The intrinsic geometry on bounded pseudoconvex domains

TL;DR

This paper investigates how the Diederich--Forn ext{æ}ss index influences the geometric properties of bounded pseudoconvex domains, providing intrinsic boundary conditions and necessary criteria, especially for index 1 cases.

Contribution

It introduces intrinsic geometric conditions on domain boundaries related to the Diederich--Forn ext{æ}ss index, focusing on necessary conditions for domains with index 1 in ^2.

Findings

Derived geometric boundary conditions for arbitrary indices.

Established a necessary condition for domains with index 1.

Simplified conditions when the Levi-flat set is a closed Riemann surface.

Abstract

The Diederich--Forn\ae ss index has been introduced since 1977 to classify bounded pseudoconvex domains. In this article, we derive several intrinsic, geometric conditions on boundary of domains for arbitrary indexes. Many results, in the past, by various mathematicians estimated the index by assuming some properties of domains. Our motivation of this paper is, the other way around, to look for how the index effects properties and shapes of domains. Especially, we look for a necessary condition of all bounded pseudoconvex domains $\Omega\subset\mathbb{C}^2$ with the Diederich--Forn\ae ss index 1. We also show that, when the Levi-flat set of $\partial\Omega$ is a closed Riemann surface, then the necessary condition can be simplified.