Quotients of gravitational instantons

TL;DR

This paper classifies Ricci-flat anti-self-dual 4-manifolds with Euclidean ends, showing they are either hyperk"ahler or cyclic quotients of Gibbons-Hawking spaces, and proves all such quotients are K"ahler.

Contribution

It provides a complete classification of certain gravitational instantons and characterizes their quotients, extending understanding of their geometric and topological properties.

Findings

Classified Ricci-flat anti-self-dual 4-manifolds as hyperk"ahler or Gibbons-Hawking quotients.

Proved all quotients are K"ahler.

Identified fundamental groups at infinity to exclude non-quotient cases.

Abstract

A classification result for Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds is obtained: they are either hyperk\"ahler (one of the gravitational instantons classified by Kronheimer), or they are a cyclic quotient of a Gibbons-Hawking space. The possible quotients are described in terms of the monopole set in R^3, and it is proved that every such quotient is actually K\"ahler. The fact that the Gibbons-Hawking spaces are the only gravitational instantons to admit isometric quotients is proved by examining the possible fundamental groups at infinity: most can be ruled out by the classification of 3-dimensional spherical space form groups, and the rest are excluded by a computation of the Rohklin invariant (in one case) or the eta invariant (in the remaining family of cases) of the corresponding space forms.