On the Szegő metric (Q471989)

Let \(\Omega=\{\rho<0\} \subset \mathbb C^n\) be a strongly pseudoconvex bounded domain. In the present article the authors introduce a Szegő kernel \(S_{\Omega}\) with respect to the Fefferman area measure \(d\sigma_F\), which is defined by \[ d\sigma_F := c_n \root n+1 \of {-\det \left( \begin{matrix} 0& \rho_{\bar k}\\ \rho_j & \rho_{j\bar k} \end{matrix} \right)_{1,j,k\leq n} } \,\frac{d \sigma_E}{\|d \rho\|} \] Then they prove a transformation formula for the new Szegő kernel under suitable biholomorphic mappings. In a next step a Szegő differential metric \(F_S\) is defined as the Kähler metric with potential \(\log S_{\Omega} (z,z)\). Its biholomorphic invariance is established. Let \(K_\Omega\) denote the Bergman kernel and \(F_B\) the Bergman differential metric. In the last part, results on the asymtotic behavior at \(\partial \Omega\) are proved for the quantities \[ SK_\Omega (z,w):= \frac{S_\Omega (z,w)^{n+1}}{K_\Omega(z,w)^n}\quad\text{and}\quad E(z,\xi):= (n+1)\left(F_S (z,\xi)\right)^2 - n F_B (z,\xi) . \] Futhermore the metric \(F_S\) is compared with \(F_B\) and the Carathéodory differential metric \(F_C\), namely it is shown that \(F_S \geq F_C\), and that an estimate of the form \[ m_\Omega F_S \leq F_B \leq M_\Omega F_S\leqno{(1)} \] holds with constants \(m_\Omega, M_\Omega\) that depend only on \(\Omega\). Finally it is shown that one cannot choose the constants in (1) uniformly with respect to \(\Omega\). This is done by explicit computations for \(F_B\) and \(F_S\) in the annuli \(\{r<|z|<1\} \), where \(r\) ranges within \((0,1)\).