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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]]].
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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 [[#References|[1]]].
  
 
====References====
 
====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>
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<table>
 
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<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>
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<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>
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<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>
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<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>
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<TR><TD valign="top">[2d]</TD> <TD valign="top">  L. Bers,  "Simultaneous uniformization"  ''Bull. Amer. Math. Soc.'' , '''66'''  (1960)  pp. 94–97</TD></TR>
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<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>
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<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>
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<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>
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</table>
  
  
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====References====
 
====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>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  F.P. Gardiner,  "Teichmüller theory and quadratic differentials" , Wiley  (1987)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Lehto,  "Univalent functions and Teichmüller spaces" , Springer  (1987)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Nag,  "The complex analytic theory of Teichmüller spaces" , Wiley  (1988)</TD></TR>
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</table>
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Latest revision as of 20:20, 22 September 2017

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].

References

[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)


Comments

References

[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)
How to Cite This Entry:
Teichmüller space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Teichm%C3%BCller_space&oldid=15836
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article