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Kobayashi, Carathéodory and Sibony metric. (English) Zbl 1166.32010
The order of growth for the Sibony differential metric \(F_{\Omega}^S\) on a \(C^2\)-smooth bounded domain \(\Omega\) in \(\mathbb C^n \) is determined near a pseudoconcave boundary point and compared with that of the differential metrics of Caratheodory and Kobayashi, which will be denoted by \(F_{\Omega}^C\) and \(F_{\Omega}^K\), respectively. Let \(A_{\Omega}(P)\) denote for a point \(P \in \Omega\) the set of all plurisubharmonic functions \(u:\Omega \to [0,1]\), such that \(u(P)=0\), and \(\log u\) is also plurisubharmonic. Then the metric \(F_{\Omega}^S\) is given by
\[ F_{\Omega}^S (P;X) := \sup \{ \sqrt{(\partial \bar \partial u (P) X,X)} \,|\, u \in A_{\Omega}(P)\} ,\qquad P \in \Omega,\,\,X \in \mathbb C^n. \]
The result is as follows:
Theorem: Let \(P \in \partial \Omega\) be a non-pseudoconvex point and \(P_\delta\) the point on the inner normal to \(\partial \Omega\) at \(P\) with boundary distance \(\delta\) and \(\nu\) the inner unit normal at \(P\). Then
\[ F_{\Omega}^S(P_\delta, \nu ) \approx \frac{1}{\sqrt{\delta}},\qquad F_{\Omega}^C(P_\delta, \nu ) \approx 1. \]
Together with the estimate
\[ F_{\Omega}^K(P_\delta, \nu ) \approx \frac{1}{\delta^{3/4}} \]
which is due to S. Krantz, it follows that the orders of growth for these differential metrics are pairwise different in this situation.
The main lemma in the proof of the above theorem is the following estimate on \(F_{\Omega}^S\) for the ring domain
\[ \Omega_m :=\{ \frac{1}{4} < |z|^2+|w|^m <3\},\qquad m \geq 2 \] at the point \(p_\delta := ( \frac{1}{2}+\delta , 0)\), namely
\[ F_{\Omega_m}^S (p_\delta , \left(\begin{matrix} 1 \\ 0 \end{matrix} \right) \,) \approx \frac{1}{\delta^{1-\frac{1}{m}}}. \]
This result is generalized to the higher dimensional ring domains \[ \Omega' :=\{ \frac{1}{4} < |z_1|^2+|z_2|^{m_2}+\dots+ |z_n|^{m_n} <1\},\qquad 2 \leq m_2\leq \cdots\leq m_n \] which serve as suitable local domains of comparison at pseudoconcave boundary points.

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
32U05 Plurisubharmonic functions and generalizations
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References:
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