The Azukawa Metric and the Pluricomplex Green Function.

Abstract

This thesis grew out of a study of the Azukawa metric, which is defined in terms of the pluricomplex Green function. The two main questions we study are upper semicontinuity of the Azukawa metric and plurisubharmonicity of a generalized Green function. We also construct a number of examples: a domain where the Azukawa metric and the closely related Sibony metric are different; a domain where the Azukawa and Sibony metrics have different pointwise hyperbolicity behavior; a domain where the Green function can not be extended continously to the boundary; and a class of domains where we find an explict formula for the Auzkawa metric. Finally, in a joint theorem with Lina Lee we show that on a Skwarczynski complete domain, the Bergman space is infinite dimensional.