Wyniki wyszukiwania

1

On the Bergman distance on model domains in ℂⁿ

100%

Herbort G.

Annales Polonici Mathematici

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2016

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tom 116

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nr 1

1-36

EN

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.

2

Estimation of the Carathéodory distance on pseudoconvex domains of finite type whose boundary has Levi form of corank at most one

100%

Herbort G.

Annales Polonici Mathematici

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2013

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tom 109

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nr 3

209-260

EN

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 in dimension two.

3

An example of a pseudoconvex domain whose holomorphic sectional curvature of the Bergman metric is unbounded

100%

Herbort G.

Annales Polonici Mathematici

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2007

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tom 92

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nr 1

29-39

EN

Let a and m be positive integers such that 2a < m. We show that in the domain $D:= {z ∈ ℂ³ | r(z):= ℜ z₁ + |z₁|² + |z₂|^{2m} + |z₂z₃|^{2a} + |z₃|^{2m} <0}$ 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.

4

On the Green function on a certain class of hyperconvex domains

100%

Herbort G.

Annales Polonici Mathematici

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2008

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tom 94

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nr 2

149-185

EN

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 $|ψ| ≤ Cexp(-C'(log(1/δ_D))^α)$ 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.

5

The pluricomplex Green function on some regular pseudoconvex domains

100%

Herbort G.

Annales Polonici Mathematici

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2014

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tom 110

|

nr 3

209-226

EN

Let D be a smooth bounded pseudoconvex domain in ℂⁿ of finite type. We prove an estimate on the pluricomplex Green function $𝒢_D(z,w)$ 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.

Ograniczanie wyników

5 Annales Polonici Mathematici

5 Herbort G.

1 2016

1 2014

1 2013

1 2008

1 2007

On the Bergman distance on model domains in ℂⁿ

Estimation of the Carathéodory distance on pseudoconvex domains of finite type whose boundary has Levi form of corank at most one

An example of a pseudoconvex domain whose holomorphic sectional curvature of the Bergman metric is unbounded

On the Green function on a certain class of hyperconvex domains

The pluricomplex Green function on some regular pseudoconvex domains