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Construction of gravitational instantons with non-maximal volume growth via codimension-1 collapse

Willem Adriaan Salm

Abstract

In this paper, we construct families of gravitational instantons of type ALG, ALG*, ALH and ALH* using a gluing construction. Away from a finite set of exceptional points, the metric collapses with bounded curvature to a quotient of $\mathbb{R}^3$ by $\mathbb{Z}_2$ and a lattice of rank one or two. Depending on whether the gravitational instantons are of type ALG/ALG* or ALH/ALH*, there are either two or four exceptional points respectively that are modelled on the Atiyah-Hitchin manifold. The other exceptional points are modelled on the Taub-NUT metric. There are at most four, respectively eight, of these points in each case.

Construction of gravitational instantons with non-maximal volume growth via codimension-1 collapse

Abstract

In this paper, we construct families of gravitational instantons of type ALG, ALG*, ALH and ALH* using a gluing construction. Away from a finite set of exceptional points, the metric collapses with bounded curvature to a quotient of by and a lattice of rank one or two. Depending on whether the gravitational instantons are of type ALG/ALG* or ALH/ALH*, there are either two or four exceptional points respectively that are modelled on the Atiyah-Hitchin manifold. The other exceptional points are modelled on the Taub-NUT metric. There are at most four, respectively eight, of these points in each case.

Paper Structure

This paper contains 12 sections, 29 theorems, 110 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Results
  3. The approximate solution
  4. The bulk space
  5. The interpolation of the Kähler forms
  6. The deformation problem
  7. Weighted analysis of functions
  8. The constant part
  9. The linearised equation
  10. The non-linear part and existence
  11. Global properties $M_{B,n}$
  12. The moduli space

Key Result

Theorem \oldthetheorem

Let $L \subset \mathbb{R}^3$ be a lattice of rank one or two and consider the $\mathbb{Z}_2$ action on $\mathbb{R}^3 / L$ that is induced by the antipodal map on $\mathbb{R}^3$. Let $\{p_i\}$ be a configuration of $n$ distinct points in $(\mathbb{R}^3/L - \operatorname{Fix}(\mathbb{Z}_2))/ \mathbb{Z

Figures (3)

  • Figure 1: The multi-Taub-NUT space retracts to a wedge sum of 2-spheres.
  • Figure 2: The underlying manifold $M_{\mathbb{R}^3, n}$ can be seen as the union of the Atiyah-Hitchin manifold and the Multi-Taub-NUT space.
  • Figure 3: Depiction of the 2-cycles with self-intersection -2 inside $P|_{X} \cup \widetilde{P|_{Y}} \cup \: \mathbb{Z}_2 \cdot \widetilde{P|_{Y}}$. The grey planes depict the boundary between these regions. The dark-blue and green spheres form a basis of $H_2(M_{\mathbb{R}^3,5})$ such that its intersection matrix is the negative Cartan matrix of $D_5$. The light-blue spheres are the $\mathbb{Z}_2$ images of the other spheres.

Theorems & Definitions (56)

  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • Remark \oldthetheorem
  • proof
  • ...and 46 more