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

## TL;DR

This paper proves a Liouville-type theorem for Ricci-flat Kähler metrics on product manifolds, showing under certain bounds such metrics must be product forms, extending understanding of complex Monge-Ampère equations.

## Contribution

It establishes a rigidity result for Ricci-flat Kähler metrics on product manifolds, characterizing when such metrics are necessarily product forms under boundedness conditions.

## Key findings

- Ricci-flat Kähler metrics are product forms under bounds
- The result applies to manifolds with a Calabi-Yau factor
- Provides a Liouville-type theorem for complex Monge-Ampère equations

## 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 $\mathbb{C}^m \times Y$.