We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in ${ℂ}^n,n\ge 2$, is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic metric over a Stein space with a spherical boundary.

中文翻译：

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Bergman–Einstein度量，具有球面边界的Kerner定理和Stein空间的推广

对于1979年提出的Cheng的猜想，我们给出肯定的解决方案，该猜想断言一个光滑有界强伪凸域的Bergman度量在 ${ℂ}^{ñ}，ñ\ge 2$，并且仅当该域对球是全同形时，才是Kähler–Einstein。我们建立了带有孤立奇异性的Stein空间的经典Kerner定理的一个版本，该定理具有立即应用，可以在具有球形边界的Stein空间上构造双曲度量。