HYPERBOLIC CURVATURE AND CONFORMAL MAPPING.pdfVIP

HYPERBOLIC CURVATURE AND CONFORMAL MAPPING BARBARA BROWN FLINN AND BRAD G. OSGOOD 1. Introduction The connection between the second derivative of a conformal mapping and curvature has been used in a number of ways. In this note we give an intrinsic formulation of this as a kind of Schwarz lemma for hyperbolic curvature. Throughout, D will denote a simply connected domain in C with at least two boundary points and K Z K z D( i) = xD ( i) will denote the geodesic curvature of a smooth curve y in D at ZE y measured in the hyperbolic metric X \ dz \ of D (constant curvature — 1). D THEOREM 1. Iff is a conformal mapping of D into itself then maix{K (J{z),f[y)),2} ^ max{K (Z,y),2}. D D It will be seen that this is equivalent to the classical coefficient inequality | a | ^ 2 2 for the class S and that the constant 2 is the same, thus establishing the sharpness. What we shall show is that | a | 2 is equivalent to the Schiffer-Tammi inequality for 2 functions mapping the unit disc into itself, which in turn is equivalent to Theorem 1. In §3 we characterize curves y with KD (Z, y) ^ 2 for all zey in terms of their mapping properties and their euclidean geometry. We recall a few necessary facts about geodesic curvature. As a generalization of euclidean curvature, geodesic curvature can be most directly defined as dd/ds where s is arclength in the metric and 0 is the angle between the tangent vector field to the curve and any vector field along the curve which remains pa