Estimation on invariant distances on pseudoconvex domains of finite type in dimension two. (English) Zbl 1081.32007
Let \(D=\{r<0\}\subset\subset\mathbb C^2\) be a smooth pseudoconvex domain such that all boundary points are of finite type. The author studies effective bounds for the inner Carathéodory \(d_D^{\text{Cara,int}}\), Kobayashi \(d_D^{\text{Kob}}\), and Bergman \(d_D^{\text{Berg}}\) distances. He proves that there exists a constant \(C_\ast>0\) such that \(C_\ast\varphi\leq d_D^{\text{Cara,int}}\leq d_D^{\text{Kob}}\leq (1/C_\ast)\varphi\) and \(C_\ast\varphi\leq d_D^{\text{Berg}}\leq(1/C_\ast)\varphi\), where the function \(\varphi\) is given effectively in terms of the defining function \(r\).
Reviewer: Marek Jarnicki (Kraków)
MSC:
| 32F45 | Invariant metrics and pseudodistances in several complex variables |
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