A maximum rank theorem for solutions to the homogenous complex Monge–Ampère equation in a \(\mathbb {C}\)-convex ring

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