Geometric Function Theory and Non-linear Analysis by Tadeusz Iwaniec

By Tadeusz Iwaniec

This specific booklet explores the connections among the geometry of mappings and plenty of very important parts of recent arithmetic comparable to Harmonic and non-linear research, the idea of Partial Differential Equations, Conformal Geometry and Topology. a lot of the e-book is new. It goals to supply scholars and researchers in lots of parts with a entire and recent account and an outline of the topic as an entire.

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The function A(r) is defined to be the average of n(r, y) = n(B(r), y) with respect to the spherical measure on C. An important result in the area is Ahlfors's theorem Introduction and overview 24 concerning the so-called defect relation. 29) is called the defect of a, Roughly, for r off a thin set E the function f covers each point ag the correct "average number" of times on the ball of radius r. Picard's theorem is a direct consequence if we put in the three omitted values for the numbers a,.

1. The most elementary of these is the following. 2 Fix K. Let 4: H -+ R' be a sequence of K-quasiconformal mappings converging pointwise to f: 11 - R". Then one of the following occurs: f is a K-quasiconformal embedding and the convergence is uniform on compact subsets. f(S1) is a doubleton with one value attained only once. f is constant. Basically it is true that any sufficiently normalized family of quasiconformal mappings forms a normal family (bounded sequences admit locally uniformly convergent subsequences).

Such mappings are known as similarities. Of course Dg = AO. More interesting examples are furnished by reflections. The reflection in a sphere S"-1(a, r) is the mapping g: R' Ht" defined by 9(x) = a+'1= Z-al" x¢{a,oo} 00 x = a a x = 00 It is clear that go g = I and g(x) = x if and only if x E S"-'(a. r). 2 The stereographic projection of the sphere Sn-1(a, r), r =jai, is the restriction of the inversion in Sn-1(2a, 2r). In particular, the Jacobian determinant is negative: J(x, g) = -r2nlx - al-2" and Dtg(x)Dg(x) = r4 lx - a 14 1 Thus every reflection in a sphere is orientation-reversing and conformal.

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