The conformal radius as a function and its gradient image

Abstract

Let Ω be a domain in\(\bar {\mathbb{C}}\) with three or more boundary points in\(\bar {\mathbb{C}}\) andR(w, Ω) the conformal, resp. hyperbolic radius of Ω at the pointw e Ω/{∞}. We give a unified proof and some generalizations of a number of known theorems that are concerned with the geometry of the surface\(s_\Omega = \{ (w,h)|w \in \Omega ,h = R(w,\Omega )\} \) in the case that the Jacobian of ∇R(w, Ω), the gradient ofR, is nonegative on Ω. We discuss the function ∇R(w, Ω) in some detail, since it plays a central role in our considerations. In particular, we prove that ∇R(w, Ω) is a diffeomorphism of Ω for four different types of domains.