A Liouville theorem for the complex Monge-Ampère equation on product manifolds

Abstract

Let $Y$ be a closed Calabi-Yau manifold. Let $\omega$ be the Kähler form of a Ricci-flat Kähler metric on ${\mathbb{C}}^m\times Y$. We prove that if $\omega$ is uniformly bounded above and below by constant multiples of ${\omega }_{{\mathbb{C}}^m}+{\omega }_Y$, where ${\omega }_{{\mathbb{C}}^m}$ is the standard flat Kähler form on ${\mathbb{C}}^m$ and ${\omega }_Y$ is any Kähler form on $Y$, then $\omega$ is actually equal to a product Kähler form, up to a certain automorphism of ${\mathbb{C}}^m\times Y$.

arXiv comment9 pages