Let D be a smoothly bounded pseudoconvex domain in C^n, n > 1. Using G(z, p), the Green function for D with pole at p in D, associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kahler metric (the so-called Lambda–metric) using the Robin function arising from G(z, p). The purpose of this thesis is to study this metric by deriving its boundary asymptotics and using them to calculate the holomorphic sectional curvature along normal directions. It is also shown that the Lambda–metric is comparable to the Kobayashi (and hence to the Bergman and Caratheodory metrics) when D is strongly pseudoconvex. The unit ball in C^n is also characterized among all smoothly bounded strongly convex domains on which the Lambda–metric has constant negative holomorphic sectional curvature. This may be regarded as a version of Lu-Qi Keng’s theorem for the Bergman metric.

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Last updated: 04 Aug 2020