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

Abstract

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with C1 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 C1 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 C2 case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.