# zbMATH — the first resource for mathematics

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
##### Keywords:
Bergman metric; Monge-Ampere equation
Full Text: