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On the structure of locally conformally flat orbifolds and ALE manifolds

Xiaokang Wang

TL;DR

The paper develops a coherent structure theory for locally conformally flat orbifolds and ALE manifolds with nonnegative scalar curvature, linking conformal blow-up/compactification to a Kleinian framework. It proves that LCF orbifolds with positive orbifold Yamabe invariant admit manifold covers and that ALE ends injectively embed their end groups into the ambient fundamental group, enabling precise low-dimensional classifications. A positive mass theorem for LCF, ALE ends is established, together with optimal decay rates, and these tools yield solvability of the orbifold Yamabe problem in the LCF setting. The results yield detailed moduli-space descriptions, rule out certain GH limits (e.g., the football orbifold in the oriented case), and extend to nonorientable examples, providing a unified picture of end structures, singularities, and deformations in these geometric settings.

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 $\mathbb{S}^4/Γ$ 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.

On the structure of locally conformally flat orbifolds and ALE manifolds

TL;DR

The paper develops a coherent structure theory for locally conformally flat orbifolds and ALE manifolds with nonnegative scalar curvature, linking conformal blow-up/compactification to a Kleinian framework. It proves that LCF orbifolds with positive orbifold Yamabe invariant admit manifold covers and that ALE ends injectively embed their end groups into the ambient fundamental group, enabling precise low-dimensional classifications. A positive mass theorem for LCF, ALE ends is established, together with optimal decay rates, and these tools yield solvability of the orbifold Yamabe problem in the LCF setting. The results yield detailed moduli-space descriptions, rule out certain GH limits (e.g., the football orbifold in the oriented case), and extend to nonorientable examples, providing a unified picture of end structures, singularities, and deformations in these geometric settings.

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 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.

Paper Structure

This paper contains 20 sections, 34 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Orbifold results
  3. ALE results
  4. Preliminaries
  5. Yamabe invariant
  6. Locally conformally flat manifolds and Kleinian structure
  7. ALE ends and orbifold singularities
  8. Totally umbilic submanifolds in Rn
  9. Harmonic functions on ALE manifolds
  10. The local isotropy group and developbility
  11. The connection with the orbifolds
  12. Some classification results
  13. Relation with the conformal group
  14. Low dimension classifications and the proof of Theorem \ref{['classification']}
  15. Application to the moduli space
  16. ...and 5 more sections

Key Result

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.

Figures (4)

  • Figure 1: A sequence $\{(M_i,g_i)\}$, where $(M_i,g_i)\in \mathfrak{M}'(4,\mu_0, C_0)$. The limit $(M_\infty, g_\infty)$ is a multifold with one multifold singularity. The rescaled limit will be a 2-end ALE manifold $(\hat{M}_\infty, \hat{g}_\infty)$.
  • Figure 2: Conformal mapping $\psi : E \to \mathbb{R}^n/\Gamma$, from the ALE end to the flat cone.
  • 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.
  • Figure 4: Two examples of where irreducible multifolds can be realized as a limit in the non-orientable case. The first row illustrates Example \ref{['e:non-orientable 1']}, where the limit is a non-orientable cap, and one non-orientable Schwarzschild ALE bubbles off. The second row illustrates Example \ref{['e: non-orientable 2']}, where the limit is a football metric, and two non-orientable Schwarzschild ALEs bubble off.

Theorems & Definitions (80)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5: Iz1
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 70 more