Suppose $\Omega_0,\Omega_1$ are two bounded strongly $\mathbb{ C}$-convex domains in $\mathbb{ C}^n$, with $n\geq 2$ 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-Amp\`ere 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.
Categories: math.CV math.AP