  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.
