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Resolution of compact Einstein orbifolds in general dimensions

Yichen Yao

Abstract

Given a noncollapsing sequence of m-dimensional compact Einstein manifolds with a uniform energy bound, the Gromov-Hausdorff limit is a compact Einstein orbifold with at most finitely many singularities. Conversely, starting with a compact Einstein orbifold, we are interested in whether there exists a sequence of smooth Einstein metrics converging to it. In this paper, we provide a negative answer. We give an explicit obstruction for a negative Einstein orbifold appearing as a noncollapsing limit of compact Einstein manifolds, which does not vanish for hyperbolic orbifolds. This work extends the work of Ozuch in dimension 4, with significant technical simplifications.

Resolution of compact Einstein orbifolds in general dimensions

Abstract

Given a noncollapsing sequence of m-dimensional compact Einstein manifolds with a uniform energy bound, the Gromov-Hausdorff limit is a compact Einstein orbifold with at most finitely many singularities. Conversely, starting with a compact Einstein orbifold, we are interested in whether there exists a sequence of smooth Einstein metrics converging to it. In this paper, we provide a negative answer. We give an explicit obstruction for a negative Einstein orbifold appearing as a noncollapsing limit of compact Einstein manifolds, which does not vanish for hyperbolic orbifolds. This work extends the work of Ozuch in dimension 4, with significant technical simplifications.
Paper Structure (34 sections, 29 theorems, 220 equations, 3 figures)

This paper contains 34 sections, 29 theorems, 220 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Noncollapsing limits of compact Einstein manifolds
  3. Resolution of Einstein orbifolds
  4. Outline of the paper
  5. Index of notations
  6. Preliminaries and Set-up
  7. ALE manifolds
  8. ALE coordinates and weighted norms
  9. Intrinsic geometric quantities
  10. Gluing construction
  11. Gluing different regions
  12. Weighted Hölder norms on glued manifold
  13. Approximating a resolution sequence
  14. Gauge fixing
  15. Linear estimate for Bianchi gauge
  16. ...and 19 more sections

Key Result

Theorem 1.1

For any sequence $(M_i,g_i)_i$ in $\mathcal{M}(m,\Lambda,D,V,E)$, we can extract a subsequence converging to a compact metric space $(M_0,d)$ in Gromov-Hausdorff topology. Moreover,

Figures (3)

  • Figure 1: Cone of metrics on $M=M_0\#N$
  • Figure :
  • Figure :

Theorems & Definitions (63)

  • Theorem 1.1: AndersonBKN
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1: ALE spaces
  • Lemma 2.4: Joyce
  • Theorem 2.5: WangYin
  • Definition 2.6
  • Proposition 2.7
  • ...and 53 more