Moduli and Limits of Minimal Surfaces

AbstractAward：DMS-0505557首席研究员：Matthias Weber该研究项目旨在联合收割机两个强大的新方法来研究完全的模空间，适当地嵌入在欧几里得空间中的极小曲面及其极限：平锥度量是表示黎曼曲面的年龄度量方法以及（可能是多值的）亚纯1-形式，给出关于形式周期的直接信息。与Teichmuller理论相结合，几何学已被应用于拟曲面的存在性和分类问题。节点曲面是黎曼曲面在曲线共形pinching下的自然极限。它们已被用来构造家庭的极小曲面退化到一个适当的noded限制，使用隐函数定理。我们的目标是将极小曲面的节点极限描述为平锥度量的几何极限。这增加了第三种类型的限制，以前考虑的几何和保形限制。保形极限忽略了极小曲面的信息，只考虑了黎曼曲面的极限，而几何极限则保留了极小曲面的性质，但通过在空间中重新缩放曲面，使保形和拓扑信息变得松散。新的锥度量极限将包含这两种类型的信息。我们希望这一发现将导致新的例子和分类结果。这项研究将得到基于目前正在开发的最小表面库的数值和图形实验的支持。最小表面是对自然界中不同尺度下出现的二维形状的数学抽象：我们都熟悉肥皂膜实验，但这种表面也被观察到在纳米尺度上作为嵌段共聚物之间的界面。它们的数学性质对于理解新织物的物理性质是很重要的。最小化表面张力的物理目标转化为250多年来一直令人感兴趣的数学方程：最小表面方程恰好处于我们通过一般理论理解的和我们只能数值分析的边界。在这一点上的任何进展都很可能对数学、物理和工程中的其他方程产生影响。用于研究极小曲面的方法从几何分析到数值数学都有。最近从偏微分方程和Teichmuller理论的理论进展使我们能够研究“极端”极小曲面，它类似于在变形下几乎破裂的肥皂膜。理解这些极端曲面不仅有助于我们通过将它们分解成更简单的片段来分析我们所拥有的例子，而且还允许构造of exciting令人兴奋new新surfaces表面by putting放suitablepieces块together一起.我们进行的计算机实验需要对所涉及的公式进行精心的符号操作，高精度的数值计算以获得精确的三维表面数据，以及高性能的计算机图形来可视化实际表面。

AbstractAward: DMS-0505557Principal Investigator: Matthias WeberThis research project aims to combine two powerful new methods toinvestigate moduli spaces of complete, properly embedded minimalsurfaces in euclidean space and their limits: Flat cone metrics are ageometric way to represent Riemann surfaces together with a (possiblymultivalued) meromorphic 1-form, giving immediate information aboutthe periods of the form. In combination with Teichmuller theory, conemetrics have been applied to existence and classification problems ofmimimal surfaces. Noded surfaces are natural limits of Riemannsurfaces under conformal pinching of curves. They have been used toconstruct families of minimal surfaces that degenerate to a suitablenoded limit, using the implicit function theorem. We aim for adescription of the noded limits of minimal surfaces as geometriclimits of flat cone metrics. This adds a third type of limit to theformerly considered geometric and conformal limits. The conformallimits ignore the minimal surface information and consider only theRiemann surface limit, while the geometric limit retain the minimalsurface nature but loose conformal and topological information byrescaling the surfaces in space. The new cone metric limit willincorporate both types of information. We hope that this descriptionwill lead to new examples and classification results. This researchwill be backed by numerical and graphical experiments based on aminimal surface library currently under development.Minimal surfaces are mathematical abstractions of 2-dimensional shapesthat arise at different scales in nature: We are all familiar withsoap film experiments, but such surfaces also have been observed atthe nano scale as interfaces between block copolymers. Theirmathematical properties are important for understanding the physicalnature of new fabrics. The physical goal to minimize surface tensiontranslates into a mathematical equation which has been of interest forover 250 years: The minimal surface equation is just at the borderbetween what we understand by general theory and what we only cananalyze numerically. Any advance at this point will most likely haveits effects on other equations from mathematical physics andengineering. The methods which are being used to investigate minimalsurfaces range from geometric analysis to numerical mathematics.Recent theoretical advances from partial differential equations andTeichmuller theory allow us to study 'extreme' minimal surfaces whichcomparable to soap films that nearly break under deformations.Understanding these extreme surfaces not only helps us to analyze theexamples we have by breaking them apart into simpler pieces but alsoallows the construction of exciting new surfaces by putting suitablepieces together. The computer experiments we conduct require elaboratesymbolic manipulations of the formulas involved, high-precisionnumerical computations to get accurate 3-dimensional surface data, andhigh performance computer graphics to visualize the actual surfaces.