Mathematical Sciences: Conformal Geometry and Geometric Function Theory

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This project continues mathematical research into problems of complex function theory, concentrating on geometric ideas which use differential geometric tools. The common feature in the investigation of all these problems is the use of conformal metrics, especially the hyperbolic, euclidean and spherical metrics. One is interested in obtaining information about one conformal geometry, say hyperbolic, in terms of another conformal (usually euclidean) geometry or in studying analytic functions viewed as mappings from one geometry to another. For instance, from this perspective, Bloch functions are precisely those analytic functions from hyperbolic to euclidean geometry with bounded distortion. Problems of interest include two-point comparison theorems for euclidean and hyperbolic geometry, generalized convexity-concavity properties of the hyperbolic metric, Bloch and linearly invariant functions, quasi-invariant domain constants, and a study of various conformal metrics and distance functions defined by families of holomorphic or harmonic functions. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications of the theory to potential theory and fluid dynamics are standard in engineering circles.