The Bergman metric on a Stein manifold with a bounded plurisubharmonic function

Chen, Bo-Yong; Zhang, Jinhao

doi:10.1090/s0002-9947-02-02989-6

Cited by 28 publications

(25 citation statements)

References 20 publications

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“…A similar criterion holds true for the existence of Bergman metric. It is a consequence of Theorem 1, p. 2998, and Observation 2, p. 3002, in [7].…”

Section: Theorem 2 Let M Be a Complex Manifold Which Has A Bounded Comentioning

confidence: 75%

On the existence of Kobayashi and Bergman metrics for Model domains

Shcherbina

2020

Math. Ann.

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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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“…A similar criterion holds true for the existence of Bergman metric. It is a consequence of Theorem 1, p. 2998, and Observation 2, p. 3002, in [7].…”

Section: Theorem 2 Let M Be a Complex Manifold Which Has A Bounded Comentioning

confidence: 75%

On the existence of Kobayashi and Bergman metrics for Model domains

Shcherbina

2020

Math. Ann.

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“…-Proof of Theorem 1. The existence of the Bergman metric of a hyperconvex manifold was shown in [4]. Take a smooth function χ on R such that χ = 1 on (−∞, −1] and χ = 0 on [0, ∞).…”

Section: Proposition 4 (Cf[1])mentioning

confidence: 99%

“…Chen [3] and Herbort [8] independently), and this property was verified by Blocki-Pflug [2] (independently Herbort [8]). The case of hyperconvex Riemann surfaces was shown in [4].…”

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confidence: 99%

“…Hence In [7], Bergman completeness has been shown in the case when M is a simply-connected complete Kähler manifold such that the sectional curvature is suitably negatively pinched, for instance, pinched between negative constants. Their result was extended in [4] by only assuming that the curvature is bounded from above by −A/r 2 . This work was done during the author's stay at Nagoya University.…”

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confidence: 99%

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Bergman completeness of hyperconvex manifolds

Chen¹

2004

Nagoya Mathematical Journal

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“…Most of the results on Bergman completeness of domains is restricted to bounded ones. However, recently some papers appeared where the problem of Bergman completeness of unbounded domains was studied (see [Chen 2004], [Chen-Zhang 2002], [Juc 2004], [Chen-Kam-Ohs 2004], [Chen-Zhang 2004]). Motivated by examples from the last two papers we present a new class of unbounded domains admitting the Bergman metric, which are Bergman exhaustive (recall that the domain D is Bergman exhaustive if for any sequence (z ν ) ⊂ D without accumulation points in D the convergence lim ν→∞ K D (z ν ) = ∞ holds) and Bergman complete (see Theorem 6).…”

mentioning

confidence: 99%