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Семинар лаборатории теории функций "Современные проблемы комплексного анализа"
20 февраля 2020 г. 12:00–13:00, г. Ташкент, Национальный университет Узбекистана, Математический факультет, аудитория А-304 (ул. Университетская, 4)
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On Kobayashi and Bergman metrics for Model domains
N. V. Shcherbina University of Wuppertal
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Эта страница: | 104 |
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Аннотация:
We prove that for a pseudoconvex domain of the form $\mathfrak{A} = \{(z, w) \in \mathbb C^2 : v > F(z, u)\}$, where $w = u + iv$ and F is a continuous function on ${\mathbb C}_z \times {\mathbb R}_u$, the following conditions are equivalent: - The domain $\mathfrak{A}$ is Kobayashi hyperbolic.
- The domain $\mathfrak{A}$ is Brody hyperbolic.
- The domain $\mathfrak{A}$ possesses a Bergman metric.
- The domain $\mathfrak{A}$ possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $\mathfrak{c}(\mathfrak{A})$ of $\mathfrak{A}$ is empty.
- The graph $\Gamma(F)$ of $F$ can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $\Gamma({\mathcal H})$ of just one entire function ${\mathcal H} : {\mathbb C}_z \to {\mathbb C}_w$.
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