Compactness of the $\bar{\partial}$-Neumann problem on domains with bounded intrinsic geometry

TL;DR

This paper introduces a new class of complex domains with bounded intrinsic geometry, establishing equivalences between compactness of the $ar{ ext{d}}$-Neumann operator, boundary properties, and metric relations.

Contribution

It defines a biholomorphically invariant domain class and proves key equivalences linking operator compactness, boundary geometry, and metric estimates.

Findings

Compactness of the $ar{ ext{d}}$-Neumann operator is equivalent to boundary not containing $q$-dimensional analytic varieties.

The Bergman metric is equivalent to the Kobayashi metric on these domains.

The pluricomplex Green function satisfies specific local estimates in terms of the Bergman metric.

Abstract

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, $\mathbb{C}$-convex domains, and homogeneous domains. For this class of domains, we show that compactness of the $\bar{\partial}$-Neumann operator on $(0,q)$-forms is equivalent to the boundary not containing any $q$-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.