Local Plurisubharmonic Defining Functions on the Boundary

TL;DR

This paper investigates conditions under which a domain in complex space admits a local plurisubharmonic defining function on its boundary, providing necessary criteria and an algorithm for construction, with some cases where these are also sufficient.

Contribution

It introduces necessary conditions and an algorithm for constructing local plurisubharmonic defining functions on boundary, advancing understanding of boundary regularity in complex analysis.

Findings

Necessary conditions for existence are established.

An explicit algorithm for construction is provided.

In some cases, conditions are also sufficient.

Abstract

Necessary conditions for a domain $\Omega\subset \mathbb C^n$ admitting a local plurisubharmonic defining function on the boundary are given. In tandem, we give an algorithm to construct a local plurisubharmonic defining function on the boundary when one exists. In some cases we show that the necessary conditions are also sufficient.