Rigidity for Multi-Taub-NUT metrics

TL;DR

This paper classifies certain hyperk"ahler manifolds with cubic volume growth, showing they are either standard Euclidean space times a circle or multi-Taub-NUT manifolds, with specific complex structures.

Contribution

It provides a classification of gravitational instantons with cubic volume growth and cyclic fundamental group at infinity, identifying them as either standard or multi-Taub-NUT manifolds.

Findings

Complete hyperk"ahler manifolds asymptotic to circle fibrations are classified.

Such manifolds are either ^3 imes S^1 or multi-Taub-NUT.

The complex structures are either  imes /\u211e or minimal resolutions of cyclic Kleinian singularities.

Abstract

This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperk\"ahler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard $\rl^3 \times \sph^1$ or a multi-Taub-NUT manifold. In particular, the underlying complex manifold is either $\cx \times \cx/\ir$ or a minimal resolution of a cyclic Kleinian singularity.