A Maximum Rank Theorem for Solutions to the Homogenous Complex Monge-Ampère Equation in a $\mathbb{C}$-Convex Ring

Abstract

Suppose ${\Omega }_0,{\Omega }_1$ are two bounded strongly $\mathbb{C}$-convex domains in ${\mathbb{C}}^n$, with $n\ge 2$ and ${\Omega }_1\supset \overline{{\Omega }_0}$. Let $\mathcal{R}={\Omega }_1\backslash \overline{{\Omega }_0}$. We call $\mathcal{R}$ a $\mathbb{C}$-convex ring. We will show that for a solution $\Phi$ to the homogenous complex Monge-Ampère equation in $\mathcal{R}$, with $\Phi =1$ on $\partial {\Omega }_1$ and $\Phi =0$ on $\partial {\Omega }_0$, $\sqrt{-1}\partial \overline{\partial }\Phi$ has rank $n-1$ and the level sets of $\Phi$ are strongly $\mathbb{C}$-convex.