Metrics associated with the Hurwitz metric

Abstract

KEYWORDS: The hyperbolic metric, the Hurwitz netric, the Kobayashi metric, the Carath´eodory metric, the Gardiner-Lakic metric, Holomorphic covering map, Lipschitz domain, hyperbolic domain, hyperbolically covered do main, Hurwitz covering, generalized Hurwitz metric, M¨obius invariant metric, conformal map, domain monotonicity, Generalized Schwarz-Pick lemma, local uniform convergence. In this thesis, we study the Hurwitz metric and introduce metrics in connection with the Hurwitz metric by adopting the idea of the Kobayashi, Carath´eodory, and Gardiner Lakic metrics. We prove that the space with the distance induced by the Hurwitz metric is a complete metric space. Unit disk automorphism plays a crucial role to give a character ization of the Hurwitz metric through which we could define a generalized Hurwitz metric in the sense of Kobayashi in arbitrary subdomains of the complex plane. We study several important properties of this generalized metric, for instance, distance decreasing property, domain monotonicity etc. We establish that the Kobayashi density of the Hurwitz metric always exceeds the Hurwitz metric while the Carath´eodory density of the Hurwitz density trails the Hurwitz density. We also study the situations where they coincide with each other. We define a subclass of the class of hyperbolic domains namely the class of hyper bolically covered domains. In the sequel, we study the local uniform convergence of the Hurwitz metric in a sequence of hyperbolically covered domains. Estimations of quotients of the Hurwitz metrics with of the hyperbolic metrics play important roles in this inves tigation. Furthermore, we study the continuity of the Hurwitz metric in arbitrary proper subdomains of the complex plane and introduce a new M¨obius invariant metric which is bi-Lipschitz equivalent to the Hurwitz metric in hyperbolic domains. In addition, the lower semi-continuity of this M¨obius invariant metric followed by bi-Lipschitz equivalence of this metric with the (quasi) hyperbolic metrics are investigated.