The log-concavity of eigenfunction to complex Monge-Amp\`ere operator in $\mathbb{C}^2$

TL;DR

This paper proves the log-concavity of solutions to the Dirichlet eigenvalue problem for the complex Monge-Ampère operator in a convex domain in ^2, advancing understanding of convexity properties in complex analysis.

Contribution

It establishes the log-concavity of eigenfunctions for the complex Monge-Ampère operator, using the constant rank theorem and deformation method, extending previous work.

Findings

Proves log-concavity of solutions in ^2

Uses constant rank theorem and deformation method

Advances convexity analysis in complex Monge-Ampère equations

Abstract

Following the authors' recent work \cite{Zhang-Zhou2025}, we further explore the convexity properties of solutions to the Dirichlet problem for the complex Monge-Amp\`ere operator. In this paper, we establish the $\log$-concavity of solutions to the Dirichlet eigenvalue problem for the complex Monge-Amp\`ere operator on bounded, smooth, strictly convex domain in $\mathbb{C}^2$. The key ingredients consist of the constant rank theorem and the deformation method.