# Uniqueness of some Calabi-Yau metrics on $\mathbf{C}^n$

## TL;DR

This paper proves the uniqueness, up to scaling and isometry, of certain Calabi-Yau metrics on complex Euclidean spaces with a specific tangent cone at infinity, and discusses potential generalizations.

## Contribution

It establishes the uniqueness of specific Calabi-Yau metrics on ^n with a given tangent cone, extending understanding of their geometric structure.

## Key findings

- Uniqueness of the Calabi-Yau metrics up to scaling and isometry.
- Characterization of tangent cones at infinity for these metrics.
- Discussion of potential generalizations to other manifolds.

## Abstract

We consider the Calabi-Yau metrics on $\mathbf{C}^n$ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone $\mathbf{C}\times A_1$ at infinity for the $(n-1)$-dimensional Stenzel cone $A_1$. We show that up to scaling and isometry this Calabi-Yau metric on $\mathbf{C}^n$ is unique. We also discuss possible generalizations to other manifolds and tangent cones.