Collaborative Research: FRG: Minimal Surfaces, Moduli Spaces, and Computation

合作研究：FRG：最小曲面、模空间和计算

• 批准号: 0139887
• 负责人: Michael Wolf
• 金额: $ 42.92万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139887&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Minimal; Surfaces

The global theory of minimal surfaces in space is in a phase ofexplosive growth. Many new methods of constructing completeembedded minimal surfaces have recently been found; in place of adearth of examples just a few years ago, we now have a quitevaried collection of surfaces, including infinite families. Abasic problem is to classify these examples, i.e. collect theminto families with common properties and understoodlimits. Fruitful approaches have recently been developed thatcombine numerical simulation with methods from the theory ofgeometric structures on surfaces and classical complex analysis,notably Teichmuller theory. Some of the problems the team willattack are: Are there embedded minimal surfaces with oneheliciodal end and arbitrary genus? Is the classical Scherksurface the unique desingularization of a pair of planes? Ofwhat families is the Scherk surface the limit point? At the sametime, the group hopes to make progress on simulation of minimalsurfaces. For example, we hope to set up a library of Weierstrassrepresentations of minimal surfaces which is reproducible, fullydocumented, and useful as a research tool.A guiding philosophy in many areas of science, from physics tobiochemistry to ecology, is that nature is maximally efficient;indeed, many explanations of natural phenomena have at theirfoundation the assumption that the phenomenon has optimized someor several of its features in the expression we witness. At itsbase, this philosophical principle is mathematical in nature: wesearch for principles in science that can be formulated asextremal problems. In mathematics, we can make this assumption ofoptimality very rigorous by expressing it as an equation. Thisleaves us with the problem of understanding all of the solutionsof that equation. In this project, we aim to study one very richtype of optimization problem, the minimal surface problem, whichis already known to have a number of quite subtlecharacteristics. (A minimal surface is one for which each smallpiece has less area than any other surface with the sameboundary.) The study of these surfaces has its origins inphysical problems studied first by Euler; then, a century later,the problem also arose in the studies of the behavior of rotatingdroplets and soap films by F. Plateau. Today the applicationsrange from cosmology to the understanding of the structure ofstable periodic structures in compound copolymers. As in manyother optimization problems, for the minimal surface problem, wedo not have much general information about solutions to theequation expressing extremality. At present though, we do have awide variety of examples which help to guide our intuition, andwhich we are beginning to organize. It is thus a good modelproblem, enriching our understanding of all optimizationproblems.

空间最小曲面的全球理论正处于爆炸性增长的阶段。 最近发现了许多构建完整嵌入最小曲面的新方法；几年前我们还缺乏例子，现在我们拥有了各种各样的表面集合，包括无限的族。基本问题是对这些示例进行分类，即将它们收集到具有共同属性和可理解的限制的族中。最近开发出了富有成效的方法，将数值模拟与表面几何结构理论和经典复分析的方法（特别是 Teichmuller 理论）相结合。该团队将解决的一些问题是：是否存在具有单螺旋末端和任意亏格的嵌入式最小曲面？经典的 Scherk 曲面是一对平面的唯一去奇异化吗？ Scherk 曲面的极限点是哪些族？与此同时，该小组希望在最小曲面的模拟方面取得进展。例如，我们希望建立一个Weierstrass最小曲面表示库，它是可重复的、有完整记录的，并且可以作为研究工具使用。从物理学到生物化学到生态学的许多科学领域的指导思想是，自然是最大效率的；事实上，对自然现象的许多解释都以这样的假设为基础：该现象已经优化了我们所目睹的表达中的一些或几个特征。 从本质上讲，这一哲学原理本质上是数学原理：我们寻找可以表述为极值问题的科学原理。在数学中，我们可以通过将其表达为方程来使这种最优性假设变得非常严格。 这给我们留下了理解该方程的所有解的问题。在这个项目中，我们的目标是研究一种非常丰富的优化问题，即最小曲面问题，已知它具有许多非常微妙的特征。 （最小曲面是指每个小块的面积都小于具有相同边界的任何其他曲面的面积。）这些曲面的研究起源于欧拉首先研究的物理问题；一个世纪后，F. Plateau 对旋转液滴和肥皂膜行为的研究也出现了这个问题。如今，其应用范围从宇宙学到理解复合共聚物中稳定周期结构的结构。与许多其他优化问题一样，对于最小曲面问题，我们没有太多关于表达极值方程的解的一般信息。但目前，我们确实有各种各样的例子可以帮助指导我们的直觉，并且我们正在开始组织这些例子。因此，这是一个很好的模型问题，丰富了我们对所有优化问题的理解。