Hyperbolic metric, punctured Riemann sphere, and modular functions
HTML articles powered by AMS MathViewer
- by Junqing Qian PDF
- Trans. Amer. Math. Soc. 373 (2020), 8751-8784 Request permission
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
- Stephen Agard, Distortion theorems for quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 413 (1968), 12. MR 0222288
- Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
- Lars V. Ahlfors, Quasiconformal mappings, Teichmüller spaces, and Kleinian groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 71–84. MR 562598
- Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300. MR 348097, DOI 10.1112/blms/4.3.257
- Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600. MR 297992, DOI 10.2307/1970638
- Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 1431508, DOI 10.1017/CBO9780511609572
- Gunhee Cho and Junqing Qian, The Kobayashi-Royden metric on punctured spheres, Forum Mathematicum , posted on (2020)., DOI 10.1515/forum-2019-0297
- Robert Fricke and Felix Klein, Lectures on the theory of automorphic functions. Vol. 2, CTM. Classical Topics in Mathematics, vol. 4, Higher Education Press, Beijing, 2017. Translated from the German original [ MR0183872] by Arthur M. DuPre. MR 3838412
- Phillip A. Griffiths, Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. of Math. (2) 94 (1971), 21–51. MR 310284, DOI 10.2307/1970733
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
- Irwin Kra, Accessory parameters for punctured spheres, Trans. Amer. Math. Soc. 313 (1989), no. 2, 589–617. MR 958896, DOI 10.1090/S0002-9947-1989-0958896-0
- John McKay and Abdellah Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), no. 2, 255–275. MR 1795562, DOI 10.1007/s002080000116
- James S. Milne, Modular functions and modular forms (v1.31), 2017. Available at www.jmilne.org/math/.
- Zeev Nehari, Conformal mapping, Dover Publications, Inc., New York, 1975. Reprinting of the 1952 edition. MR 0377031
- Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 0279280
- Frédéric Rochon and Zhou Zhang, Asymptotics of complete Kähler metrics of finite volume on quasiprojective manifolds, Adv. Math. 231 (2012), no. 5, 2892–2952. MR 2970469, DOI 10.1016/j.aim.2012.08.005
- Georg Schumacher, Asymptotics of Kähler-Einstein metrics on quasi-projective manifolds and an extension theorem on holomorphic maps, Math. Ann. 311 (1998), no. 4, 631–645. MR 1637968, DOI 10.1007/s002080050203
- Abdellah Sebbar, Torsion-free genus zero congruence subgroups of $\textrm {PSL}_2(\Bbb R)$, Duke Math. J. 110 (2001), no. 2, 377–396. MR 1865246, DOI 10.1215/S0012-7094-01-11028-4
- Abdellah Sebbar, Modular subgroups, forms, curves and surfaces, Canad. Math. Bull. 45 (2002), no. 2, 294–308. MR 1904094, DOI 10.4153/CMB-2002-033-1
- J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
- Elias M. Stein and Rami Shakarchi, Fourier analysis, Princeton Lectures in Analysis, vol. 1, Princeton University Press, Princeton, NJ, 2003. An introduction. MR 1970295
- A. B. Venkov, Accessory coefficients of a second-order Fuchsian equation with real singular points, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 129 (1983), 17–29 (Russian, with English summary). Automorphic functions and number theory. I. MR 703005
- Wenxiang Wang, On the smooth compactification of Siegel spaces, J. Differential Geom. 38 (1993), no. 2, 351–386. MR 1237488
- Damin Wu, Higher canonical asymptotics of Kähler-Einstein metrics on quasi-projective manifolds, Comm. Anal. Geom. 14 (2006), no. 4, 795–845. MR 2273294
- Damin Wu and Shing-Tung Yau, Complete Kähler-Einstein metrics under certain holomorphic covering and examples, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2901–2921 (English, with English and French summaries). MR 3959099
- Damin Wu and Shing-Tung Yau, Invariant metrics on negatively pinched complete Kähler manifolds, J. Amer. Math. Soc. 33 (2020), no. 1, 103–133. MR 4066473, DOI 10.1090/jams/933
- S.-T. Yau, Review of geometry and analysis, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 353–401. MR 1754787, DOI 10.1016/s0378-3758(98)00182-7
- Shing-Tung Yau, Métriques de kähler-einstein sur les variétés ouvertes, Premiére Classe de Chern et courbure de Ricci: Preuve de la conjecture de Calabi, volume 58 of Séminaire Palaiseau (1978), 163–167.
- Shing-Tung Yau and Yi Zhang, The geometry on smooth toroidal compactifications of Siegel varieties, Amer. J. Math. 136 (2014), no. 4, 859–941. MR 3245183, DOI 10.1353/ajm.2014.0024
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