$. By the assumption of positive scalar curvature, there exists a positive Green’s function of the conformal Laplacian blow-up at the orbifold point $x_{i}$. The superposition of all such Green’s functions, $G$, will give us the conformal factor such that $(M\setminus\{x_{i}\},G^{\frac{4}{n-2}}g)$ is a multi-ALE manifold of order $n-2$. Note $\pi_{1}^{orb}(M)\cong\pi_{1}(M\setminus\{x_{i}\})$.

From Theorem 3.6 and Corollary 3.7, each isotropy group $\Gamma_{x_{i}}$ injects into the fundamental group $\pi_{1}(M\setminus\{x_{i}\})$. Thus, by a well-known theorem of orbifolds, an orbifold is good if and only if each isotropy group injects into the orbifold fundamental group (follow the same spirit of Theorem 1.12). Hence done. ∎

Remark 3.12.

We use the ALE structure (Theorem 3.6 and Corollary 3.7) to prove the good orbifold theorem (Theorem 1.4). Note that the proof of Theorem 3.6 uses the local structure of flat cone and the LCF condition as well. So the order of the proof does not matter, if we can show an orbifold version of Theorem 2.4. Starting with the ALE manifolds seems more natural to the author.

4. Some classification results

4.1. Relation with the conformal group

From the previous discussion, the compactified nonnegative scalar curvature ALE space, $M^{\prime}$, can be viewed as the quotient of the compactified universal cover $\overline{M}$ by $\rho(\pi_{1}(M))$, where $\rho(\pi_{1}(M))$ acts on $\overline{M}$ properly discontinuously, with isolated fixed points, and the corresponding isotropy group is finite. To understand this quotient, we need to understand the subgroup of the conformal group of $\mathbb{S}^{n}$.

To study the local isotropy subgroups, we first show that we can conjugate the local group $\Gamma_{i}$ to be a subgroup of ${\rm{O}}(n)$.

Lemma 4.1.

Let $\Gamma\leq C(n)$ be a local group corresponding to the orbifold point $x$, then $\exists\gamma\in C(n)$ such that $\gamma^{-1}\Gamma\gamma\in{\rm{O}}(n)$.

Proof.

Let $g\in\Gamma$. Since $g$ fixes $x$, $g$ is of finite order, then $g$ is elliptic. Thus, $\Gamma$ is an elliptic subgroup. Note that we can always consider the chart where $\Gamma$ acts as the finite subgroup of the linear group fixing the origin (linear chart). Thus, $\Gamma$ will fix some $K\cong\mathbb{S}^{n-1}$ in the local universal cover.

Now, there exists $\gamma\in C(n)$ such that $\gamma^{-1}g\gamma\in{\rm{O}}(n+1)$ ([conformalbook], Chapter 2). Since $g$ fixes $x$, then $\gamma^{-1}g\gamma\in{\rm{O}}(n)$. By ${\rm{O}}(n+1)$ acting transitively on $\mathbb{S}^{n}$, we can also compose $\gamma$ with elements in ${\rm{O}}(n+1)$ such that $\gamma$ fixes $x$. Hence $\gamma^{-1}g\gamma$ also fixes $-x$, the antipodal point, and we can write $g\in C(\mathbb{R}^{n})$, the conformal group of $\mathbb{R}^{n}$, with the following expression:

(1) $$\gamma^{-1}g\gamma(v)=A(v)$$

for $A\in{\rm{O}}(n)$, for some $v\in\mathbb{R}^{n}$. (We let $x$ be $\infty$ and $-x$ be $0$.)

Now, $\forall f\in\Gamma$, $\gamma^{-1}f\gamma$ also fixes $x$. Thus, we write $\gamma^{-1}f\gamma(v)=cB(v)+a$. $c$ is a constant, $B\in{\rm{O}}(n)$, and $a\in\mathbb{R}^{n}$. Since $f$ is of finite order, then $c=1$. $\gamma^{-1}f\gamma(v)=B(v+b)-b$.

Since $f$ and $g$ map some $K\cong\mathbb{S}^{n-1}$ to themselves, then $\gamma^{-1}f\gamma$ and $\gamma^{-1}g\gamma$ map $\gamma^{-1}(K)\cong\mathbb{S}^{n-1}$ to itself. Now, from (1), the only fixed $\mathbb{S}^{n-1}$ are

$$S_{R}=\{v\in\mathbb{R}^{n}:|v|=R\}.$$

Then $\gamma^{-1}f\gamma$ fixes $S_{R}$ if and only if $b=0$. Thus, $\gamma^{-1}f\gamma\in{\rm{O}}(n)$. Hence $\gamma^{-1}\Gamma\gamma\leq{\rm{O}}(n)$. In particular, $\gamma^{-1}\Gamma\gamma$ fixes $x$, $-x$. ∎

Thus, for a subgroup $\Gamma\leq{\rm{O}}(n)$ acting on $\mathbb{S}^{n}$, if it has a fixed point, it fixes the antipodal point as well. It seems that the non-trivial orbifold points must appear in pairs. This will be the case if this subgroup does not contain parabolic elements. The proof of the following theorem can be found in Appendix LABEL:orbifold_points_in_the_quotient.

