Mathematical Sciences: Differential Geometry and Function Theory

这个项目继续对双曲几何在复变函数论中的作用进行数学研究。这项工作的目标是将双曲几何与欧几里德量联系起来，例如双曲度量的欧几里得估计和双曲测地线的欧几里得性质的描述。这类结果适用于单叶函数的扭曲问题以及Bloch常数和Landau常数的估计。对称性和反射的几何概念在这项研究中发挥了核心作用。要解决的典型问题是确定与函数类有关的自然常量的存在性和精确值。例如，Bloch常数测量函数的Riemann像面中最大的Schlicht圆盘，而Marden常数测量函数必须是单叶的区域中最大圆盘的大小。对于某些类，这些基本值已经得到，但对于一些更重要的单叶函数，却没有。在研究区域的双曲几何方面的工作也在继续，其中结果指向对相乘连通区域的适当推广。这里追求两个主要目标，第一个目标是根据到边界的距离找到双曲度规密度的下界。第二个目标是获得双曲测地线的欧几里得曲率和曲率中心的估计。这方面的大部分工作最初将集中在特殊区域上，例如那些球面k-凸或双曲k-凸域，然后再转移到多连通区域。

This project continues mathematical studies on the role of hyperbolic geometry in the theyro of complex function theory. The goals of the work are to relate hyperbolic geometry to Euclidean quantities such as Euclidean estimates of the hyperbolic metric and descriptions of Euclidean properties of hyperbolic geodesics. Results of this type have application to questions on distortion of univalent functions and estimates for Bloch and Landau constants. Geometric notions of symmetry and reflection play a central role in this research. Typical of the type of problems to be undertaken is establishing the existence and exact values of natural constants associated with classes of functions. The Bloch constant, for instance, measures the largest schlicht disk in the Riemann image surface of a function, whereas the Marden constant measures the size of the largest disk in the domain where a function must be univalent. These fundamental values have been obtained for some classes but not for some of the more important univalent functions. Work also continues in the study of hyperbolic geometry of regions where results point to appropriate generalizations to multiply connected regions. Two main goals are pursued here, the first being one of finding lower bounds for the density of the hyperbolic metric in terms of the distance to the boundary. The second goal is to obtain estimates for the Euclidean curvature and center of curvature for a hyperbolic geodesic. Much of this work will concentrate initially on special domains such as those which are spherically k-convex or hyperbolically k-convex before moving on to multiply connected ones.