Mathematical Sciences: Problems in Complex Analysis

这个数学研究的中心焦点是关于复函数理论和拟共形映射的问题。这项工作的主要目标之一是对欧几里得空间的某些同胚群进行分类。最容易理解的群是所谓的莫比乌斯群，这些群的元素由最简单的线性（分数）变换表示。分类问题是确定哪些其他群在拓扑上与莫比乌斯群共轭，并了解可能阻止这种群同构的障碍是什么。在目标群中有一群被称为收敛群，它具有拟共形映射群的许多性质。许多收敛群已知是与莫比乌斯群共轭的，但到目前为止，还没有发现完整的分类。我们将努力确定这些群在多大程度上享有与Mobius群相同的属性。第二项研究涉及局部一元函数的注入准则。本工作寻求保证局部一元亚纯映射从域的邻域扩展到整个复平面的拟共形映射的条件。要进行扩展，使复杂的膨胀保持在一个固定数量的范围内。关于这类问题有相当多的文献。本工作的一个具体目标是消除亚纯函数在域的邻域内是全纯的条件，而只在闭域内具有这种性质。如果去掉初始函数为亚纯的条件，则可拓问题就变得过于一般化了。然而，圆盘内复函数的交叉比的自然条件导致了一个更严格的类，其中拟共形扩展（注入问题）已被证明存在。将继续确定保证注入能力的交叉比率的精确条件。//

The central focus of this mathematical research concerns problems in complex function theory and quasiconformal mappings. One of the primary objectives of the work is that of classifying certain groups of homeomorphisms of Euclidean space. Among the best understood groups are the so-called Mobius groups, those groups whose elements are represented by simplest linear (fractional) transformations. The classification problem is one of determining which other groups are topologically conjugate to a Mobius group and to understand what the obstruction is that might prevent such a group isomorphism. Among the target groups is one known as the convergence group which has many properties of groups of quasiconformal mappings. Many convergence groups are known to be conjugate to Mobius groups, but to date, no complete classification has been found. Work will be done in an effort to determine the extent to which these groups all enjoy the same properties relative to Mobius groups. A second line of investigation concerns injectivity criteria for locally univalent functions. This work seeks conditions which guarantee the extension of locally univalent meromorphic mappings from neighborhoods of a domain to a quasiconformal map of the entire complex plane. The extension is to be carried so that the complex dilatation remains bounded by a fixed amount. There is a sizeable literature concerning questions of this type. One specific objective of this work is to remove the condition that the meromorphic function be holomorphic in a neighborhood of a domain and only assume this property within a closed domain. If one drops the condition that the initial function be meromorphic, then the extension problem becomes too general. However, a natural condition on the cross-ratio of a complex function within a disc leads to a more restrictive class where quasiconformal extensions (the injectivity problem) have been shown to exist. Work will continue in determining precise conditions on cross-ratios which will guarantee injectivity. //