An essay on Bergman completeness

Bo-Yong Chen Department of Mathematics, Tongji University

TBD mathscidoc:1701.333179

Arkiv for Matematik, 51, (2), 269-291, 2011.4
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.
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  title={An essay on Bergman completeness},
  author={Bo-Yong Chen},
  booktitle={Arkiv for Matematik},
Bo-Yong Chen. An essay on Bergman completeness. 2011. Vol. 51. In Arkiv for Matematik. pp.269-291.
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