Mathematical Sciences: Conformal Geometry and Geometric Function Theory

该项目继续对复函数理论的问题进行数学研究，集中于使用微分几何工具的几何思想。研究这些问题的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。一个人感兴趣的是用另一个共形几何（通常是欧几里得几何）来获取关于一个共形几何（比如双曲）的信息，或者研究从一个几何到另一个几何的映射的解析函数。例如，从这个角度来看，布洛赫函数正是那些从双曲几何到欧几里得几何具有有界畸变的解析函数。感兴趣的问题包括欧几里得几何和双曲几何的两点比较定理，双曲度规的广义凹凸性，Bloch和线性不变函数，拟不变域常数，并研究了由全纯函数族或调和函数族所定义的各种共形度量和距离函数。复变函数理论包括复变量的可微函数和相关的函数类，如调和函数和拟共形映射的研究。这个主题是高度几何化的；许多问题涉及由上述类之一的函数变换的各种集合的属性。该理论在位势理论和流体力学中的应用是工程界的标准。

This project continues mathematical research into problems of complex function theory, concentrating on geometric ideas which use differential geometric tools. The common feature in the investigation of all these problems is the use of conformal metrics, especially the hyperbolic, euclidean and spherical metrics. One is interested in obtaining information about one conformal geometry, say hyperbolic, in terms of another conformal (usually euclidean) geometry or in studying analytic functions viewed as mappings from one geometry to another. For instance, from this perspective, Bloch functions are precisely those analytic functions from hyperbolic to euclidean geometry with bounded distortion. Problems of interest include two-point comparison theorems for euclidean and hyperbolic geometry, generalized convexity-concavity properties of the hyperbolic metric, Bloch and linearly invariant functions, quasi-invariant domain constants, and a study of various conformal metrics and distance functions defined by families of holomorphic or harmonic functions. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications of the theory to potential theory and fluid dynamics are standard in engineering circles.