By Matti Vuorinen

This ebook is an creation to the idea of spatial quasiregular mappings meant for the uninitiated reader. whilst the publication additionally addresses experts in classical research and, particularly, geometric functionality idea. The textual content leads the reader to the frontier of present learn and covers a few latest advancements within the topic, formerly scatterd in the course of the literature. an important position during this monograph is performed via yes conformal invariants that are ideas of extremal difficulties relating to extremal lengths of curve households. those invariants are then utilized to end up sharp distortion theorems for quasiregular mappings. this type 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 received can be of substantial curiosity even within the two-dimensional specific case. This booklet combines the gains of a textbook and of a examine monograph: it's the first advent to the topic to be had in English, includes approximately 100 routines, a survey of the topic in addition to an intensive bibliography and, eventually, an inventory of open difficulties.

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**Additional resources for Conformal Geometry and Quasiregular Mappings **

**Example text**

37 we see that g E ~ J ~ ( B n+l) is a euclidean isometry iff g(0) = 0 . Next we we shall reformulate this fact for maps in f f ~ ( R '~) . Let p be the reflection in the hyperplane x n + l = 0 and f l the inversion in S n ( e n + l , v / - 2 ) , and set f = f l o p . Then f R ~ +1 = B '~+1 and f ( e n + l ) = 0 , q ( x , y ) = ½ 1 f ( x ) - f ( y ) ] now that h E ~ ( ~ n ) We see t h a t p = foho is given and that h E f f j q ( ~ n + l ) f - 1 E f f ~ ( B "+1) for all x , y e R ' ~ . Assume is its Poincar~ extension.

4r ~2 \l--r 2] " Hence s = 1 6 s h 2 ( l p ( b , c ) ) c h 2 (¼ p ( b , c ) ) = 4 s h 2 (½ p ( b , c ) ) = 2 ( c h p ( b , c ) a n d s B , ( b , c ) = c h p ( b , c ) - 1 as desired. 26 holds for the h a l f - s p a c e H n , too. 27) chp(b,c)=l+suP{½ta, b,d, c I f a , c , d , b [ : a, d e S ~ - l } . 19). " with card(R. '~ \ G) > 2. 28) w h e n b, c E G . 29. R e m a r k . 28) is a metric. 31. 30. E x e r c i s e . 1 Assume that a > 0 and define b by c h b = 1 + 7a . 31. E x e r c i s e . 23).

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. E x e r c i s e . 20) provides a connection between the hyperbolic geometries (B~,p) and ( R ~_+ I ,p_) and the spherical geometry of (R'~, q).