Non-quadratic solutions to the Monge-Ampère equation

TL;DR

This work studies the complex Monge-Ampère equation $\det(\partial\bar{\partial}u)=1$ in $\mathbb{C}^2$ by focusing on real-valued solutions quadratic in one complex variable. It derives a tractable semilinear system for the coefficients of a quadric form in $z$, proves solvability on cylindrical domains via nonlinear Poisson-system theory, and analyzes when the resulting Kähler metric is flat. The authors produce entire solutions with flat metrics, construct cylindrical-solutions with nowhere-flat metrics, and classify radially symmetric solutions with varied singularities, while proving a rigidity phenomenon for more general non-quadratic-in-$z$ forms and extending results to Donaldson’s equation. Collectively, the paper advances understanding of when Calabi’s flat-metric question holds, reveals rich families of non-quadratic solutions, and connects to known singular examples in the literature. The methods combine semilinear reductions, global solvability theorems, and careful curvature analysis to map out the landscape of solutions and their geometric consequences.

Abstract

We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Ampère equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on $\mathbb C^2$ induce flat Kahler metrics, as expected by a question of Calabi. In contrast, those on cylindrical domains produce a family of nowhere flat Kahler metrics. Beyond these smooth solutions, we also classify solutions that are radially symmetric in one variable, which exhibit various types of singularities. Finally, we explore analogous solutions to Donaldson's equation motivated by a result of He.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• proof : Proof of Theorem \ref{['obp']}:
• Theorem 3.1
• proof
• Example 1
• Corollary 3.2