Arkiv för Matematik

  • Ark. Mat.
  • Volume 51, Number 2 (2013), 269-291.

An essay on Bergman completeness

Bo-Yong Chen

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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.


Supported by Chinese NSF grant No. 11031008 and Fok Ying Tung Education Foundation grant No. 111004. Partially supported by Chinese NSF grant No. 11171255.

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Ark. Mat., Volume 51, Number 2 (2013), 269-291.

Received: 26 April 2011
First available in Project Euclid: 1 February 2017

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2012 © Institut Mittag-Leffler


Chen, Bo-Yong. An essay on Bergman completeness. Ark. Mat. 51 (2013), no. 2, 269--291. doi:10.1007/s11512-012-0174-8.

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