Abstract This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere ℂ ⁢ ℙ 1 ∖ { 0 , 1 , ∞ } {\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 ℂ ⁢ ℙ 1 ∖ { a 1 , … , a n } {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}} , n ≥ 3 {n\geq 3} , as well, and the metric on ℂ ⁢ ℙ 1 ∖ { 0 , 1 3 , - 1 6 ± 3 6 ⁢ i } {\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.

中文翻译：

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穿孔球体的 Kobayashi-Royden 度量

摘要 本文给出了小林-罗伊登度量在穿孔球体 ℂ ⁢ ℙ 1 ∖ { 0 , 1 , ∞ } {\mathbb{CP}^{1}\setminus\{0,1 ,\infty\}} 的指数贝尔多项式。我们证明了小皮卡德定理的局部定量版本作为渐近展开式的应用。此外，论文中的方法导致了一个有趣的结果，即渐近展开中的系数是有理数。此外，度量的显式公式和关于系数的结论适用于更一般的情况 ℂ ⁢ ℙ 1 ∖ { a 1 , … , an } {\mathbb{CP}^{1}\setminus\{a_{1 },\ldots,a_{n}\}} , n ≥ 3 {n\geq 3} ，以及 ℂ ⁢ ℙ 1 ∖ { 0 , 1 3 , - 1 6 ± 3 6 ⁢ i } {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},