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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

The quotient of the Szego and Bergman kernels for a smooth bounded pseudoconvex domains in C n is bounded from above by a constant multiple of δ | log δ | p for any p > n , where δ is the distance to the boundary. For a class of domains that includes those of DʼAngelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by a constant multiple of δ | log δ | p for any p − 1 . Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of δ.

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

The spectrum of the ∂-Neumann Laplacian on a polydisc in C" is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of infinite multiplicity.

Formulas that relate the Bergman kernel and projection of a bounded Reinhardt domain whose closure does not intersect the coordinate planes to those of its covering tube domain are obtained via the Poisson summation formula.

In this paper, we study the problem of classification of piecewise smooth domains in with non-compact automorphism groups. We prove that a simply-connected domain in with generic piecewise smooth Levi-flat boundary and non-compact automorphism group is biholomorphic to a bidisc.

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.

The asymptotic behavior of the holomorphic sectional curvature of the Bergman metric on a pseudoconvex Reinhardt domain of finite type in ℂ2 is obtained by rescaling locally the domain to a model domain that is either a Thullen domain Ω m = {(z 1,z 2); ¦z 1¦2 + ¦z 2¦2m < 1 or a tube domainT m = {(z 1,z 2); Imz1 + (Imz2)2m < 1}. The Bergman metric for the tube domainT m is explicitly calculated by using Fourier-Laplace transformation. It turns out that the holomorphic sectional curvature of the Bergman metric on the tube domainT m at (0, 0) is bounded above by a negative constant. These results are used to construct a complete Kahler metric with holomorphic sectional curvature bounded above by a negative constant for a pseudoconvex Reinhardt domain of finite type in ℂ2.

We discuss and expand upon our previous results on comparison of the Bergman and the Szego kernels. We also study the relations between the Bergman growth exponent and the finite type conditions.

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.

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.

We study the classification of smoothly bounded domains in complex manifolds that cover compact sets. We prove that a smoothly bounded domain in a hyperbolic complex surface that covers a compact set is either biholomorphic to the ball or covered by the bidisc.