Injective Hyperbolicity for quotients of balls and polydisks
In this article we study the injective Kobayashi metric on complex surfaces.
math.CV↗
arXiv subjects
Maria Trybula
Publications and source records attributed to Maria Trybula.
Gromov hyperbolicity and the Kobayashi metric on "convex" sets
In this paper we study the global geometry of the Kobayashi metric on "convex" sets. We provide new examples of non-Gromov hyperbolic domains in $\mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon \newline -vex, bounded and unbounded. Our first aim is to prove that if $\Omega$ is a bounded weakly linearly convex domain in $\mathbb{C}^n,\,n\geq 2,$ and $S$ is an affine complex hyperplane intersecting $\Omega,$ then the domain $\Omega\setminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we show that Gromov hyperbolicity of every open set of the form $\Omega\setminus S',$ where $S'$ is relatively closed in $\Omega$ and $\Omega$ is a convex domain, depends only on that how $S'$ looks near the boundary, i.e., whether $S'$ near $\partial\Omega$ (Theorem 1.7). We close the paper with a general remark on Hartogs type domains. The paper extends in an essential way results in [6].
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Boundary behavior of invariant functions on planar domains
Precise behavior of the Caratheodory, Kobayashi and Bergman metrics and distances near smooth boundary points of domains in C is found under different assumptions of regularity.
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The Kobayashi balls of C-convex domains
A pure geometric description of the Kobayashi balls of C-convex domains is given in terms of the so-called minimal basis.
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Gromov (non)hyperbolicity of certain domains in $\mathbb{C}^{2}$
We prove the non-hyperbolicity of the Kobayashi distance for $\mathcal{C}^{1,1}$-smooth convex domains in $\mathbb{C}^{2}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. Moreover, examples of smooth, non pseudoconvex, Gromov hyperbolic domains are given; we prove that the symmetrized polydisc and the tetrablock are not Gromov hyperbolic and write down some results about Gromov hyperbolicity of product spaces.
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Proper holomorphic mappings, Bells formula and the Lu Qi-Keng problem on tetrablock
We consider a proper holomorphic map form D to G domains in C^n and show that it induces a unitary isomorphism between the Bergman space A^2(G) and some subspace of A^2(D). Using this isomorphism we construct orthogonal projection onto that subspace and we derive Bells transformation formula for the Bergman kernel under proper holomorphic mappings. As a consequence of the formula we get that the tetrablock is not a Lu Qi-Keng domain.
math.CV↗