Applications of Modulus Inequalities

Abstract

In this chapter we shall give our first applications of the inequalities for moduli of path families proved in the preceding chapter. Further applications will be given in Chapters IV, V, and VII. We start with some global distortion results and continue by proving, among other things, that a nonconstant qr mapping of ℝⁿ into itself omits at most a set of zero capacity. A local form of the latter result will be used in the proof of a Picard-type theorem in Chapter IV. Next, we shall establish a generalization of the theorem of V.A. Zorich which was mentioned in the introduction. In all these results Poletskiĭ’s K_I-inequality is used. The rest of this chapter is devoted to local questions. There the sharper Väisälä’s inequality, or its capacity variant due to O. Martio, and the K _O -inequality play essential roles for the problems treated.