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，即把它们收集到具有共同性质和已知极限的族中。最近已经开发出富有成效的方法，将数值模拟与表面几何结构理论和经典复分析方法相结合，特别是Teichmuller理论。这个小组将要解决的问题有：是否存在具有一个螺旋端点和任意亏格的嵌入极小曲面？经典Scherk曲面是一对平面的唯一去奇异化吗？ Scherk曲面是什么族的极限点？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，我们希望建立一个魏尔斯特拉斯最小曲面表示的图书馆，它是可复制的，有充分的文献记载，并可作为一种有用的研究工具。从物理学到生物化学到生态学，许多科学领域的指导思想是，自然是最有效的;事实上，许多对自然现象的解释都是以这样一种假设为基础的，即这种现象在我们的表达中优化了它的某些或几个特征。证人 在其基础上，这一哲学原则是数学性质的：我们认为科学中的原则可以表述为极端问题。在数学中，我们可以通过将其表达为一个方程来使这种最优性假设非常严格。 这就给我们留下了一个问题，那就是如何理解这个方程的所有解。在这个项目中，我们的目标是研究一个非常丰富的优化问题，最小曲面问题，这已经知道有一些相当微妙的特点。（一个最小曲面是这样的曲面，在这个曲面上，每一小片的面积都小于具有相同边界的任何其他曲面。）对这些表面的研究起源于欧拉首先研究的物理问题;然后，世纪后，这个问题也出现在F.高原今天，它的应用范围从宇宙学到对化合物共聚物中稳定周期结构的理解.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。目前，我们确实有各种各样的例子来帮助引导我们的直觉，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.