# Spectral Stability of the $\bar\partial-$Neumann Laplacian: Domain Perturbations

## TL;DR

This paper investigates how the eigenvalues of the $ar ext{d}$-Neumann Laplacian change when the domain in complex space is slightly altered, providing continuity results and quantitative estimates for various domain classes.

## Contribution

It establishes semi-continuity properties and quantitative estimates for the spectral stability of the $ar ext{d}$-Neumann Laplacian under domain perturbations in complex analysis.

## Key findings

- Upper semi-continuity of eigenvalues on pseudoconvex domains
- Lower semi-continuity on domains satisfying property (P)
- Quantitative estimates on smooth pseudoconvex domains of finite D'Angelo type

## Abstract

We study spectral stability of the $\bar\partial$-Neumann Laplacian on a bounded domain in $\mathbb{C}^n$ when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $\bar\partial$-Neumann Laplacian on bounded pseudoconvex domains in $\mathbb{C}^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property ($P$), and quantitative estimates on smooth bounded pseudoconvex domains of finite D'Angelo type in $\mathbb{C}^n$.