On the existence of toric ALE and ALF gravitational instantons

TL;DR

This paper proves the existence and uniqueness of certain Ricci-flat, toric gravitational instantons with ALE and ALF asymptotics, and classifies all such self-dual solutions as multi-Eguchi-Hanson or multi-Taub-NUT.

Contribution

It establishes the existence and uniqueness of toric ALE and ALF gravitational instantons for all admissible structures, and classifies all toric self-dual instantons.

Findings

Existence of unique Ricci-flat, toric ALE and ALF instantons for each admissible structure.

Any toric ALE or ALF self-dual instanton is a multi-Eguchi-Hanson or multi-Taub-NUT solution.

Provides an elementary proof of the classification of these instantons.

Abstract

We establish existence and uniqueness results for asymptotically locally Euclidean (ALE) and asymptotically locally flat (ALF) gravitational instantons. In particular, we prove the existence of a unique, Ricci-flat, toric ALE and ALF gravitational instanton, for every admissible rod structure, that is smooth up to possible conical singularites. We also give an elementary proof that any toric ALE or ALF self-dual instanton is a multi-Eguchi-Hanson or multi-Taub-NUT solution.