A Liouville Theorem for the Complex Monge-Amp\`ere Equation

TL;DR

This paper proves a Liouville theorem for the complex Monge-Ampère equation, showing that solutions close to quadratic polynomials at infinity must be quadratic polynomials themselves.

Contribution

It establishes a new Liouville theorem characterizing entire solutions of the complex Monge-Ampère equation based on their asymptotic behavior.

Findings

Solutions differing from quadratic polynomials by o(|x|^2) are quadratic polynomials.

The theorem applies to solutions with constant right-hand side.

Provides a classification of solutions based on growth at infinity.

Abstract

In this note, we derive a Liouville theorem for the complex Monge-Amp\`ere equation. Our result states that if the global solution $u$ of the complex Monge-Amp\`ere equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then $u$ is a quadratic polynomial.