Normal Families of Möbius Maps

Several necessary and sufficient conditions for a family of Möbius maps to be a normal family in the extended complex plane $$\mathbb {C}_\infty $$ C ∞ are established. Each of these conditions involves collections of two or three points which may vary with the Möbius maps in the family, provided the points satisfy a uniform separation condition. In addition, we derive a sufficient condition for the normality of a family of Möbius maps in terms of the average value of the reciprocal of the chordal derivative.

SPHERICALIZATION AND FLATTENING PRESERVE UNIFORM DOMAINS IN NONLOCALLY COMPACT METRIC SPACES

The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.

Some Invariant Properties of Quasi-Möbius Maps

We investigate properties which remain invariant under the action of quasi-Möbius maps of quasimetric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)Möbius maps. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)Möbius maps as well.

MAPPING PROPERTIES OF A SCALE INVARIANT CASSINIAN METRIC AND A GROMOV HYPERBOLIC METRIC

We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps.