The Diederich--Forn\ae ss index II: for domains of trivial index

TL;DR

This paper investigates conditions under which bounded pseudoconvex domains in complex Euclidean spaces have a trivial Diederich--Forn ext{ae}ss index, linking geometric, analytical, and topological aspects through new methods.

Contribution

It introduces new analytical and geometric criteria for trivial Diederich--Forn ext{ae}ss index and connects the index to differential equations and topological conditions, expanding understanding of complex domain geometry.

Findings

Necessary conditions for trivial index identified

Geometric sufficient conditions established

Connection between topology and differential equations demonstrated

Abstract

We study bounded pseudoconvex domains in complex Euclidean spaces. We find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial Diederich--Forn\ae ss index (i.e. the index equals to 1). We also connect a differential equation to the index. This reveals how a topological condition affects the solution of the associated differential equation and consequently obstructs the index being trivial. The proofs relies on a new method of study of the complex geometry of the boundary. The method was motivated by geometric analysis of Riemannian manifolds. We also generalize our main theorems under the context of de Rham cohomology.