9401693 Duren This work will continue mathematical research on problems arising in the theory of analytic functions and related questions involving harmonic mappings. To a large degree, the effort continues several lines of research which have opened new directions of investigation. Work will be done extending recent results on zero-divisors for functions in Bergman spaces defined in a disc. These are spaces of analytic functions whose norms are defined with respect to area integration of their powers. The long-sought solution to the zero divisor problem was obtained by Hedenmalm; it opens the door to investigations into the classification of invariant subspaces analogous to the classical result of Beurling for the Hardy spaces. Considerable work still remains, almost nothing has been established about these spaces to date, especially on the question of existence of extremal functions. Work will also be done investigating plane harmonic mappings, i.e. mappings for which each component is harmonic. Such mappings are directly related to solutions of minimal surface problems. Questions of concern center on finding good estimates for the Gaussian curvature of a nonparametric minimal surface above the disc in terms of the so-called dilatation of a harmonic map and on determining the shape of images of extremal harmonic maps. Classical function theory has introduced geometric techniques which have proved to have far wider applicability than first imagined. The subject exploits the interplay between geometric interpretation with powerful analytic techniques to yield some of the most complete mathematical results in the field of analysis.In this project, variational techniques and geometric reasoning are expected to lead to results of valuable physical significance. ***
