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We study the class of smooth bounded weakly pseudoconvex domains D ⊂ ℂⁿ whose boundary points are of finite type (in the sense of J. Kohn) and whose Levi form has at most one degenerate eigenvalue at each boundary point, and prove effective estimates on the invariant distance of Carathéodory. This completes the author's investigations on invariant differential metrics of Carathéodory, Bergman, and Kobayashi in the corank one situation and on invariant distances on pseudoconvex finite type domains...

Let P be a real-valued and weighted homogeneous plurisubharmonic polynomial in ${ℂ}^{n-1}$ and let D denote the “model domain” z ∈ ℂⁿ | r(z):= Re z₁ + P(z’) < 0. We prove a lower estimate on the Bergman distance of D if P is assumed to be strongly plurisubharmonic away from the coordinate axes.

Let D be a smooth bounded pseudoconvex domain in ℂⁿ of finite type. We prove an estimate on the pluricomplex Green function ${}_D(z,w)$ of D that gives quantitative information on how fast the Green function vanishes if the pole w approaches the boundary. Also the Hölder continuity of the Green function is discussed.

We study the behavior of the pluricomplex Green function on a bounded hyperconvex domain D that admits a smooth plurisubharmonic exhaustion function ψ such that 1/|ψ| is integrable near the boundary of D, and moreover satisfies the estimate $\left|\psi \right|\le Cexp(-C^{\text{'}}{\left(log(1/{\delta }_D)\right)}^{\alpha })$ at points close enough to the boundary with constants C,C’ > 0 and 0 < α < 1. Furthermore, we obtain a Hopf lemma for such a function ψ. Finally, we prove a lower bound on the Bergman distance on D.

Let a and m be positive integers such that 2a < m. We show that in the domain $D:={z\in ℂ³|r\left(z\right):=\Re z₁+|z₁|²+|z₂|}^{2m}+{\left|z₂z₃\right|}^{2a}+{\left|z₃\right|}^{2m}<0$ the holomorphic sectional curvature $R_D(z;X)$ of the Bergman metric at z in direction X tends to -∞ when z tends to 0 non-tangentially, and the direction X is suitably chosen. It seems that an example with this feature has not been known so far.

Let $D\subset {ℂ}^n$ be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by $G_D(.,.)$. In this article we give for a compact subset $K\subset D$ a quantitative upper bound for the supremum ${sup}_{z\in K}\left|G_D(z,w)\right|$ in terms of the boundary distance of $K$ and $w$. This enables us to prove that, on a smooth bounded regular domain $D$ (in the sense of Diederich-Fornaess), the Bergman differential metric $B_D(w;X)$ tends to infinity, for $X\in {ℂ}^n/\left\{O\right\}$, when $w\in D$ tends to a boundary point....