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.
Categories: math.AP