# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] E. T. Bell, Exponential polynomials, Ann. of Math. (2) 35 (1934), no. 2, 258-277. · Zbl 0009.21202 [2] J. Carlson and P. Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. (2) 95 (1972), 557-584. · Zbl 0248.32018 [3] G. Cho, Invariant metrics on the complex ellipsoid, J. Geom. Anal. (2019), 10.1007/s12220-019-00333-w. [4] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, enlarged ed., D. Reidel Publishing, Dordrecht, 1974. [5] J.-P. Demailly, L. Lempert and B. Shiffman, Algebraic approximations of holomorphic maps from Stein domains to projective manifolds, Duke Math. J. 76 (1994), no. 2, 333-363. · Zbl 0861.32006 [6] R. E. Greene, K.-T. Kim and S. G. Krantz, The Geometry of Complex Domains, Progr. Math. 291, Birkhäuser, Boston, 2011. [7] P. A. Griffiths, Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. of Math. (2) 94 (1971), 21-51. · Zbl 0221.14008 [8] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, extended ed., De Gruyter Exp. Math. 9, Walter de Gruyter, Berlin, 2013. · Zbl 1273.32002 [9] H. Kang, L. Lee and C. Zeager, Comparison of invariant metrics, Rocky Mountain J. Math. 44 (2014), no. 1, 157-177. · Zbl 1293.32014 [10] J. Qian, Hyperbolic metric, punctured Riemann sphere and modular funtions, preprint (2019), https://arxiv.org/abs/1901.06761. [11] A. Sebbar, Modular subgroups, forms, curves and surfaces, Canad. Math. Bull. 45 (2002), no. 2, 294-308. · Zbl 1010.20035 [12] D. Wu and S.-T. Yau, Invariant metrics on negatively pinched complete Kähler manifolds, J. Amer. Math. Soc. (2019), 10.1090/jams/933. · Zbl 07171893 [13] H. Wu, Old and new invariant metrics on complex manifolds, Several Complex Variables (Stockholm 1987/1988), Math. Notes 38, Princeton University Press, Princeton (1993), 640-682. · Zbl 0773.32017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.