A remark on holomorphic isometries with respect to the induced bergman metrics

On strictly pseudoconvex domains with Kähler-Einstein Bergman metrics

Mathematical Research Letters ◽

10.4310/mrl.1997.v4.n5.a7 ◽

1997 ◽

Vol 4 (5) ◽

pp. 697-703 ◽

Cited By ~ 20

Author(s):

Siqi Fu ◽

Bun Wong

Keyword(s):

Pseudoconvex Domains ◽

Bergman Metrics

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Optimal convergence speed of Bergman metrics on symplectic manifolds

Journal of Symplectic Geometry ◽

10.4310/jsg.2020.v18.n4.a5 ◽

2020 ◽

Vol 18 (4) ◽

pp. 1091-1126

Author(s):

Wen Lu ◽

Xiaonan Ma ◽

George Marinescu

Keyword(s):

Symplectic Manifolds ◽

Convergence Speed ◽

Optimal Convergence ◽

Bergman Metrics

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On the existence of Kobayashi and Bergman metrics for Model domains

Mathematische Annalen ◽

10.1007/s00208-020-02074-6 ◽

2020 ◽

Author(s):

Nikolay Shcherbina

Keyword(s):

Entire Function ◽

Continuous Function ◽

Pseudoconvex Domain ◽

Plurisubharmonic Function ◽

Holomorphic Curves ◽

List Type ◽

Bergman Metric ◽

The Core ◽

Bounded Smooth ◽

Bergman Metrics

Abstract We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$ A = { ( z , w ) ∈ C 2 : v > F ( z , u ) } , where $$w = u + iv$$ w = u + i v and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$ C z × R u , the following conditions are equivalent: The domain $$\mathfrak {A}$$ A is Kobayashi hyperbolic. The domain $$\mathfrak {A}$$ A is Brody hyperbolic. The domain $$\mathfrak {A}$$ A possesses a Bergman metric. The domain $$\mathfrak {A}$$ A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$ c ( A ) of $$\mathfrak {A}$$ A is empty. The graph $$\Gamma (F)$$ Γ ( 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 $$\Gamma ({\mathcal H})$$ Γ ( H ) of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$ H : C z → C w .

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