Mathematical Sciences: Topics in Geometric Function Theory

这项数学研究的目标是应用 微分几何工具的研究中出现的问题， 几何函数论 作品的一个共同特点是 共形度量的使用，特别是双曲、欧几里德 和球形度量。 其中一个主要的兴趣领域将是重新检查 根据最近的一个结果， 这是五十年来下界的第一次改进。 而 改进小，方法新。 此外，新的 几何推导的改进界限打开了 进一步改进和应用可能性 其他极端问题（其中一些已经被 made）。 这项工作的扩展到几个复杂的变量也 将被考虑。 第二条调查路线涉及推导 的解析表达式是必要的和充分的， 定义域上的一个单叶函数。 这些都是在 两点失真定理的术语，其中有很多。 工作中的两个具体目标涉及的条件， 非单连通整环的单价 并确定是否可以找到一个表达式， 满足的所有单叶函数，给出了有关的信息 底层域的连通性。 将圆填充作为一种近似方法 保形映射将继续。 这个相对较新的 五年前，W.瑟斯顿，并导致了 函数论中的几条新的研究路线。 在 在这个项目中，工作将在最近成功的基础上扩大， n-连通共形映射的逼近 域映射的情况下，黎曼曲面。 同样，注意力将集中在使用圆填料上 把一个圆盘的覆盖映射近似到一个 连通性。

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.