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

Published online by Cambridge University Press:  11 January 2016

Zbigniew Błocki
Zbigniew Błocki*
Affiliation:
Jagiellonian University, Institute of Mathematics, Reymonta 4 30-059 Kraków, Poland, Zbigniew.Blocki@im.uj.edu.pl
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Abstract

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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.

Keywords

30C4031A35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 185 , 2007 , pp. 143 - 150
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

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