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

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