On completeness of the Bergman metric and its subordinate metric.
It is proved that on any bounded domain in the complex Euclidean space C(n) the Bergman metric is always greater than or equal to the Carathéodory distance. This leads to a number of interesting consequences. Here two such consequences are given. (i) The Bergman metric is complete whenever the Carathéodory distance is complete on a bounded domain. (ii) The Weil-Petersson metric is not uniformly equivalent to the Bergman metric in the Teichmüller space T(g) of any Riemann surface of genus g >/= 2.
Hahn
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Department of Mathematics, The Pennsylvania State University, University Park, Pa. 16802.