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In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich–Fornaess exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.

Let \(\Omega ={\widetilde{\Omega }}{\setminus } \overline{D}\) where \({\widetilde{\Omega }}\) is a bounded domain with connected complement in \({\mathbb {C}}^n\) (or more generally in a Stein manifold) and D is relatively compact open subset of \({\widetilde{\Omega }}\) with connected complement in \(\widetilde{\Omega }\). We obtain characterizations of pseudoconvexity of \({\widetilde{\Omega }}\) and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups of \(\Omega \) on various function spaces. In particular, we show that if the boundaries of \({\widetilde{\Omega }}\) and D are Lipschitz and \(C^2\)-smooth respectively, then both \({\widetilde{\Omega }}\) and D are pseudoconvex if and only if 0 is not in the spectrum of the \(\overline{\partial }\)-Neumann Laplacian of \(\Omega \) on (0, q)-forms for \(1\le q\le n-2\) when \(n\ge 3\); or 0 is not a limit point of the spectrum of the \(\overline{\partial }\)-Neumann Laplacian on (0, 1)-forms when \(n=2\).

We study the Diederich–Fornaess exponent and relate it to non-existence of Stein domains with Levi-flat boundaries in complex manifolds. In particular, we prove that if the Diederich–Fornaess exponent of a smooth bounded Stein domain in an $$n$$ -dimensional complex manifold is greater than $$k/n$$ , then it has a boundary point at which the Levi-form has rank greater than or equal to k.

We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective version of Rhodes' result is given for a tower of coverings on a compact Riemann surface of genus greater than or equal to 2. Stability of the Bergman kernel is established for towers of coverings on hyperbolic Riemann surfaces and on complete Kaehler manifolds satisfying certain potential conditions. As a consequence, stability of the Bergman kernel is established for any tower of coverings of Riemann surfaces when the top manifold is simply-connected., Comment: 22 pages

Estimates of the Kobayashi metric in the normal direction are used to study the mapping problem in several complex variables., Comment: 13 pages; to appear in the special issue of Complex Variables and Elliptic Equations on holomorphic mappings edited by Steven Krantz and John Erik Fornaess

A smooth bounded pseudoconvex domain in C2 is of finite type if and only if the number of eigenvalues of its a-Neumann Laplacian that are less than or equal to ? has at most polynomial growth as ? goes to infinity.

Compactness of the Neumann operator in the ∂ ¯ -Neumann problem is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth domain is constructed for which condition (P) fails to hold, yet the Kohn Laplacian still has compact resolvent. The main result, in contrast, is that for smoothly bounded Hartogs domains, the well-known sufficient condition (P) is equivalent to compactness. The analyses of compactness and condition (P) boil down to the asymptotic behavior of the lowest eigenvalues of two related sequences of Schrodinger operators, one with a magnetic field and one without, parametrized by a Fourier variable resulting from the Hartogs symmetry. The nonsmooth example is based on the Aharonov–Bohm phenomenon of quantum physics. For smooth domains not satisfying (P), we prove that there always exists an exceptional sequence of Fourier variables for which the Aharonov–Bohm effect is weak and thence that compactness fails to hold. This sequence can be very sparse, so that the lack of compactness is due to a rather subtle effect.

We prove that, if ρ is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in , then the variety type at ρ is identical to the regular type.

In this paper we obtain the asymptotic expansions of the Carathéodory and Kobayashi metrics of strictly pseudoconvex domains with C∞ smooth boundaries in ℂn. The main result of this paper can be stated as following:Main Theorem. Let Ω be a strictly pseudoconvex domain with C∞ smooth boundary. Let FΩ(z,X) be either the Carathéodory or the Kobayashi metric of Ω. Let δ(z) be the signed distance from z to ∂Ω with δ(z) < 0 for z ∊ Ω and δ(z) ≥ 0 for z ∉ Ω. Then there exist a neighborhood U of ∂Ω, a constant C > 0, and a continuous function C(z,X):(U ∩ Ω) × ℂn -> ℝ such that and|C(z,X)| ≤ C|X| for z ∊ U ∩ Ω and X ∊ ℂn

A bounded domain in several complex variables with connected Lipschitz boundary is pseudoconvex if and only if the bottom of the (essential) spectrum of the Kohn Laplacian is positive on all (0, q)-forms with square-integrable coefficients., 8 pages

We give an explicit description of smoothly bounded Reinhardt domains with noncompact automorphism groups. In particular, this description confirms a special case of a conjecture of Greene/Krantz.

The ∂-Neumann operator on (0,q)-forms (1⩽q⩽n) on a bounded convex domainΩin Cnis compact if and only if the boundary ofΩcontains no complex analytic (equivalently: affine) variety of dimension greater than or equal toq.

We study the asymptotic behavior, in a “semi-classical limit,” of the first eigenvalues (i.e., the groundstate energies) of a class of Schrodinger operators with magnetic fields and the relationship of this behavior with compactness in the ∂ -Neumann problem on Hartogs domains in C 2 .