Nonuniqueness of Calabi-Yau metrics with maximal volume growth

TL;DR

This paper constructs a family of inequivalent Calabi-Yau metrics on complex three-space with the same asymptotic cone, providing new examples and insights into their classification and nonuniqueness.

Contribution

It presents the first known family of Calabi-Yau metrics on ^3 with fixed asymptotic cone and fixed complex structure, challenging existing uniqueness conjectures.

Findings

Constructed a family of inequivalent Calabi-Yau metrics on ^3

Metrics are asymptotic to  d7 A_2 at infinity

Proposed a refinement of Sze9kelyhidi's conjecture

Abstract

We construct a family of inequivalent Calabi-Yau metrics on $\mathbf{C}^3$ asymptotic to $\mathbf{C} \times A_2$ at infinity, in the sense that any two of these metrics cannot be related by a scaling and a biholomorphism. This provides the first example of families of Calabi-Yau metrics asymptotic to a fixed tangent cone at infinity, while keeping the underlying complex structure fixed. We propose a refinement of a conjecture of Sz\'ekelyhidi addressing the classification of such metrics.