View source for Teichmüller space

A metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923301.png" /> with as points abstract Riemann surfaces (that is, classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923302.png" /> of conformally-equivalent Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923303.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923304.png" /> (cf. [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]]) with singled out equivalent (with respect to the identity mapping) systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923305.png" /> of generators of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923306.png" />, and in which the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923307.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t0923309.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t09233010.png" />, where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t09233011.png" /> is the dilatation of the Teichmüller mapping (of the quasi-conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092330/t09233012.png" /> giving the smallest maximum dilatation among all such mappings). Introduced by O. Teichmüller [[#References|[1]]].

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.P. Gardiner,   "Teichmüller theory and quadratic differentials" , Wiley  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Lehto,   "Univalent functions and Teichmüller spaces" , Springer  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Nag,   "The complex analytic theory of Teichmüller spaces" , Wiley  (1988)</TD></TR></table>