Research in Classical Minimal Surface Theory

极小曲面在三维拓扑和黎曼几何的研究中起着重要的作用。该方案中的研究涉及嵌入极小曲面的全局性质以及这些结果在三维拓扑和几何基础研究中的可能应用。研究人员将研究三维欧氏空间中适当嵌入的极小曲面的几何、渐近行为、共形结构和拓扑。该方案的主要目的之一是对欧氏平面上所有可由区域参数化的适当嵌入的极小曲面进行分类，并描述所有有限亏格实例的渐近几何。还将研究局部有界亏格极小曲面的紧性、正则性和收敛的相关理论技巧。作为他最近与Charles Froman关于极小曲面拓扑分类的联合手稿的产物，研究人员提议证明双曲三维空间中的Bryant曲面是无纽结的。经典的极小曲面理论起源于18世纪和19世纪的数学。极小曲面是变分法的第一个重要例子，约在1735年由欧拉首次描述。物理上最小的曲面可以局部建模为电线上的肥皂膜，或者通过相对于其边界面积最小的曲面来建模。极小曲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。这项建议中的工作将有助于将可能出现的物理形状归类为此类界面。许多已知的极小曲面的例子都是在物理上观察到的，因此有一个严格的定理来预测可能出现的形状是很有意义的。在某种程度上，由于与其他数学领域的重要联系，以及因为可以为经典例子制作美丽的计算机图形图像，极小曲面仍然是科普文章和公共科学展览的主要主题之一。因此，间接地，这项提案中概述的令人兴奋的研究问题有助于将许多年轻科学家和数学家带到研究的前沿。

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.