Plurisubharmonic polynomials and bumping

TL;DR

This paper investigates the problem of bumping pseudoconvex domains in complex spaces while preserving pseudoconvexity and explicitly controlling contact orders, focusing on non-semiregular domains in C^3 and perturbing plurisubharmonic polynomials.

Contribution

It introduces methods to perturb homogeneous plurisubharmonic polynomials to achieve controlled bumping in complex domains, especially for non-semiregular cases in C^3.

Findings

Explicit bumping with minimal contact orders achieved for certain domains.

Perturbation techniques preserve plurisubharmonicity of polynomials.

Analysis of non-semiregular domain behavior in C^3.

Abstract

We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain \Omega\subset C^n in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with bdy(\Omega), at the site of the bumping, are explicitly realised. Generally, when \Omega\subset C^n, n\geq 3, the known methods lead to bumpings with high orders of contact -- which are not explicitly known either -- at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in C^3. These families are identified by the behaviour of the least-weight plurisubharmonic polynomial in the Catlin normal form. Accordingly, we study how to perturb certain homogeneous plurisubharmonic polynomials without destroying plurisubharmonicity.