Research in Classical Minimal Surface Theory

极小曲面在三维拓扑学和黎曼几何研究中起着重要的作用。本提案的研究涉及嵌入最小表面的全局特性以及这些结果在三维拓扑和几何基础研究中的可能应用。研究者将研究三维欧氏空间中适当嵌入的最小曲面的几何、渐近行为、共形结构和拓扑结构。该建议的主要目标之一是对欧几里得平面上所有可被域参数化的适当嵌入的最小曲面进行分类，并描述所有有限属样例的渐近几何。研究了局部有界属最小曲面的紧性、正则性和收敛性的相关理论技术。作为他最近与Charles Froman关于最小曲面拓扑分类的联合手稿的一个成果，研究者提出证明双曲三维空间中的Bryant曲面是解结的。经典极小曲面理论起源于18、19世纪的数学。最小曲面是所谓变分法的第一个重要例子，它最早是由欧拉在1735年左右描述的。物理上最小的表面可以局部建模为电线上的肥皂膜或相对于其边界的最小面积表面。最小表面代表静止的流体界面，因此它们的形状出现在许多物理问题中。本提案中的工作将有助于对可能出现的这种界面的物理形状进行分类。许多已知的最小曲面的例子都是在物理上观察到的，所以有一个严格的定理来预测可能出现的形状是很有趣的。部分原因是由于它与其他数学领域的重要联系，也因为可以用经典例子制作漂亮的计算机图形图像，最小曲面继续成为科普文章和公共科学展览的主要主题之一。因此，本提案中概述的令人兴奋的研究问题间接地帮助许多年轻的科学家和数学家走向研究的前沿。

Minimal surfaces play an important role as a tool in the study of three-dimensional topology and Riemannian geometry. The research in this proposal concerns global properties of embedded minimal surfaces and possible applications of these results to basic research in three-dimensional topology and geometry. The researcher will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal surfaces in three-dimensional Euclidean space. One of the main goals of the proposal is to classify all of the properly embedded minimal surfaces which can be parametrized by domains in the Euclidean plane and to describe the asymptotic geometry of all finite genus examples. Related theoretical techniques concerning compactness, regularity and convergence of minimal surfaces of locally bounded genus will be investigated as well. As an outgrowth of his recent joint manuscript with Charles Froman on the topological classification for minimal surfaces, the researcher proposes to prove that Bryant surfaces in hyperbolic three-space are unknotted.Classical minimal surface theory has its roots in 18-th and 19-th century mathematics. Minimal surfaces are the first important examples of what is called the calculus of variations, first described by Euler around 1735. Physically minimal surfaces can be modeled locally as soap films on wires or by surfaces of least-area relative to their boundaries. Minimal surfaces represent stationary fluid interfaces, and so their shapes arise in many physical problems. The work in this proposal will help classify the possible physical shapes which might occur as such interfaces. Many of the known examples of minimal surfaces are observed physically, and so it is of interest to have a rigorous theorem which predicts the shapes which can occur. In part because of important connections with other areas of mathematics and because it is possible to make beautiful computer graphics pictures of classical examples, minimal surfaces continue to be one of the principal topics for popular science articles and public science exhibits. Thus, indirerctly, the exciting research problems outlined in this proposal help bring many young scientists and mathematicians to the froniters of research.