Dynamical Properties of Spaces of Minimal Surfaces

AbstractAward：DMS 1307953，主要研究者：Jacob Bernstein该项目将研究经典极小曲面的性质-即欧氏三空间中的曲面，它们是面积的临界点。 重点将放在理解最小曲面的有趣类的全局属性上，例如，紧性这将通过研究类元素的几何结构来完成。 这种方法的动机是具有深远意义的工作的Colding和Minicozzi谁的特点嵌入最小磁盘的结构。 他们的工作是该领域最近取得的许多进展的基础。 PI将主要通过使用起源于可积系统理论的方法来解决这些问题-特别是Korteweg-de弗里斯（KdV）方程的几何理论。具体来说，PI将开展初步工作，这表明，许多空间的最小曲面具有动态的KdV方程建模。最小曲面和KdV方程之间的联系最好使用来自射影几何的语言和思想来理解。因此，该项目的一个主要目标是用射影几何的语言形式化这些观察，并理解最小曲面的动态和几何之间的相互作用。 一个近期的目标是将开发的技术应用于自由边界极小曲面的空间。这些特别适合于这些方法，因此可以作为良好的模型空间。受到Meeks、Perez和Ros关于零亏格嵌入极小曲面的工作的启发，一个长期目标是将这种观点与更成熟的方法相结合，以回答关于有限亏格嵌入极小曲面的问题。其中最主要的问题是由Colding和Minicozzi提出的，有限亏格的嵌入极小曲面是否一定是完全嵌入的。极小曲面在数学上模拟了跨越固定线框的肥皂膜的形状。这是因为，粗略地说，这种膜的“能量”由其表面积给出，因此稳定的构型是那些具有最小面积的构型，即，最小的表面。因此，极小曲面理论与物理学、化学、生物学和材料科学中出现的问题直接相关。 更广泛地说，极小曲面是许多几何变分问题的重要模型-这是一个问题，人们试图找到和研究的性质的几何对象是最佳的在某种意义上。 除了作为一个基本原则，在物理科学，这样的变分问题出现在不同领域的纯数学和应用数学。 该项目的一个具体目标是了解最小曲面和KdV方程之间的关系-一个在流体动力学中很重要的方程。 与静态的最小表面相反，KdV方程模拟动态物理现象，即某些一维水波的运动。KdV方程具有许多显著的数学性质，在物理学的其他领域也很重要，特别是在弦理论中。 KdV方程和最小曲面方程之间的关系知之甚少，该项目可能在这方面的任何见解都应该在数学和物理学中具有广泛的重要性。

AbstractAward: DMS 1307953, Principal Investigator: Jacob BernsteinThe project will study properties of classical minimal surfaces - that is, surfaces in Euclidean three-space which are critical points for area. The focus will be on understanding global properties of interesting classes of minimal surfaces, e.g., compactness properties. This will be done by studying the geometric structure of elements of the class. This approach is motivated by far-reaching work of Colding and Minicozzi who characterized the structure of embedded minimal disks. Their work has been the basis for much recent progress in the field. The PI will attack these questions primarily by using methods originating in the theory of integrable systems - especially the geometric theory of the Korteweg-de Vries (KdV) equation. Specifically, the PI will develop preliminary work which suggests that many spaces of minimal surfaces possess dynamics which are modeled on the KdV equation. The connection between minimal surfaces and the KdV equation is best understood using language and ideas coming from projective geometry. As such, one over-arching goal of the project is to formalize these observations in the language of projective geometry and to understand the interaction between the dynamics and the geometry of the minimal surfaces. A near-term goal is to apply the techniques developed to spaces of free-boundary minimal surfaces. These are particularly amenable to the methods and so act as good model spaces. Motivated by work of Meeks, Perez and Ros on properly embedded minimal surfaces of genus-zero, a longer-term goal is to blend this perspective with more established methods in order to answer questions about embedded minimal surfaces of finite genus. Chief among these is the question, raised by Colding and Minicozzi, whether a embedded minimal surface of finite genus which is complete is necessarily properly embedded.A minimal surface mathematically models the shape of a soap film spanning a fixed wire frame. This is because, roughly speaking, the "energy" of such a film is given by its surface area and so stable configurations are those with least area - i.e., minimal surfaces. As such, the theory of minimal surfaces directly connects to problems arising in physics, chemistry, biology and materials science. More broadly, minimal surfaces are an important model for many geometric variational problems - that is problems where one seeks to find and study the properties of geometric objects which are optimal in some sense. In addition to being a fundamental principle in the physical sciences, such variational problems arise in diverse areas of pure and applied mathematics. A specific goal of the project is to understand the relationship between minimal surfaces and the KdV equation - an equation important in fluid dynamics. In contrast to minimal surfaces, which are static, the KdV equation models a dynamic physical phenomena, namely the motion of certain one-dimensional water waves. The KdV equation possesses many remarkable mathematical properties and is important in other areas of physics, notably in string theory. The relationship between the KdV equation and the minimal surface equation is poorly understood and any insight the project may shed on this connection should have broad importance both in mathematics and in physics.