Mathematical Sciences: Topics in Geometric Function Theory

这项数学研究的目标是将微分几何工具应用于研究几何函数论中出现的问题。这项工作的一个共同特点是使用了共形度量，特别是双曲、欧几里得和球面度量。主要感兴趣的领域之一将是根据最近的结果重新检查布洛赫常数，该结果在50年来首次改善了下限。虽然改进很小，但这种方法是新的。此外，改进的界的一个新的几何推导为进一步改进以及应用于其他极值问题(其中一些已经被做了)提供了可能性。还将考虑将这项工作扩展到几个复变量。第二条研究路线涉及到解析表达式的推导，解析表达式是给定函数在区域上单叶的必要条件和充分条件。这些都是以两点偏差定理的形式给出的，其中有很多这样的定理。工作中的两个特定目标涉及到在不是单连通的域中隐含单价的条件，以及确定是否可以找到一个表达式，当该表达式被所有单叶函数满足时，给出关于底层域的连通性的信息。圆填充作为近似共形映射的一种方法的工作将继续进行。这一相对较新的观点是由W.瑟斯顿在五年前提出的，并导致了函数论中的几条新的研究路线。在这个项目中，我们将把有限连通区域的共形映射的逼近推广到黎曼曲面上的映射。同样，人们的注意力将集中在使用圆填充来近似圆盘到有限连通区域的覆盖映射上。

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.