Hartogs domains and the Diederich–Fornæss index

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–Fornæ ss index. Using this property, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich–Fornæss 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–Fornæss index for a Hartogs domain is equal to one, then the domain admits a Stein neighborhood basis.