A note on plurisubharmonic defining functions in $\mathbb{C}^n$

TL;DR

This paper investigates the properties of smoothly bounded domains in complex spaces of dimension greater than 2, showing that if such a domain admits a boundary plurisubharmonic defining function, then certain complex-analytic properties follow, including the ability to approximate the Diederich-Fornaess exponent and the existence of Stein neighborhood bases.

Contribution

It establishes a connection between boundary plurisubharmonic defining functions and the approximation of the Diederich-Fornaess exponent, as well as Stein neighborhood basis existence.

Findings

Diederich-Fornaess exponent can be made arbitrarily close to 1.

Closure of the domain admits a Stein neighborhood basis.

Boundary conditions influence complex-analytic properties of the domain.

Abstract

Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.