A Liouville theorem for the complex Monge-Amp\`ere equation on product manifolds

Abstract

Let $Y$ be a closed Calabi-Yau manifold. Let $\omega$ be the K\"ahler form of a Ricci-flat K\"ahler 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 K\"ahler form on $\mathbb{C}^m$ and $\omega_Y$ is any K\"ahler form on $Y$, then $\omega$ is actually equal to a product K\"ahler form, up to a certain automorphism of $\mathbb{C}^m \times Y$.