Power convexity of solutions to complex Monge-Ampère equation in $\mathbb{C}^2$

Abstract

The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of solutions to the Dirichlet problem for the complex Monge-Ampère equation on a bounded, smooth, strictly convex domain in $\mathbb{C}^2$. Our approach is based on the constant rank theorem and the deformation process. A key obstacle in establishing the constant rank theorem lies in the construction of suitable auxiliary functions for deriving the associated differential inequalities. To address this difficulty, we refine the auxiliary function introduced by Bian and Guan \cite{Bian-Guan2009}.