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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hyperbolic metric, punctured Riemann sphere, and modular functions


Author: Junqing Qian
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
Published electronically: October 5, 2020
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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.


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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
Email: jqian20@unm.edu

DOI: https://doi.org/10.1090/tran/8175
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.
Article copyright: © Copyright 2020 American Mathematical Society