Comparison of invariant metrics (Q2453637)

Let \(D \subset \mathbb C^n\) be a domain. By \(F_B^D\) and \(F_K^D\) we denote the pseudodifferential metrics of Bergman and Kobayashi, respectively. The authors estimate precisely the quantity \(F_K^{{\mathbb C}\setminus \{0,1\}}\) and compare the Bergman differential metrics of the unit ball \(B_n\) in \( {\mathbb C}^n\) and the ring domain \(\Omega_r=\{ z \in {\mathbb C}^n \,|\, r<|z|<1\}\), for \(r \in (0,1)\). Here are the results: Theorem 1. Let \(p \in \mathbb C\setminus \{0,1\}\) and \(\delta= \text{dist}\,(p,0)\) and \(\xi =1\). Then we have for small enough \(\delta\) \[ F_K^{{\mathbb C}\setminus \{0,1\}}(p,\xi) \approx \frac{1}{\delta \,\log (1/\delta)} \,. \] Theorem 2. Let \(p \in \Omega_r\) and \(\xi \in T_p \Omega_r\) be a tangent vector such that \(p \cdot \overline{\xi} =0\). Then \[ F_B^{\Omega_r} (p, \xi) \lneq F_B^{B_n} (p, \xi) \] for all \(p \in \Omega_r\). The restriction \(p \cdot \overline{\xi} =0\) can be let away in dimension two, if \(p\) lies on the normal to a point on the inner boundary of \(\Omega_r\). Theorem 3. If \(n=2\) and \(p=(r+\varepsilon , 0)\) for small \(\varepsilon>0\), then we have for small enough \(r\) and for arbitrary \(\xi \in {\mathbb C}^2\) that \[ F_B^{\Omega_r} (p, \xi) \lneq F_B^{B_2} (p, \xi) \]