By Matti Vuorinen

This booklet is an advent to the speculation of spatial quasiregular mappings meant for the uninitiated reader. whilst the booklet additionally addresses experts in classical research and, particularly, geometric functionality conception. The textual content leads the reader to the frontier of present study and covers a few most up-to-date advancements within the topic, formerly scatterd in the course of the literature. an immense position during this monograph is performed by means of convinced conformal invariants that are recommendations of extremal difficulties concerning extremal lengths of curve households. those invariants are then utilized to end up sharp distortion theorems for quasiregular mappings. this kind of extremal difficulties of conformal geometry generalizes a classical two-dimensional challenge of O. Teichmüller. the unconventional characteristic of the exposition is the way conformal invariants are utilized and the pointy effects acquired might be of substantial curiosity even within the two-dimensional specific case. This ebook combines the good points of a textbook and of a learn monograph: it's the first creation to the topic on hand in English, includes approximately 100 routines, a survey of the topic in addition to an intensive bibliography and, ultimately, an inventory of open difficulties.

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O(~,y). 42. E x e r c i s e . 41(2) with the help of the hyperbolic metric. e. 43. E x e r c i s e . ] For an open set D in R ~, D ~ R ~D(x,y)----log(l+max{ ,x--y[ ~,let [x--y[ 2 } ) Show that jD(x,y) ~ +D(X,y) ~_ 2JD(X,y ) . 44. E x e r c i s e . (1) Observe PH" first that, for t E (0, 1), llh (ten, en) = PH, (ten, S n - l ( l~e n, -~jj (cf. 8)). , log ~). tc(0j) (2) For p > 0 and t :> 0 let A(t) = flH n ((0, tP), (t,tv)). 0 A(t) and limt~o~ A(t) in the three cases p < 1, p = 1, and p > 1. 45.

In particular, A/a has an upper bound depending only on M . 32. E x e r c i s e . Let x 0 E B '~, M > 0 and v = min{ Iz - =01: p(=o,z) = M } , V = max{ l z - xol: p(xo, z) = M } . 33. Exercise. 43(1) . 19) using the identity 2sh2A = ch 2A - 1. 19) where x . and y, are the "end-points" of a geodesic segment containing x and y . Sometimes it will be convenient to express p(x,y) in a different way without refering to the points x , and y, at all. Such an expression can be achieved by exploiting an extremal property of p(x, y) as we shall show in the next section (see also Section 8).

27) Ix - vl -< 2th ¼p(~,v). Equality holds here if x = - y . (~, v) for x, y E B '~. 29. E x e r c i s e . Verify the following elementary relations. (1) 1 - e - s < t h s < 1 - e -28 for s > 0 . (2) If s > 0 , then th 2s ths = 1 + ~/1 - th 2 2s Further, if u E [ 0 , 1 ] and 2 s = a r t h u , t h e n th~ - u 1 + ~/i - ~ < ½(u + u 2 ) . - (3) l o g t h 8 = - 2 a r t h e -28 , s > 0. 32]. 1 + thpx 1-thpx for p =- 1 , 2 , . . - (1 + t h x ~ p ~l--thx ) and x > 0 . 30. E x e r c i s e . 31. Exercise.

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