Research in Classical Minimal Surface Theory

极小曲面在三维拓扑学和黎曼几何的研究中起着重要的作用。本文的研究内容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。研究者将研究三维欧氏空间中适当嵌入极小曲面的几何、渐近性态、共形结构和拓扑。该建议的主要目标之一是分类的所有适当嵌入的极小曲面，可以参数化的域在欧几里德平面和描述的渐近几何的所有有限属的例子。同时也将研究局部有界亏格极小曲面的紧性、正则性和收敛性等相关理论技巧。作为他最近与Charles Froman共同撰写的关于极小曲面拓扑分类的论文的一个成果，作者提出了双曲三维空间中Bryant曲面是无结曲面的证明。经典极小曲面理论起源于18、19世纪数学。极小曲面是所谓的变分法的第一个重要例子，最早由欧拉在1735年左右描述。物理上的最小曲面可以局部建模为导线上的肥皂膜或相对于其边界的最小面积曲面。极小曲面表示静止的流体界面，因此它们的形状出现在许多物理问题中。在这个建议中的工作将有助于分类可能出现的物理形状，作为这样的接口。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。部分原因是与其他数学领域的重要联系，因为它可以使美丽的计算机图形图片的经典例子，最小的表面仍然是一个主要议题的科普文章和公共科学展览。因此，间接地，在这个建议中概述的令人兴奋的研究问题有助于把许多年轻的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.