Hyperbolic Distortion of Conformal Maps at Corners

We consider conformal self-maps φ of the unit disk onto simply connected domains. We assume φ is continuous in a neighborhood of a point , with φ(ζ) of modulus one, and that has a corner at φ(ζ). We prove that the modulus of the hyperbolic derivative of φ tends to a limit along certain simple curves in the disk that end at ζ non-tangentially. Moreover, we prove that the value of this limit depends only on the geometry of the corner and on the angle of approach to ζ. Our proof is based on a constructive approximation of the domain by more special domains. This research in its different stages was supported partially by MEC grants MTM2006-14449-C02-02 and MTM2006-26627-E (Acciones Complementarias), Spain; and also by “Ingenio Mathematica (i-MATH)” CSD2006-00032 (Consolider—Ingenio 2010), from MCyT, Spain; as well as by the MEC/Fulbright Fellowship 2007-0752.