In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the K\"ahler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,\epsilon}$ regularity.

中文翻译：

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覆盖有限体积流形的光滑有界域

在本文中，我们证明：如果一个具有 $C^2$ 边界的有界域覆盖了一个相对于 Bergman 体积、K\"ahler-Einstein 体积或 Kobayashi-Eisenman 体积具有有限体积的流形，那么域对于单位球是双全纯的。这回答了 Yau 的一个老问题。此外，当域是凸的时，我们可以假设边界只有 $C^{1,\epsilon}$ 正则性。