By Yoichi Imayoshi, Masahiko Taniguchi

This publication deals a simple and compact entry to the speculation of Teichm?ller areas, ranging from the main basic points to the latest advancements, e.g. the position this conception performs in regards to thread conception. Teichm?ller areas provide parametrization of the entire advanced constructions on a given Riemann floor. This topic is said to many various components of arithmetic together with advanced research, algebraic geometry, differential geometry, topology in and 3 dimensions, Kleinian and Fuchsian teams, automorphic kinds, advanced dynamics, and ergodic thought. lately, Teichm?ller areas have all started to play an enormous position in string idea. Imayoshi and Taniguchi have tried to make the booklet as self-contained as attainable. They current a variety of examples and heuristic arguments with a purpose to support the reader clutch the information of Teichm?ller idea. The booklet should be a great resource of data for graduate scholars and reserachers in complicated research and algebraic geometry in addition to for theoretical physicists operating in quantum idea.

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**Example text**

With "I1(Z) (i)' 2'11"i. (i)' r f = ("11) (rr) with 71(z) zz + (ii)' (ii)' r f ==( r("11) r ) w with i t h 7"I1(Z) 1 ( z )== Z z+ * 11.. - zz exp groupof order n. ordern. finitegroup is aa finite (iii)' r (2'11"ijn) , which is exp(2riln),which with "I1(Z) ('rt) with 1= ("11) 71(z) = )2. with "I1(Z) (iv)' AZ. (iu)' r r = ("11) (zr) with 71(z) "fz(z)= Z z+ a,nd"I2(Z) where"I1(Z) (T,72) -- r f,,T , where + r. 2. Construction Surfaces Covering Surfaces of Universal tJniversal Covering Construction of means aa path on surface R R means First of on aa Riemann Riemann surface definitions.

Given at a point p and whose whose electric potential potential on R where where a positive charge charge is given boundary is earthed. Mathematically, when z is a local coordinate around p is is earthed. ,p) on R with pole at as the minimal ,R we define Green are harmonic function in the family of positive superharmonic superharmonic functions which are -loglz p. R - {p} {p} and "capacity" of the boundary of R is indepent of Green rt and is depends on "capacity" Green function depends with disk Li 4 with the choice example, the Green Green function on the unit disk choice of aa point p.

Mappings. We We shall shall In this chapter, we only consider consider smooth quasiconformal chapter, we quasiconformal mappings 4. study more general general quasiconformal mappings in Chapter 4. 4) lprl(z)l of the = diffeomorpJQ)dzldz an orientation-preserving Beltrami = III (z) di/ dz of an orientation-preserving diffeomorcoefficient III Beltrami coefficient W is aa - S l?. Thus III phism f: I II is local coordinates coordinates on R. 9 does lpy I : R --+ get we get r? is is compact, compact, we I I| < 11 on continuous and III on R.