On the Bergman invariant and curvatures of the Bergman metric (Q1922056)

Let \(A^2(\Omega)\) be the Bergman space on a bounded domain \(\Omega \subseteq \mathbb{C}^n\). If \(\{\varphi_j\}\) is a complete orthonormal system for \(A^2\) then set \[ K(z, \zeta) = \sum^\infty_{j = 1} \varphi_j (z) \overline {\varphi_j (\zeta)}. \] This is the Bergman kernel. We set \(K_\Omega (z) = K(z,z)\). Now define the (infinitesimal) Bergman metric by \[ g_{\alpha \overline \beta} (z) = {\partial^2 \log K_\Omega (z) \over \partial z_\alpha \partial \overline {z_\beta}}. \] We let \(G_\Omega (z)\) denote the matrix \((g_{\alpha \overline \beta})\). The Bergman canonical invariant is defined to be \(J_\Omega (z) = {\text{det} G_\Omega (z) \over K_\Omega (z)}\). The Ricci curvature tensor \(R_\Omega = \sum R_{\alpha \overline \beta} dz_\alpha d \overline {z_\beta}\) is defined by \[ R_{\alpha \overline \beta} = - {\partial^2 \log \text{det} G_\Omega \over \partial z_\alpha \partial \overline {z_\beta}}. \] Both \(J_\Omega\) and \( R_\Omega\) are invariant under biholomorphic mappings. The two invariants are closely related. The authors study the boundary behavior of these invariants on some natural classes of weakly pseudoconvex domains. The method used is to study certain minimal integrals.