A maximum rank theorem for solutions to the homogenous complex Monge-Ampere equation in a C-convex ring

Suppose Omega(0),Omega(1) are two bounded strongly C-convex domains in C-n, with n >= 2 and Omega(1)superset of(Omega(0)) over bar. Let R=Omega(1)\(Omega(0)) over bar. We call R a C-convex ring. We will show that for a solution Phi to the homogenous complex Monge-Ampere equation in R, with Phi=1 on partial derivative Omega(1) and Phi=0 on partial derivative Omega(0), root-1 partial derivative partial derivative Phi has rank n-1 and the level sets of Phi are strongly C-convex.