Moduli and Limits of Minimal Surfaces

摘要：项目负责人：Matthias weber本研究项目旨在结合两种强大的新方法来研究欧氏空间中完备的、适当嵌入的极小曲面的模空间及其极限：平锥度量是表示黎曼曲面和（可能是多值的）亚纯1型的几何方法，给出了关于形式周期的直接信息。结合Teichmuller理论，将几何理论应用于最小曲面的存在性和分类问题。结点曲面是黎曼曲面在保角缩紧曲线下的自然极限。它们被用来构造退化到合适的极限的最小曲面族，使用隐函数定理。我们的目的是将最小曲面的节点极限描述为平锥度量的几何极限。这在以前考虑的几何和保形极限之外增加了第三种极限。保形极限忽略了最小曲面信息，只考虑了黎曼曲面极限，而几何极限保留了最小曲面性质，但通过在空间中重新缩放曲面，保留了松散的保形和拓扑信息。新的圆锥公制限制将包含这两种类型的信息。我们希望这一描述将导致新的示例和分类结果。本研究将以目前正在开发的基于最小曲面库的数值和图形实验为基础。最小表面是自然界在不同尺度上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非常重要。将表面张力最小化的物理目标转化为250多年来一直感兴趣的数学方程：最小表面方程正好处于我们通过一般理论理解的和我们只能通过数值分析的边界上。在这一点上的任何进步都很可能对数学、物理和工程中的其他方程产生影响。用于研究最小曲面的方法范围从几何分析到数值数学。来自偏微分方程和teichmuller理论的最新理论进展使我们能够研究“极端”最小表面，其可与变形下几乎破裂的肥皂膜相媲美。了解这些极端表面不仅有助于我们通过将它们分解成更简单的部分来分析我们所拥有的例子，而且还可以通过将合适的部分组合在一起来构建令人兴奋的新表面。我们进行的计算机实验需要对所涉及的公式进行详细的符号操作，高精度的数值计算以获得准确的三维表面数据，以及高性能的计算机图形以可视化实际表面。

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.