A Maximum Rank Theorem for Solutions to the Homogenous Complex Monge-Amp\`ere Equation in a $\mathbb{C}$-Convex Ring

TL;DR

This paper proves that solutions to the homogeneous complex Monge-Ampère equation in a $ ext{C}$-convex ring have a specific rank property and strongly $ ext{C}$-convex level sets, revealing geometric structure of solutions.

Contribution

It establishes a maximum rank theorem for solutions in $ ext{C}$-convex rings, showing the rank of the complex Hessian is $n-1$ and level sets are strongly $ ext{C}$-convex, which is a new geometric insight.

Findings

The complex Hessian of solutions has rank $n-1$.

Level sets of solutions are strongly $ ext{C}$-convex.

The result applies to solutions with specified boundary conditions.

Abstract

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.