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

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.