Geometry of reinhardt domains of finite type in C²

The asymptotic behavior of the holomorphic sectional curvature of the Bergman metric on a pseudoconvex Reinhardt domain of finite type in C² is obtained by rescaling locally the domain to a model domain that is either a Thullen domain Ω_m = {(z₁, Z₂); [Z₁]² + [Z₂]^2m < 1} or a tube domain T_m = {(z₁, Z₂); Imz₁ + (Imz₂)^2m < 1}. The Bergman metric for the tube domain T_m is explicitly calculated by using Fourier-Laplace transformation. It turns out that the holomorphic sectional curvature of the Bergman metric on the tube domain T_m at (0, 0) is bounded above by a negative constant. These results are used to construct a complete Kahler metric with holomorphic sectional curvature bounded above by a negative constant for a pseudoconvex Reinhardt domain of finite type in C².