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

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 bounded, smooth, strictly convex domain in ${\mathbb{C}}^2$. Our approach is based on the constant rank theorem and the deformation process.