Applied math. Part 1: Integral by Bocconi

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14) on C. It also expresses in a specific way the analytic dependence of the solution / on \i. This analytic dependence was emphasized by Ahlfors and Bers in [AB61] and is essential in determining a complex structure for Teichmuller space. 14) depends holomorphically on \i and for any R > 0 there exists S > 0 and C(R) > 0 such that \f^(z)-z-tV{z)\

2). 4 Teichmiiller Space Definition 5 A Beltrami differential p(z)j^ on a Riemann surface R is an assignment to each chart za on Ua an L^ complex-valued function pa defined on za(Ua) such that ^ a ) = / ( ^ ) ^ | Z . 3) dza Note that | |p| |oo — s u P a 11Ma I loo is defined independently of coordinate charts. The space of essentially bounded, complex-valued, measurable Beltrami differentials on R is a Banach space, which we denote by LOQ(R). We denote the open unit ball in LQO^R^ by M(R). ) are called Beltrami coefficients because, by the mapping theorem of Chapter 1, corresponding to any p with ||p||oo < 1> there is a quasiconformal homeomorphism g = ga defined on each za(Ua) such that g-z(z) = pa(z)gz(z).

22), observe that d(Am o A-1^),^) < d(A m o A-l(rk),Am +d(Am o A ' V n ) , ^ ) + Since Am o Aml o A-\rn)) d(rn,rk). is an isometry, we get d{AmoA~1{rk),rk) < 2d(rn,rk) + d{Arn{pn),rn). But d{Am(pn),rn) < d(Am(pn),Am(pm)) + d(Am(pm),rn) - d(p n ,p m ) + d ( r m , r n ) . 22). To obtain the converse part of the lemma, notice that since a single point set {p} is compact, the proper discontinuity implies that its isotropy group is finite. Secondly, for an arbitrary point p, the local compactness of X implies there exists e > 0 such that K = {q\d(p, q) < e} is compact.

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