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

空间最小曲面的全局理论正处于爆发式增长的阶段。最近发现了许多构造完整嵌入最小曲面的新方法；就在几年前，我们还没有足够的例子，而现在我们有了各种各样的曲面集合，包括无限的种类。基本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的族中并了解其限制。最近，数值模拟与曲面几何结构理论和经典复合体分析方法（特别是Teichmuller理论）相结合的方法取得了丰硕的成果。该团队将解决的一些问题是：是否存在嵌入的具有一个螺旋末端和任意属的最小表面？经典的舍克曲面是一对平面的独特的去物化吗？Scherk曲面是什么族的极限点？同时，该小组希望在最小表面的模拟方面取得进展。例如，我们希望建立一个最小曲面的weierstrassrepresentation库，它是可复制的，完整记录的，并且作为研究工具有用。从物理学到生物化学再到生态学，许多科学领域的指导思想都是：自然是最有效的；的确，许多对自然现象的解释都基于这样一个假设，即自然现象在我们所看到的表达中优化了它的某些或几个特征。从根本上说，这一哲学原理本质上是数学的：我们在科学中寻找可以公式化为极值问题的原理。在数学中，我们可以把这个最优性假设用方程来表示，使它非常严格。这就给我们留下了理解这个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，最小曲面问题，它已经知道有许多非常微妙的特征。（最小表面是指每个小块的面积小于具有相同边界的任何其他表面的表面。）这些表面的研究起源于欧拉首先研究的物理问题；然后，一个世纪之后，F. Plateau对旋转液滴和肥皂膜行为的研究也出现了这个问题。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。正如在许多其他优化问题中一样，对于最小曲面问题，我们没有太多关于表示极值的方程解的一般信息。目前，我们确实有很多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.