Theorem 4.2.

Let $G\leq C(n)$, a discrete subgroup acting on $\mathbb{S}^{n}$ properly discontinuously. Denote the limiting set of $G$ as $\Lambda$. Assume there are no parabolic elements in $G$ and the Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}(\Lambda)<\frac{n-2}{2}$, then the fixed points of the local isotropy subgroup $\Gamma$ must appear in pairs unless the quotient is non-orientable.

In particular, the above theorem applies when the fundamental group is a Schottky group. We have the following corollary regarding the number of orbifold points and the corresponding non-trivial ALE ends.

Corollary 4.3.

Let $(M,g)$ be an ALE LCF manifold. If $M$ is orientable and there is no parabolic element in $\pi_{1}(M)$, then the ALE ends with nontrivial group must appear in pairs.

Proof.

Let $M^{\prime}$ be the compactified ALE manifold. By Theorem 3.9, we have $Y_{orb}(M^{\prime})>0$. Thus, the compact manifold cover $N$, as in Proposition 3.8, has $Y(N)>0$. By Proposition 4.7 in [SYconformal], $\operatorname{dim}_{\mathcal{H}}(\Lambda)\leq\frac{n-2}{2}$. Since $(N,g^{\prime})$ has a positive Yamabe constant, by Corollary 3.4 in [Nayatani], $\operatorname{dim}_{\mathcal{H}}(\Lambda)<\frac{n-2}{2}$. Thus, we can apply Theorem 4.2. ∎

Corollary 4.3 will be used to prove the ALE end structure part in Theorem 1.6.

4.2. Low dimension classifications and the proof of Theorem 1.6

From Theorem 3.9, there is a one-to-one correspondence between LCF orbifolds with positive Yamabe invariant and LCF, scalar-flat ALE spaces. We obtain some classification theorems of the orbifolds in low dimensions. Thus, we also classify the ALE spaces in these cases.

Specifically, when $n\leq 4$, we have a classification of the LCF orbifolds with positive scalar curvature with the help of Ricci flow.

Theorem 4.4 (3D, [GL],[Ilimit],[KL]).

The 3-D LCF orbifolds with positive scalar curvature are diffeomorphic to the connected sum of the quotients $\mathbb{S}^{3}/\Gamma$ and $\mathbb{S}^{1}\times\mathbb{S}^{2}/\Gamma$, i.e.,

$$M\cong\mathbb{S}^{3}/\Gamma_{1}\#\mathbb{S}^{3}/\Gamma_{2}\#...\mathbb{S}^{3}/\Gamma_{k}\#\mathbb{S}^{2}\times\mathbb{S}^{1}/\Gamma^{\prime}_{1}\#...\mathbb{S}^{2}\times\mathbb{S}^{1}/\Gamma^{\prime}_{m},$$

where all the orbifold points are $\mathbb{Z}_{2}$-quotient singularities.

Theorem 4.5 (4D, [Ilimit], [Ha], [ChenZhu]).

The 4-D LCF orbifolds with positive scalar curvature are diffeomorphic to the connected sum of the quotients $\mathbb{S}^{4}/\Gamma$ and $\mathbb{S}^{1}\times\mathbb{S}^{3}/\Gamma$, i.e.,

$$M\cong\mathbb{S}^{4}/\Gamma_{1}\#\mathbb{S}^{4}/\Gamma_{2}\#...\mathbb{S}^{4}/\Gamma_{k}\#\mathbb{S}^{3}\times\mathbb{S}^{1}/\Gamma^{\prime}_{1}\#...\mathbb{S}^{3}\times\mathbb{S}^{1}/\Gamma^{\prime}_{m}.$$

In the compactified universal cover, $\overline{M}$, of the ALE manifold, by positive Yamabe, Proposition 4.7 in [SYconformal], and Corollary 3.4 [Nayatani], the Hausdorff dimension of the limiting set induced by $\rho(\pi_{1}(M))$, $\Lambda$, is strictly less than $\frac{n-2}{2}$. In particular, when $n=3,4$, the non-negative scalar curvature implies $\dim_{\mathcal{H}}(\Lambda)<1$.

Proof of Theorem 1.6.

