Hearing pseudoconvexity in Lipschitz domains with holes via $${\bar{\partial }}$$ ∂ ¯

Abstract

Let Ω = Ω \ D where Ω is a bounded domain with connected complement in C n (or more generally in a Stein manifold) and D is relatively compact open subset of Ω with connected complement in Ω. We obtain characterizations of pseudoconvexity of Ω and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of Ω and D are Lipschitz and C 2 -smooth respectively, then both Ω and D are pseudoconvex if and only if 0 is not in the spectrum of the ∂-Neumann Laplacian on (0, q)-forms for 1 ≤ q ≤ n − 2 when n ≥ 3; or 0 is not a limit point of the spectrum of the ∂-Neumannn Laplacian on (0, 1)-forms when n = 2. (2000): 32C35, 32C37, 32W05. Mathematics Subject Classification

Exact sequences and estimates for the $\overline{\partial}$-problem

“…Now we construct a long exact sequence associated to an annulus which relates the function theory of the annulus with that of its hole and its envelope. An immediate consequence of our result is Corollary 1.2 of the introduction, and in particular the very important exact sequence (1.7), which encompasses many of the results about L 2 -estimates on annuli as found in [LS13,FLTS17] and earlier works cited there.…”

Holomorphic Approximation via Dolbeault Cohomology

“…Following the arguments in the proof of Lemma 1.7, we can prove that if Ȟn,q Φ (X \ D) = 0 for all ≤ q ≤ n − 1, then Ȟn,q (X \ D) = 0, for any 1 ≤ q ≤ n − 2 and Ȟn,n−1 (X\D) = 0 is Hausdorff. Moreover, Theorem 3.11 in [5] implies that H n,q D,cur (X) = 0 if and only if Ȟn,q−1 (X \ D) = 0, for any 1 ≤ q ≤ n − 1. Proposition 3.7 in [5] implies that if Ȟn,n−1 (X \ D) = 0, then H n,n D,cur (X) is Hausdorff.…”

“…Moreover, Theorem 3.11 in [5] implies that H n,q D,cur (X) = 0 if and only if Ȟn,q−1 (X \ D) = 0, for any 1 ≤ q ≤ n − 1. Proposition 3.7 in [5] implies that if Ȟn,n−1 (X \ D) = 0, then H n,n D,cur (X) is Hausdorff. Therefore the corollary follows from Corollary 3.12, Theorem 3.13 in [5] and Corollary 2.5.…”

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

“…Spectral behavior of the ∂-Neumann Laplacian has also been shown to be sensitive to the geometry of the domains. Positivity of the ∂-Neumann can be used to characterize pseudoconvexity (see [17,22] and references therein). Spectral discreteness of the ∂-Neumann Laplacian can be used to determine whether the boundary of a convex domain in C n contains a complex variety ( [19,20]) and whether the boundary of a smooth bounded pseudoconvex Hartogs domain in C 2 satisfies property (P), a potential theoretic property introduced by Catlin [5] (see [9,21]).…”