Hartogs Domains and the Diederich Forn{\ae}ss Index

TL;DR

This paper investigates the relationship between the boundary geometry of Hartogs domains and the Diederich-Forn{ a}ss Index, establishing conditions under which the index equals one and linking it to the existence of good vector fields and Stein neighborhood bases.

Contribution

It provides new criteria connecting the Diederich-Forn{ a}ss Index with boundary properties and the existence of Stein neighborhood bases in Hartogs domains.

Findings

Diederich-Forn{ a}ss Index equals one under certain boundary conditions.

Equivalence between index one and existence of good vector fields.

Index one implies the domain admits a Stein neighborhood basis.

Abstract

We study a geometric property of the boundary on Hartogs domains which can be used to find upper and lower bounds for the Diederich-Forn{\ae}ss Index. Using this, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich-Forn{\ae}ss Index for a Hartogs domain is equal to one if and only if the domain admits a family of good vector fields in the sense of Boas and Straube. We also study the analogous problem for a Stein neighborhood basis, and show that under the same hypotheses if the Diederich-Forn{\ae}ss Index for a Hartogs domain is equal to one then the domain admits a Stein neighborhood basis.