A Model of the Teichmüller space of genus-zero bordered surfaces by period maps

Radnell, David; Schippers, Eric; Staubach, Wolfgang

doi:10.1090/ecgd/332

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A Model of the Teichmüller space of genus-zero bordered surfaces by period maps

Authors: David Radnell, Eric Schippers and Wolfgang Staubach
Journal: Conform. Geom. Dyn. 23 (2019), 32-51
MSC (2010): Primary 30F60, 32G15, 32G20; Secondary 30C55, 30H20, 46G20
DOI: https://doi.org/10.1090/ecgd/332
Published electronically: February 27, 2019
MathSciNet review: 3917230
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Abstract: We consider Riemann surfaces $\Sigma$ with borders homeomorphic to $\mathbb{S}^1$ and no handles. Using generalized Grunsky operators, we define a period mapping from the infinite-dimensional Teichmüller space of surfaces of this type into the unit ball in the linear space of operators on an -fold direct sum of Bergman spaces of the disk. We show that this period mapping is holomorphic and injective.

[1] Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
[2] S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1951), 205–249. MR 39812
[3] Abdullah Çavuş, Approximation by generalized Faber series in Bergman spaces on finite regions with a quasiconformal boundary, J. Approx. Theory 87 (1996), no. 1, 25–35. MR 1410610, https://doi.org/10.1006/jath.1996.0090
[4] Soo Bong Chae, Holomorphy and calculus in normed spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 92, Marcel Dekker, Inc., New York, 1985. With an appendix by Angus E. Taylor. MR 788158
[5] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
[6] Frederick P. Gardiner, Schiffer’s interior variation and quasiconformal mapping, Duke Math. J. 42 (1975), 371–380. MR 382637
[7] K.-G. Grosse-Erdmann, A weak criterion for vector-valued holomorphy, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 2, 399–411. MR 2040581, https://doi.org/10.1017/S0305004103007254
[8] J. A. Hummel, Inequalities of Grunsky type for Aharonov pairs, J. Analyse Math. 25 (1972), 217–257. MR 311893, https://doi.org/10.1007/BF02790039
[9] Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
[10] Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
[11] Subhashis Nag, A period mapping in universal Teichmüller space, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 280–287. MR 1121571, https://doi.org/10.1090/S0273-0979-1992-00273-2
[12] Subhashis Nag and Dennis Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the 𝐻^{1/2} space on the circle, Osaka J. Math. 32 (1995), no. 1, 1–34. MR 1323099
[13] Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR 0507768
[14] David Aubrey Alfred Radnell, Schiffer variation in Teichmuller space, determinant line bundles and modular functors, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. MR 2705159
[15] David Radnell and Eric Schippers, Quasisymmetric sewing in rigged Teichmüller space, Commun. Contemp. Math. 8 (2006), no. 4, 481–534. MR 2258875, https://doi.org/10.1142/S0219199706002210
[16] David Radnell and Eric Schippers, A complex structure on the set of quasiconformally extendible non-overlapping mappings into a Riemann surface, J. Anal. Math. 108 (2009), 277–291. MR 2544761, https://doi.org/10.1007/s11854-009-0025-0
[17] David Radnell and Eric Schippers, Fiber structure and local coordinates for the Teichmüller space of a bordered Riemann surface, Conform. Geom. Dyn. 14 (2010), 14–34. MR 2593332, https://doi.org/10.1090/S1088-4173-10-00206-7
[18] David Radnell, Eric Schippers, and Wolfgang Staubach, Dirichlet spaces of domains bounded by quasicircles, to appear in Commun. Contemp. Math., preprint arXiv:1705.01279v2.
[19] Eric Schippers and Wolfgang Staubach, Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks, J. Math. Anal. Appl. 448 (2017), no. 2, 864–884. MR 3582264, https://doi.org/10.1016/j.jmaa.2016.11.041
[20] Eric Schippers and Wolfgang Staubach, Riemann boundary value problem on quasidisks, Faber isomorphism and Grunsky operator, Complex Anal. Oper. Theory 12 (2018), no. 2, 325–354. MR 3756161, https://doi.org/10.1007/s11785-016-0598-4
[21] YuLiang Shen, Faber polynomials with applications to univalent functions with quasiconformal extensions, Sci. China Ser. A 52 (2009), no. 10, 2121–2131. MR 2550270, https://doi.org/10.1007/s11425-009-0062-2
[22] P. K. Suetin, Series of Faber polynomials, Analytical Methods and Special Functions, vol. 1, Gordon and Breach Science Publishers, Amsterdam, 1998. Translated from the 1984 Russian original by E. V. Pankratiev [E. V. Pankrat′ev]. MR 1676281
[23] Leon A. Takhtajan and Lee-Peng Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc. 183 (2006), no. 861, viii+119. MR 2251887, https://doi.org/10.1090/memo/0861