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The Kobayashi-Royden metric on punctured spheres. (English) Zbl 1442.32020
Summary: This paper gives an explicit formula of the asymptotic expansion of the Kobayashi-Royden metric on the punctured sphere \(\mathbb{CP}^1 \setminus \{0, 1, \infty\}\) in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of \(\mathbb{CP}^1 \setminus \{a_1, \ldots, a_n\}\), \(n \geq 3\), as well, and the metric on \(\mathbb{CP}^1 \setminus \{0, \frac{1}{3}, -\frac{1}{6} \pm \frac{\sqrt{3}}{6}i\}\) will be given as a concrete example of our results.

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32C15 Complex spaces
30E15 Asymptotic representations in the complex plane
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