Mathematical Sciences: Topics in Geometric Function Theory

这项数学研究的目标是应用微分几何工具来研究几何函数理论中出现的问题。这项工作的一个共同特点是使用共形度量，特别是双曲，欧几里得和球面度量。其中一个主要的兴趣领域将是重新检查布洛赫常数在最近的结果，提供了第一次改进的下限在五十年。虽然改进很小，但这种方法是新的。此外，改进界的一个新的几何推导为进一步改进和应用于其他极值问题（其中一些已经提出）开辟了可能性。将这项工作扩展到几个复杂变量也将被考虑。第二项研究涉及解析表达式的推导，它是给定函数在一个定义域上是一元的必要和充分条件。这些都是用两点扭曲定理给出的，其中有很多。工作中的两个特定目标涉及在非单连通域中隐含单价的条件，以及确定是否可以找到一个表达式，当所有单价函数满足时，给出有关底层域连通性的信息。作为近似保角映射的一种方法，将继续研究圆填充。这个相对较新的观点是五年前由W. Thurston提出的，并导致了函数理论中几个新的研究方向。在这个项目中，工作将扩展最近在有限连通域的共形映射近似方面的成功，以黎曼曲面上的映射为例。同样，注意将集中在使用圆填充来近似磁盘到有限连通性区域的覆盖映射。

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.