The goals of this mathematical research are to apply differential geometric tools to the study of problems arising in geometric function theory. A common feature of the work is the use of conformal metrics, especially the hyperbolic, euclidean and spherical metrics. One of the primary areas of interest will be a reexamination of the Bloch constant in light of a recent result which provided the first improvement in the lower bound in fifty years. While the improvement is small, the method is new. Moreover, a new geometric derivation of the improved bound opens up the possibility of further improvement as well as applications to other extremal problems (some of which have already been made). Extensions of this work to several complex variables also will be considered. A second line of investigation involves the derivation of analytic expressions which are necessary and sufficient for a given function to be univalent on a domain. These are given in terms of two-point distortion theorems, of which there are many. Two particular goals in the work concern conditions which imply univalence in domains which are not simply connected and establishing whether an expression can be found which, when satisfied by all univalent functions, gives information about the connectivity of the underlying domain. Work on circle packing as a means for approximating conformal mappings will continue. This relatively new point of view was introduced five years ago by W. Thurston and has led to several new lines of investigation in function theory. In this project, work will be done expanding on recent successes in the approximation of conformal maps of finitely connected domains to the case of mappings on Riemann surfaces. Equivalently, attention will focus on the use of circle packings to approximate the covering map of a disk onto a region of finite connectivity.
