2010/01/28 by Diganta Borah, Borah, Diganta, Kaushal Verma +1
Mathematics · #31B25 #31C10 #32F45 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:31B25 #msc:31C10 #msc:32F45
paper · pdf · doi:10.48550/arxiv.1001.5101
31 pages
arxiv created 2010/01/28 · openalex publication_date 2010/01/28 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let D be a smoothly bounded pseudoconvex domain in \mathbf Cn, n > 1. Using G(z, p), the Green function for D with pole at p ∈ D associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kähler metric (the so-called \La-metric) using the Robin function \La(p) arising from G(z, p). The purpose of this article 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 \La-metric is comparable to the Kobayashi (and hence to the Bergman and Carathéodory metrics) when D is strongly pseudoconvex. The unit ball in \mathbf Cn is also characterized among all smoothly bounded strongly convex domains on which the \La-metric has constant negative holomorphic sectional curvature. This may be regarded as a version of Lu-Qi Keng's theorem for the Bergman metric.