The conformal radius as a function and its gradient image

Let Ω be a domain in with three or more boundary points in andR(w, Ω) the conformal, resp. hyperbolic radius of Ω at the pointw ε Ω/{∞}. We give a unified proof and some generalizations of a number of known theorems that are concerned with the geometry of the surface 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. This work was supported by a grant of the Deutsche Forschungsgemeinschaft for F. G. Avkhadiev.