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

TL;DR

This work constructs a new family of complete Calabi–Yau metrics on $C^{N+1}$ for $N\ge 3$, extending Taub–NUT-type geometries to higher dimensions. The authors leverage a generalized Gibbons–Hawking framework and devise a multistage gluing strategy to overcome non-decaying volume-form errors along a complex discriminant locus, implemented via an induction on discriminant strata. After refining the Gibbons–Hawking data through surgeries and ensuring compatible smooth holomorphic structures, they invoke the Tian–Yau–Hein package to solve the complex Monge–Ampère equation, yielding complete CY metrics with bounded curvature and non-maximal volume growth: $Vol(B_p(R)) \sim R^{N+2}$. The resulting metrics are torus-invariant and recover the classical Taub–NUT metric when $N=1$ and the Li higher-dimensional analogue when $N=2$, situating them within a toric fibration picture with an unbounded discriminant locus.

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.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 2.1
• Remark 2.2
• proof
• Definition 2.3
• Lemma 3.1
• proof
• Proposition 3.2
• Lemma 3.3
• proof