By Matti Vuorinen

This publication is an creation to the idea of spatial quasiregular mappings meant for the uninitiated reader. even as the e-book additionally addresses experts in classical research and, specifically, geometric functionality concept. The textual content leads the reader to the frontier of present examine and covers a few latest advancements within the topic, formerly scatterd throughout the literature. an important position during this monograph is performed through definite conformal invariants that are options of extremal difficulties regarding 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 radical function of the exposition is the way conformal invariants are utilized and the pointy effects acquired may be of substantial curiosity even within the two-dimensional specific case. This booklet combines the gains of a textbook and of a learn monograph: it's the first creation to the topic to be had in English, comprises approximately 100 workouts, a survey of the topic in addition to an intensive bibliography and, eventually, an inventory of open difficulties.

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**Sample text**

Clearly, k H , ---- pH,~, and we see easily t h a t PB- --< 2 kB,, _< (cf. 15)). Hence, the geodesics of ( H ~ , k H . ) are those of ( H n , P H , ) , but it is a difficult task to find the geodesics of k D when D is given. The following m o n o t o n e p r o p e r t y of k D is clear: if D and D ~ are domains with D r c D and x , y E D ' , then k D,(x,y) > k D(x,y) . 34). The metric JD is indeed a natural choice for such a comparison function since both k D and JD are invariant under translations, stretchings and orthogonal mappings.

G. in P. Jones' works [J1], [J2] on extension operators of function spaces. An exposition of these results occurs in [GS], with several equivalent definitions of plane uniform domains. The above variant of the definition of a uniform domain is suggested by [GOS] and [VU10]. 4. Some covering problems In this section we shall consider some geometric problems related to the hyperbolic or quasihyperbolic metric. A typical question, which we are going to answer, is the following. Let X be a compact set in B '~ and let iT be a covering of X by hyperbolic balls with fixed radii.

1 + t h x ~ p ~l--thx ) and x > 0 . 30. E x e r c i s e . 31. Exercise. 23) show that if D(x,M) = B'~(y,r), then r admits an estimate (1-[yl) b