Research in Classical Minimal Surface Theory

DMS-0405836题目：经典极小曲面理论研究PI：William H. Meeks，马萨诸塞州大学（Amherst）摘要建议的研究项目摘要在这个建议中，研究者将研究三维欧氏空间中适当嵌入极小曲面的几何、渐近行为、共形结构和拓扑。 本文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极小曲面进行分类，并描述所有有限亏格实例的渐近几何，同时也研究了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术。 本文的一个应用是对三维球面上的光滑有限群作用进行分类。 作为他最近与CharlesFrohman关于极小曲面拓扑分类的联合手稿的一个衍生物，研究者提出证明双曲三维空间中的Bryant曲面是无结的。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。 物理极小曲面可以局部地建模为导线上的肥皂膜或相对于其局部边界的最小面积曲面。 极小曲面作为研究三维拓扑和黎曼几何的一个重要工具。极小曲面在数学和物理科学中有着广泛的影响。 极小曲面是静止的流体界面，因此在许多物理问题中会出现极小曲面的形状。 这个提议的工作将把可能出现的物理形状分类为无限界面。 许多已知的极小曲面的例子都是物理上可以观察到的，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。这里提出的研究强烈地影响了三维欧氏空间中曲面的经典微分几何领域。 正如几何学家所熟知的，极小曲面理论一直是并将继续是证明广义相对论和三维拓扑学定理的主要工具之一。 一个众所周知的应用是舍恩和丘的证明正质量猜想。 Gabai最近关于广义Smale猜想的工作表明了极小曲面在三维拓扑中的重要性。本文所做的大部分研究都是希望通过它能得到一个正解，并能得到具有有限基本群的三流形的分类。 这希望拓扑应用于数学中的一个突出的分类问题，其根源在于研究人员彼得斯科特，查尔斯弗罗曼和S。T.你。 部分原因是与其他领域的重要联系，以及可以轻松地将美丽的经典例子制作成计算机图形，最小表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，在这一建议中概述的令人兴奋的研究问题有助于将许多年轻的科学家和数学家带到研究的前沿。

DMS-0405836Title: Research in classical minimal surface theoryPI: William H. Meeks, University of Massachusetts (Amherst)ABSTRACTProposed Research Project AbstractIn this proposal the researcher will study the geometry, asymptoticbehavior, conformal structure and topology of properly embeddedminimal surfaces in three-dimensional Euclidean space. One of the maingoals of the proposal is to classify all of the properly embedded minimal surfaces which can be parametrized by domains in the Euclidean plane andto describe the asymptotic geometry of all finite genus examples.Related theoretical techniques concerning compactness, regularity andconvergence of minimal surfaces of locally bounded genus will beinvestigated as well. One hoped for application of this research isto classify all smooth finite group actions on the three-dimensionalsphere. As an outgrowth of his recent joint manuscript with CharlesFrohman on the topological classification for minimal surfaces, theresearcher proposes to prove that Bryant surfaces in hyperbolic three-space are unknotted.Classical minimal surface theory has its roots in 18-th and 19-thcentury mathematics. Minimal surfaces are the firstimportant examples of what is called the calculus of variations, firstdescribed by Euler around 1735. Physically minimal surfaces can bemodeled locally as soap films on wires or by surfaces of least-area relative to their local boundaries. Minimal surfaces play an importantrole as a tool in the study of three-dimensional topology andRiemannian geometry. The subject of minimal surfaces has a broad impact in mathematics andphysical sciences. Minimal surfaces are stationary fluid interfaces,sotheir shapes arise in many physical problems. The work in thisproposal would classify the possible physical shapes which could occuras infinite interfaces. Many of the known examples of minimal surfacesare observed physically, so it is of interest to have a rigoroustheorem which predicts the shapes which can occur.The research proposed here strongly impacts the area of classicaldifferential geometry of surfaces in three-dimensional Euclideanspace. As is well known to geometers, minimal surface theory has beenand continues to be one of the principal tools for proving theorems ingeneral relativity and three-dimensional topology. One well-knownsuch application is Schoen and Yau's proof of the Positive MassConjecture. Recent work of Gabai on the Generalized Smale Conjectureshows the continued importance of minimal surfaces inthree-dimensional topology. Most of the research proposed here is related to andmotivated by the hope that it will lead to a positive solution of thePitts-Rubenstein Conjecture and to the classification ofthree-manifolds with finite fundamental group. This hoped fortopological application to one of the outstanding classificationproblems in mathematics has its roots in previous joint research bythe researcher, Peter Scott, Charles Frohman and S. T. Yau. In partbecause of the important connections with other areas of mathematicsand the ease in which it is possible to make computer graphicspictures of beautiful classical examples, minimal surfaces continue tobe one of the principal topics for popular science articles and publicscience exhibits. Thus, indirectly, the exciting research problemsoutlined in this proposal help bring many young scientists andmathematicians to the frontiers of research.