Teichmüller space

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A metric space $(M_g,d)$ with as points abstract Riemann surfaces (that is, classes $\bar X$ of conformally-equivalent Riemann surfaces $X$ of genus $g$ (cf. Riemann surfaces, conformal classes of) with singled out equivalent (with respect to the identity mapping) systems $\Sigma$ of generators of the fundamental group $\pi_1(X)$, and in which the distance $d$ between $\bar X$ and $\bar X'$ is equal to $\log K$, where the constant $K$ is the dilatation of the Teichmüller mapping (of the quasi-conformal mapping $\bar X \rightarrow \bar X'$ giving the smallest maximum dilatation among all such mappings). Introduced by O. Teichmüller [1].


[1] O. Teichmüller, "Extremale quasikonforme Abbildungen und quadratische Differentialen" Abhandl. Preuss. Akad. Wissenschaft. Math.-Nat. Kl. , 22 (1939) pp. 3–197
[2a] L. Bers, "Quasi-conformal mappings and Teichmüller's theorem" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 89–119
[2b] L.V. Ahlfors, "The complex analytic structure of the space of Riemann surfaces" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 45–66
[2c] L. Bers, "Spaces of Riemann surfaces" , Proc. Intern. Congress Mathematicians, Edinburgh 1958 , Cambridge Univ. Press (1959) pp. 349–361
[2d] L. Bers, "Simultaneous uniformization" Bull. Amer. Math. Soc. , 66 (1960) pp. 94–97
[2e] L. Bers, "Holomorphic differentials as functions of moduli" Bull. Amer. Math. Soc. , 67 (1961) pp. 206–210
[2f] L. Ahlfors, "On quasiconformal mappings" J. d'Anal. Math. , 3 (1954) pp. 1–58; 207–208
[3] S.L. Krushkal, "Quasi-conformal mappings and Riemann surfaces" , Halsted (1979) (Translated from Russian)



[a1] F.P. Gardiner, "Teichmüller theory and quadratic differentials" , Wiley (1987)
[a2] O. Lehto, "Univalent functions and Teichmüller spaces" , Springer (1987)
[a3] S. Nag, "The complex analytic theory of Teichmüller spaces" , Wiley (1988)
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Teichmüller space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article