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

空间极小曲面的整体理论正处于爆炸性发展阶段。最近发现了许多构造完全嵌入极小曲面的新方法；取代了几年前的大量例子，我们现在有了各种各样的曲面集合，包括无限族。一个基本的问题是对这些例子进行分类，即将它们收集到具有共同属性和理解限制的族中。近年来，将数值模拟与曲面几何结构理论和经典复分析方法相结合，取得了丰硕的成果，其中最著名的是TeichMuller理论。该团队将解决的一些问题是：是否存在具有一个螺旋节末端和任意亏格的嵌入极小曲面？经典ScherkSurface是一对平面的唯一去单形化吗？谢尔克曲面是什么家族的极限点？与此同时，该小组希望在极小曲面的模拟方面取得进展。例如，我们希望建立一个极小表面的魏尔斯特拉表示库，它是可重现的，有完整记录的，并可用作研究工具。从物理到生物化学再到生态学，许多科学领域的指导哲学是自然是最有效的；事实上，许多对自然现象的解释都建立在这样的假设基础上，即这种现象在我们看到的表达式中优化了它的一些或几个特征。从根本上说，这一哲学原则本质上是数学的：我们在科学中寻找可以被表述为极端问题的原则。在数学中，我们可以通过将其表示为方程来使这一最优性假设非常严格。这就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类非常丰富的优化问题--极小曲面问题，它已经被认为具有许多非常微妙的特征。(最小曲面是指每个小块的面积比具有相同边界的任何其他曲面都小的曲面。)对这些表面的研究起源于欧拉首先研究的物理问题；一个世纪后，这个问题也出现在F·普拉特对旋转液滴和肥皂膜的行为的研究中。今天的应用范围从宇宙学到了解化合物共聚物中稳定的周期结构的结构。与许多其他优化问题一样，对于极小曲面问题，我们没有太多关于表示极值的方程的解的一般信息。然而，目前我们确实有各种各样的例子来帮助指导我们的直觉，我们正在开始组织这些例子。因此，它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。

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.