On strictly pseudoconvex domains with Kähler-Einstein Bergman metrics

Fu, Siqi; Wong, Bun

doi:10.4310/mrl.1997.v4.n5.a7

Cited by 40 publications

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“…Recall that the Bergman kernel of B n is given by K(z, w) = 1 (1 − z, w ) n+1 , z,w ∈ B n . We follow [3] and [8] A direct computation (see pages 22-23 of [22] for example) shows that…”

Section: Logarithmic Convexitymentioning

confidence: 99%

Volume integral means of holomorphic functions

Xiao¹,

Zhu

2011

Proc. Amer. Math. Soc.

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Abstract. The classical integral means of a holomorphic function f in the unit disk are defined by 1 2πThese integral means play an important role in modern complex analysis. In this note we consider integral means of holomorphic functions in the unit ball B n in C n with respect to weighted volume measures,where α is real, dv α (z) = (1 − |z| 2 ) α dv(z), and dv is volume measure on B n . We show that M p,α (f, r) increases with r strictly unless f is a constant, but in contrast with the classical case, log M p,α (f, r) is not always convex in log r. As an application, we show that

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“…Recall that the Bergman kernel of B n is given by K(z, w) = 1 (1 − z, w ) n+1 , z,w ∈ B n . We follow [3] and [8] A direct computation (see pages 22-23 of [22] for example) shows that…”

Section: Logarithmic Convexitymentioning

confidence: 99%

Volume integral means of holomorphic functions

Xiao¹,

Zhu

2011

Proc. Amer. Math. Soc.

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“…Cheng says that if the Bergman metric of a strictly pseudoconvex domain is Kähler-Einstein, then the domain is biholomorphic to the ball (cf. [23]). The deep works of Cheng-Yau [11] and MokYau [47] showed that on a bounded domain of holomorphy, there exists a unique biholomorphic invariant complete Kähler-Einstein metric with scalar curvature −1.…”

Section: Introductionmentioning

confidence: 99%

Bergman kernel and Kähler tensor calculus

2013

Pure and Applied Mathematics Quarterly

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Fefferman [22] initiated a program of expressing the asymptotic expansion of the boundary singularity of the Bergman kernel for strictly pseudoconvex domains explicitly in terms of boundary invariants. Hirachi, Komatsu and Nakazawa [30] carried out computations of the first few terms of Fefferman's asymptotic expansion building partly on Graham's work on CR invariants and Nakazawa's work on the asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains. In this paper, we prove a formula for coefficients in Nakazawa's asymptotic expansion as explicit summations over strongly connected graphs, and a formula expressing partial derivatives of Kähler metrics (resp. functions) as summations over rooted trees encoding covariant derivatives of curvature tensors (resp. functions). These formulae shall be useful in studying general patterns of Fefferman's asymptotic expansion.

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“…Fu and Wong [7] proved these results for simply connected domains using a weaker uniformization result of Chern and Ji [5] and stated the general case as an open question.…”

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confidence: 99%

“…It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.In this brief note we use our uniformization result from [10,11] to extend the work of Fu and Wong [7] on the relationship between two long-standing conjectures about the behaviour of the Bergman metric of a strictly pseudoconvex domain in C n , n ≥ 2. …”

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confidence: 99%

“…Suppose that the Bergman metric of a strictly pseudoconvex domain D is Kähler-Einstein. Fu and Wong [7] computed (rather ingeniously) that in this case the logarithmic coefficient ψ in the decomposition (1) Since D is the quotient of the ball by the group of holomorphic deck transformations, there is a natural complete metric of constant holomorphic sectional curvature on D obtained by taking the quotient of the standard invariant metric on the ball. According to Cheng and Yau [4], the complete Kähler-Einstein metric on D is unique up to a constant factor.…”

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Conjectures of Cheng and Ramadanov

Nemirovski¹,

Shafikov²

2006

Russ. Math. Surv.

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Abstract. It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.In this brief note we use our uniformization result from [10,11] to extend the work of Fu and Wong [7] on the relationship between two long-standing conjectures about the behaviour of the Bergman metric of a strictly pseudoconvex domain in C n , n ≥ 2.

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On strictly pseudoconvex domains with Kähler-Einstein Bergman metrics

Cited by 40 publications

References 0 publications

Volume integral means of holomorphic functions

Volume integral means of holomorphic functions

Bergman kernel and Kähler tensor calculus

Conjectures of Cheng and Ramadanov

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