# The Kobayashi-Royden metric on punctured spheres

## TL;DR

This paper derives an explicit asymptotic expansion of the Kobayashi-Royden metric on punctured spheres, revealing rational coefficients and applying the results to specific cases, with implications for the Little Picard theorem.

## Contribution

It provides the first explicit formula for the asymptotic expansion of the Kobayashi-Royden metric on punctured spheres, including rational coefficients and applications to generalized cases.

## Key findings

- Explicit asymptotic expansion in terms of exponential Bell polynomials.
- Coefficients in the expansion are rational numbers.
- Application to a specific punctured sphere example.

## Abstract

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi-Royden metric on the punctured sphere $\mathbb{CP}^1\backslash\{0,1,\infty\}$ 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. Meanwhile, the approach in the paper leads to the conclusion that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit metric formula and the conclusion regarding the coefficients apply to a more general case as well, the metric on $\mathbb{CP}^1\backslash\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}$ will be given as a concrete example of our results.