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

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.

26 pages