On plurisubharmonic defining functions for pseudoconvex domains in $\mathbb{C}^2$

TL;DR

This paper explores the existence of plurisubharmonic defining functions for pseudoconvex domains in ^2, providing counterexamples and criteria that determine when such functions exist, with implications for complex analysis.

Contribution

It introduces new counterexamples and establishes criteria for the existence of plurisubharmonic defining functions in ^2 domains, advancing understanding in complex analysis.

Findings

Counterexamples to smooth plurisubharmonic defining functions

Criteria equivalent to existence of such functions

Analysis of specific domain classes

Abstract

We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in $\mathbb{C}^2$. In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains.