Abstract
We give first of all a new criterion for Bergman completeness in terms of the pluricomplex Green function. Among several applications, we prove in particular that every Stein subvariety in a complex manifold admits a Bergman complete Stein neighborhood basis, which improves a theorem of Siu. Secondly, we give for hyperbolic Riemann surfaces a sufficient condition for when the Bergman and Poincaré metrics are quasi-isometric. A consequence is an equivalent characterization of uniformly perfect planar domains in terms of growth rates of the Bergman kernel and metric. Finally, we provide a noncompact Bergman complete pseudoconvex manifold without nonconstant negative plurisubharmonic functions.
Funding Statement
Supported by Chinese NSF grant No. 11031008 and Fok Ying Tung Education Foundation grant No. 111004. Partially supported by Chinese NSF grant No. 11171255.
Citation
Bo-Yong Chen. "An essay on Bergman completeness." Ark. Mat. 51 (2) 269 - 291, October 2013. https://doi.org/10.1007/s11512-012-0174-8
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