## Hyperbolic metric, punctured Riemann sphere, and modular functions

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## Abstract:

We derive a precise asymptotic expansion of the complete Kähler-Einstein metric on the punctured Riemann sphere with three or more omitting points. By using the Schwarzian derivative, we prove that the coefficients of the expansion are polynomials on the two parameters which are uniquely determined by the omitting points. Furthermore, we use the modular form and the Schwarzian derivative to explicitly determine the coefficients in the expansion of the complete Kähler-Einstein metric for the punctured Riemann sphere with $3, 4, 6$, or $12$ omitting points.## References

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## Additional Information

**Junqing Qian**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connectitut 06268
- Address at time of publication: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87106
- ORCID: 0000-0002-4326-8504
- Email: jqian20@unm.edu
- Received by editor(s): January 31, 2019
- Received by editor(s) in revised form: April 15, 2020
- Published electronically: October 5, 2020
- Additional Notes: This work was supported by NSF grant DMS-1611745.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 8751-8784 - MSC (2010): Primary 11F03, 30F35, 32Q20, 53C55, 54E50
- DOI: https://doi.org/10.1090/tran/8175
- MathSciNet review: 4177275