Suppose $\dim_{\mathcal{H}}(\Lambda)<1$, then, we can apply Theorem 6.2 in [Ilimit] and Selberg’s lemma (Theorem 2.8). There is a torsion-free normal subgroup of $\rho(\pi_{1}(M))$ containing no parabolic elements, and there is a finite cover of $\overline{M}/\rho(\pi_{1}(M))$ (possibly a finite cover of the manifold cover) that is diffeomorphic to

$$\mathbb{S}^{1}\times\mathbb{S}^{n-1}\#...\#\mathbb{S}^{1}\times\mathbb{S}^{n-1}=k(\mathbb{S}^{1}\times\mathbb{S}^{n-1}),$$

for some $k>0$.

The low-dimensional topological classification is given by Theorem 4.4, 4.5. For the conformal connected sum: note the above decomposition can be viewed as the connected sum of orbifolds near the manifold points. Thus, we can apply Theorem 2.7 from [Idecomposition].

For the last statement in Theorem 1.6: If we in addition assume $(M,g)$ is orientable, since $\dim_{\mathcal{H}}(\Lambda)<1$, there is no parabolic element. Then we can apply Corollary 4.3. Note, when $n$ is odd, the local isotropy group is isomorphic to $\mathbb{Z}_{2}\subseteq{\rm{O}}(n)$, which is orientation-reversing. So by Theorem 1.12, when $M$ is orientable, then it has no such orbifold points. ∎

Note in [Ilimit], the 4-D classification is actually obtained by Ricci flow on the Positive Isotropic Curvature (PIC). Thus, we can derive the corollary of the structure of 4-D PIC orbifolds.

Proof of Corollary 1.10.

By [ChenZhu], a 4-D orbifold admits a positive isotropic curvature metric if and only if it admits an LCF metric with positive scalar curvature. Since in 4-D, nonnegative scalar curvature implies $\dim_{\mathcal{H}}(\Lambda)<1$, the corollary follows from Theorem 1.6. ∎

In particular, the non-trivial ALE ends of an orientable ALE space appear in pairs. An example can be constructed as follows: on $\mathbb{S}^{n}/\Gamma$, where $\Gamma\leq{\rm{O}}(n)$ has two fixed points, $s$ and $-s$, with the standard spherical metric. Consider the Green’s function metric blown up at $s$ and $-s$, the resulting manifold $(M,G^{\frac{4}{n-2}}g_{\mathbb{S}^{n}})$ is diffeomorphic to the standard Schwarzschild metric modulo $\Gamma$. We call such a manifold the Schwarzschild ALE manifold.

In the non-orientable case, we can easily construct an LCF manifold with one end:

Example 4.6.

If the dimension $n$ is even, then the antipodal map is an orientation-reversing map. Then $\mathbb{S}^{n}/\{-Id\}$ is non-orientable. Consider the universal cover $\tilde{M}\cong\mathbb{S}^{n}\setminus\{x,-x\}$, blown up at $x$ and $-x$, then $\overline{M}/\Gamma$ is a Schwarzschild ALE manifold. If $\Gamma$ contains no $\mathbb{Z}_{2}$-rotation, we can further do a quotient (otherwise, we fix the equator). Then $(M,g)\cong(\mathbb{R}P^{n}\setminus\{p\},g_{\mathbb{R}P^{n}})/\Gamma$, which is an LCF, ALE manifold with one end. But it is not orientable. Such a one-end, non-orientable ALE is illustrated in Figure 3. We call such a metric a non-orientable Schwarzschild ALE.

Figure 3. The construction of non-orientable Schwarzschild ALE: We start with $(M,g)=(\mathbb{S}^{n}/\Gamma,g_{\mathbb{S}^{n}})$. To construct the one-end ALE manifold, we first do the quotient by $\sigma=-\operatorname{Id}$, then, we can blow up the orbifold singularity by the conformal Green’s function.

Note that the same construction does not hold when $n$ is odd. The only non-trivial $\Gamma$ is the $\mathbb{Z}_{2}$ action, then the resulting quotient contains edge singularities.

To end this section, we think that this ’pair-of-ends’ phenomenon holds even in the presence of the parabolic element. We conjecture that

Conjecture 4.7.

When $n\geq 5$, for an orientable LCF, nonnegative scalar curvature ALE manifold $(M,g)$, the number of non-trivial ALE ends is even, and they occur in orientation-reversing conjugate pairs.

5. Application to the moduli space

5.1. The oriented moduli space

Now, we consider the moduli space $\mathfrak{M}(n,\mu_{0},C_{0})$, and $\mathfrak{M}^{\prime}(n,\mu_{0},C_{0})$.

From [Aku],[TV2], $\mathfrak{M}(n,\mu_{0},C_{0})$ can be compactified under the Gromov-Hausdorff topology by adding the limits of the Gromov-Hausdorff limits. A similar result holds for $\mathfrak{M}^{\prime}(n,\mu_{0},C_{0})$. More precisely:

Theorem 5.1 (Tian-Viaclovsky).

Let $\{(M_{i},g_{i})\}$ be a sequence in $\mathfrak{M}(n,\mu_{0},C_{0})$, then we have: