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

TL;DR

This paper constructs new gravitational instantons with specific volume growth properties using a gluing method, revealing detailed local models and exceptional point configurations.

Contribution

It introduces a novel gluing construction for gravitational instantons of types ALG, ALG*, ALH, and ALH*, detailing their local models and exceptional point arrangements.

Findings

Constructed families of gravitational instantons with non-maximal volume growth.

Identified local models at exceptional points as Atiyah-Hitchin and Taub-NUT metrics.

Determined the maximum number of exceptional points in each instanton type.

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.