Teichmüller space via Kuranishi families
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 89-116.

In this partly expository note we construct Teichmüller space by patching together Kuranishi families. We also discuss the basic properties of Teichmüller space, and in particular show that our construction leads to simplifications in the proof of Teichmüller’s theorem asserting that the genus g Teichmüller space is homeomorphic to a (6g-6)-dimensional ball.

Classification : 30F60, 14H15, 32G15, 14H10
Arbarello, Enrico 1 ; Cornalba, Maurizio 2

1 Dipartimento di Matematica, “G. Castelnuovo”, Sapienza Università di Roma, Piazzale A. Moro, 2, 00185 Roma, Italia
2 Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, 27100 Pavia, Italia
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Arbarello, Enrico; Cornalba, Maurizio. Teichmüller space via Kuranishi families. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 89-116. http://www.numdam.org/item/ASNSP_2009_5_8_1_89_0/

[1] W. Abikoff, “The Real Analytic Theory of Teichmüller Space”, Lecture Notes in Mathematics, Vol. 820, Springer, Berlin, 1980. | MR | Zbl

[2] L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. | MR | Zbl

[3] L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces., In: “Analytic Functions”, Princeton Univ. Press, Princeton, N.J., 1960, 45–66. | MR | Zbl

[4] L. V. Ahlfors, Quasiconformal mappings, Teichmüller spaces, and Kleinian groups, In: “Fields Medallists’ Lectures”, World Sci. Ser. 20th Century Math., Vol. 5, World Sci. Publ., River Edge, NJ, 1997, 10–23. | Zbl

[5] L. V. Ahlfors, “Lectures on Quasiconformal Mappings”, second ed., with supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard, University Lecture Series, Vol. 38, American Mathematical Society, Providence, RI, 2006. | MR | Zbl

[6] E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, “Geometry of Algebraic Curves”, Vol. II, Grundlehren der Mathematischen Wissenschaften, Vol. 268, Springer-Verlag, New York, 2009, to appear. | MR | Zbl

[7] L. Bers, “Riemann Surfaces”, Notes by E. Rodlidtz and R. Pollack, Courant Institute of Mathematical Sciences, New York University, 1951-52.

[8] L. Bers, Quasiconformal mappings and Teichmüller’s theorem, In: “Analytic Functions”, Princeton Univ. Press, Princeton, NJ, 1960, 89–119. | MR | Zbl

[9] L. Bers, Spaces of Riemann surfaces, In: “Proc. Internat. Congress Math. 1958”, Cambridge Univ. Press, New York, 1960, 349–361. | MR | Zbl

[10] L. Bers, Uniformization and moduli, In: “Contributions to Function Theory”, Internat. Colloq. Function Theory, Bombay, 1960, Tata Institute of Fundamental Research, Bombay, 1960, 41–49. | MR | Zbl

[11] L. Bers, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215–228. | MR | Zbl

[12] L. Bers, “On Moduli of Riemann Surfaces”, Mimeographed Lecture Notes, Eidgenössische Technische Hochschule, Zürich, 1964.

[13] L. Bers, On Teichmüller’s proof of Teichmüller’s theorem, J. Anal. Math. 46 (1986), 58–64. | MR | Zbl

[14] S.-Shen Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc. 6 (1955), 771–782. | MR

[15] R. Courant and D. Hilbert, “Methods of Mathematical Physics: Partial Differential Equations”, Vol. II by R. Courant., Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. | MR | Zbl

[16] C. J. Earle, Teichmüller spaces, Ann. Acad. Sci. Fenn. Math. 13 (1988), 355–361. | MR | Zbl

[17] C. J. Earle and J. Eells, Jr., On the differential geometry of Teichmüller spaces, J. Anal. Math. 19 (1967), 35–52. | MR | Zbl

[18] D. B. A. Epstein, Curves on 2-manifolds and isotopies, Acta Math. 115 (1966), 83–107. | MR | Zbl

[19] F. P. Gardiner, “Teichmüller Theory and Quadratic Differentials”, Wiley-Interscience Publication, Wiley, New York, 1987. | MR | Zbl

