Mathematical Sciences: Differential Geometry and Function Theory

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This project continues mathematical studies on the role of hyperbolic geometry in the theyro of complex function theory. The goals of the work are to relate hyperbolic geometry to Euclidean quantities such as Euclidean estimates of the hyperbolic metric and descriptions of Euclidean properties of hyperbolic geodesics. Results of this type have application to questions on distortion of univalent functions and estimates for Bloch and Landau constants. Geometric notions of symmetry and reflection play a central role in this research. Typical of the type of problems to be undertaken is establishing the existence and exact values of natural constants associated with classes of functions. The Bloch constant, for instance, measures the largest schlicht disk in the Riemann image surface of a function, whereas the Marden constant measures the size of the largest disk in the domain where a function must be univalent. These fundamental values have been obtained for some classes but not for some of the more important univalent functions. Work also continues in the study of hyperbolic geometry of regions where results point to appropriate generalizations to multiply connected regions. Two main goals are pursued here, the first being one of finding lower bounds for the density of the hyperbolic metric in terms of the distance to the boundary. The second goal is to obtain estimates for the Euclidean curvature and center of curvature for a hyperbolic geodesic. Much of this work will concentrate initially on special domains such as those which are spherically k-convex or hyperbolically k-convex before moving on to multiply connected ones.