This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere 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 , , as well, and the metric on will be given as a concrete example of our results.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1611745
Funding statement: This work was partially supported by NSF grant DMS-1611745.
We would like to thank our advisor Professor Damin Wu for helpful discussions and the encouragement of making the collaboration. We would also like to thank Professor Keith Conrad for useful comments.
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