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Comparison of invariant metrics. (English) Zbl 1293.32014
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 $F_B^{\Omega_r} (p, \xi) \lneq F_B^{B_2} (p, \xi)$

##### MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables
##### Keywords:
Bergman metric; Kobayashi metric; modular function
Full Text:
##### References:
 [1] L. Ahlfors, Complex analysis : An introduction to the theory of analytic functions of one complex variable , Third edition, McGraw-Hill Book Company, New York, 1979. · Zbl 0395.30001 [2] K. Diederich and J.E. Fornaess, Comparison of the Bergman and the Kobayashi metric , Math. Ann. 254 (1980), 257-262. · Zbl 0429.32031 · doi:10.1007/BF01457999 · eudml:163491 [3] K. Diederich, J.E. Fornaess and G. Herbort, Boundary behavior of the Bergman metric , Proc. Symp. Pure Math. 41 (1984), 59-67. · Zbl 0533.32012 · doi:10.1090/pspum/041/740872 [4] J.E. Fornaess and L. Lee, Kobayashi, Carathéodory, and Sibony metrics , Complex Variable Elliptic Eq. 54 (2009), 293-301. · Zbl 1166.32010 · doi:10.1080/17476930902760450 [5] K.T. Hahn, Inequality between the Bergman metric and Carathéodory differential metric , Proc. Amer. Math. Soc. 68 (1978), 193-194. · Zbl 0376.32020 · doi:10.2307/2041770 [6] M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis , Walter de Gruyter, Boston, 1993. · Zbl 0789.32001 [7] S. Kobayashi, Hyperbolic complex spaces , Springer, New York, 1998. · Zbl 0917.32019 [8] S.G. Krantz, The boundary behavior of the Kobayashi metric , Rocky Mountain J. Math. 22 (1992), 227-233. \noindentstyle · Zbl 0760.32010 · doi:10.1216/rmjm/1181072807
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