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2 results

  • Some estimates for the Bergman Kernel and Metric in Terms of Logarithmic Capacity

  • Zbigniew Błocki
  • Journal:
    Nagoya Mathematical Journal / Volume 185 / 2007
    Published online by Cambridge University Press:
    11 January 2016, pp. 143-150
    Print publication:
    2007
    • Article
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  • For a bounded domain Ω on the plane we show the inequality cΩ(z)2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.

  • ELECTROSTATICS, HYPERBOLIC GEOMETRY AND WANDERING VECTORS

  • ANATOLII GRINSHPAN
  • Journal:
    Journal of the London Mathematical Society / Volume 69 / Issue 1 / February 2004
    Published online by Cambridge University Press:
    28 January 2004, pp. 169-182
    Print publication:
    February 2004

  • A family of planar discrete electrostatic systems on the unit circle with finitely atomic external fields is considered. The geometry of particles in the external field yielding a given minimum energy configuration is studied. As an application, the wandering vectors of the shift operator in the Dirichlet spaces associated with finitely atomic measures are also studied. In particular, the zero locus of a wandering vector is discussed.