Conjectures of Cheng and Ramadanov

Nemirovski, Stefan; Shafikov, Ruslan G

doi:10.1070/rm2006v061n04abeh004349

Cited by 6 publications

(2 citation statements)

References 11 publications

(11 reference statements)

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“…The aforementioned Cheng Conjecture was confirmed by S. Fu-B. Wong [15] and S. Nemirovski-R. Shafikov [27] in the two dimensional case and by X. Huang and the second author [19] in higher dimensions. X. Huang and X. Li [17] recently generalized this result to Stein manifolds with strongly pseudoconvex boundary as follows: The only Stein manifold with smooth and compact strongly pseudoconvex boundary for which the Bergman metric is Kähler-Einstein is the unit ball B n (up to biholomorphism).…”

Section: Introductionmentioning

confidence: 76%

On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

Ebenfelt¹,

Xiao²,

Xu³

2020

Preprint

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In this paper, we study the Bergman metric of a finite ball quotient B n /Γ, where Γ ⊆ Aut(B n ) is a finite, fixed point free, abelian group. We prove that this metric is Kähler-Einstein if and only if Γ is trivial, i.e., when the ball quotient B n /Γ is the unit ball B n itself. As a consequence, we establish a characterization of the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman-Einstein metric.

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Section: Introductionmentioning

confidence: 76%

On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

Ebenfelt¹,

Xiao²,

Xu³

2020

Preprint

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“…A classical and important problem is to study the geometry of the domain and its boundary in terms of properties of the Bergman kernel and metric. For example, as a consequence of a series of papers [9,19,16], the unit ball may be characterized among bounded strongly pseudoconvex domains Ω in C n in terms of the Kähler-Einstein property of its Bergman metric. This is the well-known Cheng Conjecture.…”

Section: Introductionmentioning

confidence: 99%

Algebraic Bergman kernels and finite type domains in $\mathbb{C}^2$

Ebenfelt¹,

Xiao²,

Xu³

2021

Preprint

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Let G ⊂ C 2 be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of G is algebraic of degree d. We show that the boundary ∂G is of finite type and the type r satisfies r ≤ 2d. The inequality is optimal as equality holds for the egg domains {|z| 2 + |w| 2s < 1}, s ∈ Z+, by D'Angelo's explicit formula for their Bergman kernels. Our results imply, in particular, that a smoothly bounded pseudoconvex domain G ⊂ C 2 cannot have rational Bergman kernel unless it is strongly pseudoconvex and biholomorphic to the unit ball by a rational map. Furthermore, we show that if the Bergman kernel of G is rational of the form p q , reduced to lowest degrees, then its rational degree max{deg p, deg q} ≥ 6. Equality is achieved if and only if G is biholomorphic to the unit ball by a complex affine transformation of C 2 .

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Cauchy–Riemann meet Monge–Ampère

Błocki

2014

Bull. Math. Sci.

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Conjectures of Cheng and Ramadanov

Cited by 6 publications

References 11 publications

On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

Algebraic Bergman kernels and finite type domains in $\mathbb{C}^2$

Cauchy–Riemann meet Monge–Ampère

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