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Conjectures of Cheng and Ramadanov. (English. Russian original) Zbl 1120.32006
Russ. Math. Surv. 61, No. 4, 780-782 (2006); translation from Usp. Mat. Nauk 61, No. 4, 193-194 (2006).
Let \(D\) denote a bounded strongly pseudoconvex domain in \(\mathbb{C}^n\) with a \(C^\infty\)-smooth boundary. For the Bergman kernel \(K_D\) of \(D\) the Fefferman asymptotic formula is as follows: \[ K_D(z,z) = \varphi (z) \,(\varrho (z)\,)^{-n-1} + \psi (z) \log | \varrho (z)| {(F)} \] (where \(\varphi, \psi \in C^\infty (\overline{D})\), the function \(\varphi \) has no zeros on \(\partial D\), and \(\varrho\) is a defining function for \(D\)).
In this short note the following two conjectures are discussed:
Cheng’s conjecture: \(D\) is biholomorphic to the ball if and only if its Bergman metric is Kähler-Einstein.
Ramadanov’s conjecture: If the \(\psi\)-term in Fefferman’s formula (F) vanishes to infinite order at \(\partial D\), then \(\partial D\) must be locally CR-diffeomorphic to the unit sphere.
The Ramadanov conjecture is known to be true in the case \(n=2\).
The authors prove, that, for arbitrary \(n\), Cheng’s conjecture follows from Ramadanov’s conjecture.
As a corollary they note, that in case \(n=2\) the domain \(D\) is biholomorphic to the ball if and only if the Bergman metric of \(D\) is Kähler-Einstein.
The authors finally point out that in Burn-Shnider’s example \[ D_{BS}:=\{ (z,w) \,\in \mathbb{C}^2\,\,| \,\, \sin (\log | z| ) + | w| ^2 <0\} \] the \(\psi\)-term from (F) vanishes to infinite order at the boundary, but nevertheless the Bergman metric of \(D_{BS}\) is not Kähler-Einstein.

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
32Q20 Kähler-Einstein manifolds
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