Boundary invariants and the closed range property for $\bar\partial$

TL;DR

This paper links CR geometric invariants with analytic conditions to establish the closed range property of the $ar ext{d}$ operator on complex domains, introducing new invariants and linear algebra tools.

Contribution

It introduces third and fourth order CR invariants on domain boundaries that ensure the closed range property of the $ar ext{d}$-Laplacian, bridging geometric and analytic approaches.

Findings

CR invariants determine closed range conditions

New linear algebra methods relate invariants to Levi form eigenvalues

Examples demonstrate the effectiveness of the new invariants

Abstract

This paper provides a connection between two distinct branches of research in CR geometry -- namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR manifolds. Specifically, we work on not necessarily pseudoconvex domains $\Omega\subset\mathbb{C}^n$ and define third and fourth order CR invariants on $\bd\Om$ and show that these invariants provide enough information to establish closed range for the $\bar\partial$-Laplacian in $L^2_{(0,q)}(\Omega)$ for a given, fixed $q$. The closed range estimates follow from our previously defined weak $Z(q)$ condition. We also develop powerful linear algebra machinery to translate the information from the invariants into information about the Levi form and its eigenvalues. We conclude with several examples that demonstrate the usefulness and ease of use of the new conditions.