Model pseudoconvex domains and bumping

TL;DR

This paper explores the construction of explicit model domains in complex analysis by bumping pseudoconvex domains in C^3, aiming to simplify their Levi geometry while preserving pseudoconvexity.

Contribution

It introduces a method for bumping pseudoconvex domains in C^3 beyond h-extendible cases, providing explicit models that facilitate analysis of boundary geometry.

Findings

Constructed explicit bumping models for a broader class of domains in C^3.

Demonstrated preservation of pseudoconvexity during bumping.

Achieved simpler defining functions with minimal contact orders at bumping points.

Abstract

The Levi geometry at weakly pseudoconvex boundary points of domains in C^n, n \geq 3, is sufficiently complicated that there are no universal model domains with which to compare a general domain. Good models may be constructed by bumping outward a pseudoconvex, finite-type \Omega \subset C^3 in such a way that: i) pseudoconvexity is preserved, ii) the (locally) larger domain has a simpler defining function, and iii) the lowest possible orders of contact of the bumped domain with \bdy\Omega, at the site of the bumping, are realised. When \Omega \subset C^n, n\geq 3, it is, in general, hard to meet the last two requirements. Such well-controlled bumping is possible when \Omega is h-extendible/semiregular. We examine a family of domains in C^3 that is strictly larger than the family of h-extendible/semiregular domains and construct explicit models for these domains by bumping.