Arkiv för Matematik

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

An essay on Bergman completeness

Bo-Yong Chen

Full-text: Open access

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.

Note

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

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907219

Digital Object Identifier
doi:10.1007/s11512-012-0174-8

Mathematical Reviews number (MathSciNet)
MR3090197

Zentralblatt MATH identifier
1276.32011

Rights
2012 © Institut Mittag-Leffler

Citation

Chen, Bo-Yong. An essay on Bergman completeness. Ark. Mat. 51 (2013), no. 2, 269--291. doi:10.1007/s11512-012-0174-8. https://projecteuclid.org/euclid.afm/1485907219


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References

  • Andreotti, A. and Vesentini, E., Carleman estimates for the Laplace–Beltrami equation in complex manifolds, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 81–130.
  • Beardon, A. F. and Pommerenke, C., The Poincaré metric of plane domains, J. Lond. Math. Soc. 18 (1978), 475–483.
  • Błocki, Z., The Bergman metric and the pluricomplex Green function, Trans. Amer. Math. Soc. 357 (2004), 2613–2625.
  • Błocki, Z. and Pflug, P., Hyperconvexity and Bergman completeness, Nagoya Math. J. 151 (1998), 221–225.
  • Buser, P., A note on the isoperimetric constant, Ann. Sci. Éc. Norm. Supér. 15 (1982), 213–230.
  • Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis (Princeton, NJ, 1969 ), pp. 195–199, Princeton University Press, Princeton, NJ, 1970.
  • Chen, B. Y., Completeness of the Bergman metric on non-smooth pseudoconvex domains, Ann. Polon. Math. 71 (1999), 241–251.
  • Chen, B. Y., Bergman completeness of hyperconvex manifolds, Nagoya Math. J. 175 (2004), 165–170.
  • Chen, B. Y. and Zhang, J. H., The Bergman metric on a Stein manifold with a bounded plurisubharmonic function, Trans. Amer. Math. Soc. 354 (2002), 2997–3009.
  • Demailly, J. P., Estimations L2 pour l’opérateur d’un fibré vectoriel holomorphe semi-positiv au dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér. 15 (1982), 457–511.
  • Demailly, J. P., Complex Analytic and Differential Geometry,
  • Diederich, K. and Ohsawa, T., An estimate for the Bergman distance on pseudoconvex domains, Ann. of Math. 141 (1995), 181–190.
  • Fernández, J. L., Domains with strong barrier, Rev. Mat. Iberoam. 5 (1989), 47–65.
  • Griffiths, P. and Harris, J., Principle of Algebraic Geometry, Wiley, New York, 1978.
  • Grigor’yan, A., Analytic and geometric background for recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135–249.
  • Harvey, F. R. and Wells, R. O., Holomorphic approximation and hyperfunction theory on a C1 totally real submanifold of a complex manifold, Math. Ann. 197 (1972), 287–318.
  • Herbort, G., The Bergman metric on hyperconvex domains, Math. Z. 232 (1999), 183–196.
  • Herbort, G., The pluricomplex Green function on pseudoconvex domains with a smooth boundary, Internat. J. Math. 11 (2000), 509–522.
  • Hörmander, L., An Introduction to Complex Analysis in Several Variables, 3rd ed., Elsevier, Amsterdam, 1990.
  • Jarnicki, M. and Pflug, P., Invariant Distances and Metrics in Complex Analysis, de Gruyter Expositions in Math. 9, de Gruyter, Berlin, 1993.
  • Klimek, M., Pluripotential Theory, Clarendon Press, Oxford, 1991.
  • Kobayashi, S., Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267–290.
  • Lyons, T. and Sullivan, D., Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), 299–323.
  • Pflug, P. and Zwonek, W., Logarithmic capacity and Bergman functions, Arch. Math. (Basel ) 80 (2003), 536–552.
  • Pflug, P. and Zwonek, W., Bergman completeness of unbounded Hartogs domains, Nagoya Math. J. 180 (2005), 121–133.
  • Pommerenke, C., Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel ) 32 (1979), 192–199.
  • Richberg, R., Stetige streng pseudokonvexe funktionen, Math. Ann. 175 (1968), 257–286.
  • Siu, Y. T., Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976), 89–100.
  • Stein, K., Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math. (Basel ) 7 (1956), 354–361.
  • Sugawa, T., Uniformly perfect sets: analytic and geometric aspects, Sūgaku 53 (2001), 387–402 (Japanese). English transl.: Sugaku Expositions 16 (2003), 225–242.
  • Sullivan, D., Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), 327–351.
  • Tsuji, M., Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
  • Varopoulos, N. T., A potential theoretic property of soluble groups, Bull. Sci. Math. 108 (1983), 263–273.
  • Wang, X., Bergman completeness is not a quasi-conformal invariant, to appear in Proc. Amer. Math. Soc. doi:
  • Zwonek, W., Wiener’s type criterion for Bergman exhaustiveness, Bull. Pol. Acad. Sci. Math. 50 (2002), 297–311.