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

We study spectral stability of the $${\bar{\partial }}$$ ∂ ¯ -Neumann Laplacian on a bounded domain in $${\mathbb {C}}^n$$ 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 $${\mathbb {C}}^n$$ 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 $${\mathbb {C}}^n$$ C n .

Homogeneous Plurisubharmonic Polynomials in Higher Dimensions

We prove several results on homogeneous plurisubharmonic polynomials on $${\mathbb {C}}^n$$ C n , $$n\in {\mathbb {Z}}_{\ge 2}$$ n ∈ Z ≥ 2 . Said results are relevant to the problem of constructing local bumpings at boundary points of pseudoconvex domains of finite D’Angelo 1-type in $${\mathbb {C}}^{n+1}$$ C n + 1 .