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Register a new userComplete proper holomorphic embeddings of strictly pseudoconvex domains into balls
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Barbara Drinovec Drnov\\v{s}ek
Publication date
2015
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English
Authors
Barbara Drinovec Drnovsek
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We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $mathcal{C}^2$-boundary in $mathbb{C}^n$ into the unit ball of $mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.
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We investigate regularity properties of the $overline{partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
We make several new contributions to the study of proper holomorphic mappings between balls. Our results include a degree estimate for rational proper maps, a new gap phenomenon for convex families of arbitrary proper maps, and an interesting result about inverse images.
In this paper we introduce, via a Phragmen-Lindelof type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the {sl pluricomplex Poisson kernel} because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero on the boundary except at one boundary point where it has a non-tangential simple pole, and reproduces pluriharmonic functions. We also use such a function to obtain a new intrinsic version of the classical Julias Lemma and Julia-Wolff-Caratheodory Theorem.
In this paper, we generalize a recent work of Liu et al. from the open unit ball $mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Caratheodory metric near the boundary of $C^2$ domains and the well-known Grahams estimate on the boundary behavior of the Caratheodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.
In this paper (Math. Res. Lett. 13 (2006). No 4, 509-523), the authors established a pseudo-normal form for proper holomoprhic mappings between balls in complex spaces with degenerate rank. This then was used to give a complete characterization for all proper holomorphic maps with geometric rank one, which, in particular, includes the following as an immediate application: Theorem: Any rational holomorphic map from B^n into B^N with $4le nle Nle 3n-4$ is equivalent to the DAngelo map $$F_{theta}(z,w)=(z,(costheta)w,(sintheta)z_1w, ..., (sintheta)z_{n-1}w, (sintheta)w^2, 0), 0le thetaleq pi/2.$$ It is a well-known (but also quite trivial) fact that any non-constant rational CR map from a piece of the sphere $partial {B^n}$ into the sphere $partial {B^N}$ can be extended as a proper rational holomoprhic map from $B^n$ into $B^N$ ($Nge nge 2$). By using the rationality theorem that the authors established in [HJX05], one sees that the the above theorem (and also the main theorem of the paper) holds in the same way for any non-constant $C^3$-smooth CR map from a piece of $partial {B^n}$ into $partial{B^N}$. The paper [Math. Res. Lett. 13 (2006). No 4, 509-523] was first electronically published by Mathematical Research Letters several months ago at its home website: http://www.mrlonline.org/mrl/0000-000-00/Huang-Ji-Xu2.pdf. (The pdf file of the printed journal version can also be downloaded at http://www.math.uh.edu/~shanyuji/rank1.pdf).
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