Smoothly bounded domains covering finite volume manifolds

2018/02/04 by Zimmer, Andrew
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.01178

Abstract

In this paper we prove: if a bounded domain with C² boundary covers a manifold which has finite volume with respect to either the Bergman volume, the Kähler-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,ε regularity.

Related

• Smoothly bounded domains covering compact manifolds
• Geometry of complex bounded domains with finite-volume quotients
• Kähler-Einstein Bergman metrics on pseudoconvex domains of dimension two
• A characterization of the ball
• Bergman-Einstein metrics, hyperbolic metrics and Stein spaces with spherical boundaries