By Yoichi Imayoshi, Masahiko Taniguchi

ISBN-10: 4431700889

ISBN-13: 9784431700883

This e-book bargains a simple and compact entry to the speculation of Teichm?ller areas, ranging from the main undemanding elements to the newest advancements, e.g. the function this concept performs with reference to thread thought. Teichm?ller areas supply parametrization of all of the advanced buildings on a given Riemann floor. This topic is said to many alternative parts of arithmetic together with advanced research, algebraic geometry, differential geometry, topology in and 3 dimensions, Kleinian and Fuchsian teams, automorphic types, advanced dynamics, and ergodic idea. lately, Teichm?ller areas have started to play a major position in string conception. Imayoshi and Taniguchi have tried to make the e-book as self-contained as attainable. They current a number of examples and heuristic arguments to be able to support the reader seize the guidelines of Teichm?ller idea. The publication could be an outstanding 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

If 7(oo) = oo, then in a neighborhood of oo, 7 has the Laurent expansion tQ)=",+i bnz-n, where a I 0. Then tQ) - oz is holomorphic on e , and hence the maximum principle shows that lk) - oz must be a constant function, say 6. Thus we have l @ ) = a z l b w i t h o + 0 . I f z ( o o ) = z o # @ , t h e n s e t t i n gt { z ) - t l Q - z " ) , we see that both 11 and lpl are elements of Aut(e), and 71o7(m) = oo. Thus we havefolQ) = I/QQ)z o ) = a r z * b r , w h e r e0 r , 6 r € C w i t h o , * 0 .

Is considered as Emmple y'. 1,we define similarly its fundamental domain. , (t)" give examples of fundamental domains for covering groups of (i)', . . , (t)' in Example 2, respectively' ( i ) " r - { , e C | 0 < I m z< 2 r } . ( i i ) " r - { , e H | 0 < R e z< 1 } . ( i i i ) "F - { , e c - { 0 } | 0 < a r s z< 2 r / n } . ( i v ) " . F - { z e H l I < l r l< f } . ( u ) "F - { , e C l r - a } b r , 0 < a < 1 , 0 < D < 1 } . There is a simple way to construct canonically a fundamental domain for a Fuchsian model of a Riemann surface r?.

L < 1). \), where logz denotes its principal branch. Then 11 becomes a universal covering surface of the annulus ,4. ( v ) t e t 4 b e a l a t t i c e g r o u p g e n e r a t e db y 1 a n d a p o i n t r € I / , a n d l e t r b e the projection of C onto the quotient space C f fr. Then C is a universal covering surface of the torus C/ lr. Any biholomorphic mapping '1, fr, - E wittr ro^l = z is-called a coaering transformalion of a covering (R,r,R). For a given covering (R,r, R), denote by l' the set of all its covering transformations.

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An Introduction to Teichmuller Spaces by Yoichi Imayoshi, Masahiko Taniguchi

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