[20] A. Grothendieck, “Techniques de construction en géométrie analytique. I. Description axiomatique de l’espace de Teichmüller et de ses variantes”, In: “Familles d’Espaces Complexes et Fondements de la Géométrie Analytique”, Séminaire Henri Cartan, 13ième année: 1960/61. Exposés No. 7 et 8, Sécretariat Mathématique, Paris, 1962. | EuDML | Numdam | Zbl

[21] A. Grothendieck, “Techniques de construction en géométrie analytique. X. Construction de l’espace de Teichmüller”, In: “Familles d’Espaces Complexes et Fondements de la Géométrie Analytique”, Séminaire Henri Cartan, 13ième année: 1960/61, Exposé No. 17, Sécretariat Mathématique, Paris, 1962. | EuDML | Zbl

[22] Y. Imayoshi and M. Taniguchi, “An Introduction to Teichmüller Spaces”, Springer-Verlag, Tokyo, 1992. | MR | Zbl

[23] L. Keen, Intrinsic moduli on Riemann surfaces, Ann. of Math. (2) 84 (1966), 404–420. | MR | Zbl

[24] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2) 71 (1960), 43–76. | MR | Zbl

[25] A. Korn, Zwei Anwendungen der Methode der sukzessiven Annäherungen, In: “Schwarz Festschrift”, Berlin, 1914, 215–229. | JFM

[26] O. Lehto, “Univalent Functions and Teichmüller Spaces”, Graduate Texts in Mathematics, Vol. 109, Springer-Verlag, New York, 1987. | MR | Zbl

[27] L. Lichtenstein, Zur Theorie der konformen Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete, Bull. Int. de l’Acad. Sci. Cracovie, Sér. A (1916), 192–217. | JFM

[28] A. Marden and K. Strebel, A characterization of Teichmüller differentials, J. Differential Geom. 37 (1993), 1–29. | MR | Zbl

[29] C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166. | JFM | MR

[30] E. Reich and K. Strebel, Teichmüller mappings which keep the boundary point-wise fixed, In: “Advances in the Theory of Riemann Surfaces”, Proc. Conf., Stony Brook, N.Y., 1969, Princeton Univ. Press, Princeton, N. J., 1971, 365–367. | MR | Zbl

[31] E. Reich and K. Strebel, Extremal quasiconformal mappings with given boundary values, In: “Contributions to Analysis” (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, 375–391. | MR | Zbl

[32] M. Seppälä and T. Sorvali, “Geometry of Riemann Surfaces and Teichmüller Spaces”, North-Holland Mathematics Studies, Vol. 169, North-Holland Publishing Co., Amsterdam, 1992. | MR | Zbl

[33] E. Sernesi, “Deformations of Algebraic Schemes”, Grundlehren der Mathematischen Wissenschaften, Vol. 334, Springer-Verlag, Berlin, 2006. | MR | Zbl

[34] K. Strebel, On the existence of extremal Teichmüller mappings, Complex Variables Theory Appl. 9 (1987), 287–295. | MR | Zbl

[35] O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1939 (1940), no. 22, 197. | MR

[36] O. Teichmüller, Veränderliche Riemannsche Flächen, Deutsche Math. 7 (1944), 344–359. | MR | Zbl

[37] O. Teichmüller, “Gesammelte Abhandlungen”, Springer-Verlag, Berlin, 1982, Edited and with a preface by Lars V. Ahlfors and Frederick W. Gehring. | MR | Zbl

[38] A. J. Tromba, “Teichmüller Theory in Riemannian Geometry”, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992, Lecture notes prepared by Jochen Denzler. | MR | Zbl

[39] J. H. Hubbard, “Teichmüller Theory and Applications to geometry, Topology, and Dinamics”, Vol. 1: “Teichmüller Theory”. With contributions by A. Douady, W. Dunbar, R. Roeder, S. Bonnot, D. Brown, A. Hatcher, C. Hruska and S. Mitra, Matrix Editions, Ithaca, NY, 2006. | MR