Research in Classical Minimal Surface Theory

题目：经典极小曲面理论的研究i: William H. Meeks，马萨诸塞大学（Amherst）摘要：拟研究项目摘要：在该项目中，研究者将研究三维欧几里德空间中适当嵌入极小曲面的几何、渐近行为、共形结构和拓扑结构。该方案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。本研究的一个应用希望是对三维球面上的所有光滑有限群作用进行分类。作为他最近与CharlesFrohman合著的关于最小曲面拓扑分类的手稿的一个成果，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，或者通过相对于其局部边界的最小面积表面。极小曲面作为一种工具在三维拓扑和黎曼几何的研究中起着重要的作用。最小曲面在数学和物理科学中有着广泛的影响。最小表面是静止的流体界面，因此它们的形状出现在许多物理问题中。本提案中的工作将对可能出现无限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。本文提出的研究强烈地影响了三维欧几里得空间中曲面的经典微分几何面积。正如几何学家所熟知的那样，最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的主要工具之一。一个著名的应用是Schoen和Yau对正质量猜想的证明。Gabai最近关于广义小猜想的工作显示了最小曲面在三维拓扑中的持续重要性。这里提出的大多数研究都与pitts - rubenstein猜想的正解和具有有限基本群的三流形的分类有关。这种希望将拓扑学应用于数学中突出的分类问题之一的想法，源于研究人员Peter Scott、Charles Frohman和S. T. Yau之前的联合研究。在某种程度上，由于最小曲面与其他数学领域的重要联系，以及很容易用计算机对美丽的经典例子进行图形化处理，因此它一直是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。

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.