Dynamical Properties of Spaces of Minimal Surfaces

项目编号：DMS 1307953，项目负责人：Jacob bernstein该项目将研究经典极小曲面的性质，即欧几里得三维空间中作为面积临界点的曲面。重点将放在理解有趣的最小曲面类的整体性质上，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Colding和Minicozzi的深远工作，他们描述了嵌入式最小磁盘的结构。他们的工作是该领域最近许多进展的基础。PI将主要通过使用源自可积系统理论的方法来解决这些问题-特别是Korteweg-de Vries （KdV）方程的几何理论。具体而言，PI将开展初步工作，表明许多最小曲面空间具有以KdV方程为模型的动力学。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，该项目的一个首要目标是用射影几何的语言形式化这些观察结果，并理解最小表面的动力学和几何之间的相互作用。近期目标是将开发的技术应用于自由边界最小曲面的空间。这些空间特别适合这些方法，因此可以作为很好的模型空间。受Meeks、Perez和Ros对零属的适当嵌入最小曲面的研究的启发，他们的长期目标是将这一观点与更成熟的方法相结合，以回答关于有限属的嵌入最小曲面的问题。其中最主要的是Colding和Minicozzi提出的问题，即有限属的完备嵌入最小曲面是否必然是适当嵌入的。最小表面在数学上模拟了横跨固定线框的肥皂膜的形状。这是因为，粗略地说，这种薄膜的“能量”是由它的表面积决定的，所以稳定的构型是那些面积最小的构型，即最小的表面。因此，最小表面理论与物理学、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，最小曲面是许多几何变分问题的一个重要模型，这些问题是人们试图找到和研究几何物体在某种意义上最优的性质。除了作为物理科学的基本原理外，这种变分问题还出现在纯数学和应用数学的各个领域。该项目的一个具体目标是了解最小表面和KdV方程之间的关系，KdV方程是流体动力学中的一个重要方程。与静态的最小表面不同，KdV方程模拟的是一种动态物理现象，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，特别是在弦理论中很重要。KdV方程和最小曲面方程之间的关系尚不清楚，该项目可能对这种联系产生的任何见解在数学和物理学中都具有广泛的重要性。

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.