Power Convexity of Solutions to the Complex Monge-Ampère Equation $\det(u_{i\overline{j}})=1$ in Complex Dimension Two

Abstract

In this paper, we establish the power convexity of solutions to the complex Monge-Ampère equation $\det(u_{i\overline{j}})=1$ in a convex set in the complex 2 dimensional Euclidean space. Even the convexity result is only for 2 dimension, our approach is applicable to all dimensions.