Growth of Sibony metric and Bergman kernel for domains with low regularity

Nikolai Nikolov; Pascal J. Thomas

doi:10.1016/j.jmaa.2021.125018

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arXiv · 2005.04479

Growth of Sibony metric and Bergman kernel for domains with low regularity

Nikolai Nikolov·Pascal J. Thomas

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Abstract

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $\mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general $\mathcal C^1$ boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the Blocki-Zwonek Suita-type theorem implies growth to infinity of the Bergman kernel; the fact that the Bergman kernel grows as the square of the reciprocal of the distance to the boundary, proved by S. Fu in the $\mathcal C^2$ case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.

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BibTeX RIS

Nikolai Nikolov, Pascal J. Thomas. 2020-05-09. Growth of Sibony metric and Bergman kernel for domains with low regularity. https://doi.org/10.1016/j.jmaa.2021.125018

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