Download A Crash Course on Kleinian Groups: Lectures given at a by Lipman Bers (auth.), Lipman Bers, Irwin Kra (eds.) PDF

By Lipman Bers (auth.), Lipman Bers, Irwin Kra (eds.)

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Additional info for A Crash Course on Kleinian Groups: Lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco

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Beltrami A for for z-* ¢o. * = Al_q, A * F = AI_qF If A is a M~Sbius coefficient i~ for F, is a generalized is a potential for transformation, and F a potential ]Beltrami A Li. ~ a generalized for coefficient ~, for then A IFA and 31 Lemma 4. If ~ nonelementary is a generalized Kleinian group a I, a2, .... 3) F(z) = is a potential Beltrami F with s u p p o r t in Z, are distinct points in A(F), for the a n d if then ~t(~)d~ A d~ (~-z)({-a I) ... (C-a2q_l) ( z - a 1) ,.. ( z - a 2 q _ l ) f f 2wi Z for coefficient U.

The derivative of this mapping (at the origin) can be computed and seen (Bets [8]) to be surjective. Since B(F) is a finite dimensional space, it can be shown using the implicit function theorem that ~(M(F)) = ~(T(F)) is open in B(F). 1 (Bers [8]). The Teichm~ller space T(F) has a unique complex structure so that : M(F) ~ T(F) i___ssholomorphic with local holomorphic sections. We can realize T(F) as a bounded domain (of holomorphy) i__%B(r). Further T(r) i_~stopolo~ically a cell. The fact that T(r) is a domain of holomorphy was proven by Bers-Ehrenpreiss [12].

J and, Thus by letting a.. is continuous elementary everywhere estimates and one shows 32 F(z) = 0(]z]2q-2) as variable formula, one c a n s h o w that, constant C(R) z-b ~ and, by r o u t i n e u s e of t h e c h a n g e of for every R > 0 t h e r e is a such that I F ( z ) - F ( w ) [ < C(R) [ z - w I l o g l z - w l [ whenever lzl and twt < R. It remains tions. Let support. ) to show that 8F/Sz ~0 be a test function, We must show test function ~p. Let If F~o~ dz A d---z - = that is, a - ff that = ~ F~ in the sense C° function dz A d-'~ = ff ~ p(z) ff ~ - ~ ~(~) (~-z)p(~) with compact U~ dz A d---z for p(z) = (z-a l) • • ' (z-a2q_l).

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