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

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 ${\mathbb{C}}^n$.

arXiv comment28 pages