Plurisubharmonic Defining Functions in $\mathbb C^2$

TL;DR

This paper derives a formula relating the complex Hessian determinants of different defining functions for a domain in ^2, providing criteria for when alternative defining functions are plurisubharmonic.

Contribution

It introduces a formula connecting the complex Hessians of two defining functions and establishes conditions for their local plurisubharmonicity.

Findings

Derived a formula for the determinant of the complex Hessian of alternative defining functions.

Provided necessary and sufficient conditions for local plurisubharmonicity of these functions.

Abstract

Let $\Omega=\{r<0\}\subset\mathbb C^2$, with $r$ plurisubharmonic on $b\Omega=\{r=0\}$. Let $\rho$ be another defining function for $\Omega$. A formula for the determinant of the complex Hessian of $\rho$ in terms of $r$ is computed. This formula is used to give necessary and sufficient conditions that make $\rho$ (locally) plurisubharmonic.