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

Gregor Herbort (2007)

Annales Polonici Mathematici

Let a and m be positive integers such that 2a < m. We show that in the domain D : = z ³ | r ( z ) : = z + | z | ² + | z | 2 m + | z z | 2 a + | z | 2 m < 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.

Boundary behaviour of holomorphic functions in Hardy-Sobolev spaces on convex domains in ℂⁿ

Marco M. Peloso, Hercule Valencourt (2010)

Colloquium Mathematicae

We study the boundary behaviour of holomorphic functions in the Hardy-Sobolev spaces p , k ( ) , where is a smooth, bounded convex domain of finite type in ℂⁿ, by describing the approach regions for such functions. In particular, we extend a phenomenon first discovered by Nagel-Rudin and Shapiro in the case of the unit disk, and later extended by Sueiro to the case of strongly pseudoconvex domains.

C k -estimates for the ¯ -equation on concave domains of finite type

William Alexandre (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

C k estimates for convex domains of finite type in n are known from [7] for k = 0 and from [2] for k &gt; 0 . We want to show the same result for concave domains of finite type. As in the case of strictly pseudoconvex domain, we fit the method used in the convex case to the concave one by switching z and ζ in the integral kernel of the operator used in the convex case. However the kernel will not have the same behavior on the boundary as in the Diederich-Fischer-Fornæss-Alexandre work. To overcome this problem...

Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form.

Philippe Charpentier, Yves Dupain (2006)

Publicacions Matemàtiques

In this paper, we give precise isotropic and non-isotropic estimates for the Bergman and Szegö projections of a bounded pseudoconvex domain whose boundary points are all of finite type and with locally diagonalizable Levi form. Additional local results on estimates of invariant metrics are also given.

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

Gregor Herbort (2013)

Annales Polonici Mathematici

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...

On the Bergman distance on model domains in ℂⁿ

Gregor Herbort (2016)

Annales Polonici Mathematici

Let P be a real-valued and weighted homogeneous plurisubharmonic polynomial in 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.

Quantitative estimates for the Green function and an application to the Bergman metric

Klas Diederich, Gregor Herbort (2000)

Annales de l'institut Fourier

Let D 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 D a quantitative upper bound for the supremum sup z K | G D ( z , w ) | 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 n / { O } , when w D tends to a boundary point....

Smoothness of Cauchy Riemann maps for a class of real hypersurfaces.

Hervé Gaussier (2001)

Publicacions Matemàtiques

We study the regularity problem for Cauchy Riemann maps between hypersurfaces in Cn. We prove that a continuous Cauchy Riemann map between two smooth C∞ pseudoconvex decoupled hypersurfaces of finite D'Angelo type is of class C∞.

The pluricomplex Green function on some regular pseudoconvex domains

Gregor Herbort (2014)

Annales Polonici Mathematici

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.

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