New Calabi-Yau Metrics of Taub-NUT Type on C^{N+1}

TL;DR

This paper constructs new complete Calabi-Yau metrics on complex Euclidean spaces of dimension N+1, generalizing known Taub-NUT metrics, and introduces a gluing method to handle singularities in the construction.

Contribution

It introduces a novel class of Calabi-Yau metrics on C^{N+1} for N ≥ 3, extending the Taub-NUT family and overcoming singularity issues with a gluing technique.

Findings

Constructed complete non-flat Calabi-Yau metrics on C^{N+1} for N ≥ 3.

Generalized the Taub-NUT metrics to higher dimensions.

Resolved singularities via a gluing procedure.

Abstract

We construct a class of complete non-flat Calabi-Yau metrics on C^{N+1} for every N >= 3, which generalize the Taub-NUT metrics from C^2 and C^3 and whose tangent cone at infinity is R^N. The construction relies on the generalized Gibbons-Hawking ansatz. A key obstacle is that the volume-form defect of the ansatz fails to decay near certain components of the discriminant locus, producing singularities more severe than those encountered in dimension three, we resolve this by a gluing procedure.