Plurisubharmonic Defining Functions in ${\mathbb{C}}^2$

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.