Titlebook: Conformally Invariant Metrics and Quasiconformal Mappings; Parisa Hariri,Riku Klén,Matti Vuorinen Book 2020 Springer Nature Switzerland AG

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Generalized Hyperbolic GeometriesIn geometric function theory, invariance properties of metrics are important. In our work below, two notions of invariance are most important; invariance with respect to the group of Möbius transformations and invariance with respect to the group of similarity transformations.

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Metrics and GeometryIn this chapter we shall consider some geometric issues related to the hyperbolic or quasihyperbolic metric. We begin with several comparison results for the quasihyperbolic metric. Here an important fact is that various metrics may be comparable in some but not in all domains.

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The Modulus of a Curve FamilyFor the sake of easy reference and for the reader’s convenience we shall give in this chapter the basic properties of the modulus of a curve family. The proofs of several well-known results are omitted.

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The Modulus as a Set FunctionLet . and . be compact disjoint non-empty sets in . and ..

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The Capacity of a CondenserIn the present chapter we shall introduce, as a special case of curve families and their moduli, the notion of a condenser and its capacity, and we shall examine various properties of condensers.

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Conformal InvariantsIn the preceding chapters we have studied some properties of the conformal invariant .( Δ(., .;.)). In this chapter we shall introduce two other conformal invariants, the modulus metric . and its “dual” quantity ., where . is a domain in . and ., . ∈ ..

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Hyperbolic Type MetricsIn Chaps. . and . we studied the quasihyperbolic metric .. (.) and the distance ratio metric .. (.). These two metrics are generalizations of the hyperbolic metric (Chap. .) and in this chapter we introduce other such generalizations.

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Comparison of MetricsWe start the comparison of metrics with those ones we have considered in the earlier chapters, namely the chordal metric ., (.), the hyperbolic metric ., (.), (.), the distance ratio metric ., (.) and the quasihyperbolic metric ., (.).

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Local Convexity of BallsIn this chapter we study some geometric properties of metric balls for small radii. It is natural to expect that balls of small radii are like euclidean balls whereas the geometric structure of . strongly influences on the shape of balls for large radii.

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Inclusion Results for BallsIn this chapter we consider inclusion of balls defined by several metrics.