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The boundary behavior of the Kobayashi metric. (English) Zbl 0760.32010
Let \(3/4\leq\lambda<1\). Set \(m=1/(2-2\lambda)\), and define \[ U=\{(w,z)\in\mathbb{C}^ 2: 1<| w|^ 2+| z|^ m<4\}. \] The author uses the euclidean distance from the boundary of \(U\) to estimate the Kobayashi-Royden metric \(F^ U\) in the direction \(\xi=(1,0)\), as follows. There exist positive constants \(C_ 1\) and \(C_ 2\) such that \[ C_ 1\text{dist}(P,\partial U)^{-\lambda}\leq F^ U(P,\xi)\leq C_ 2\text{dist}(P,\partial U)^{-\lambda} \] for all \(P=(- 1-\delta,0)\) with \(\delta>0\) sufficiently small.

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
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