Mean curvature flow, minimal submanifolds and Willmore surfaces

在数学上，极小曲面是指三维欧氏空间或任意维曲空间中平均曲率为零的曲面。将封闭的线框浸入肥皂溶液中，肥皂膜是一个以线框为边界的极小表面。将极小曲面推广到任意维导出极小子流形。一类独特的曲面或子流形起源于经典力学，由拉格朗日曲面或子流形组成。在弦理论--一种最新的粒子物理理论中，同时是拉格朗日的极小子流形是基本的积木。 从数学上讲，要构造这些特殊的物体是非常困难的--拉格朗日和体积最小。建立它们的一种方法是使用平均曲率流，它通过减小拉格朗日子流形的体积而保持拉格朗日子流形的拉格朗日。极小曲面在理解其周围区域的几何和拓扑方面有许多很好的应用，例如形状和曲率。一类更大的曲面，包括最小曲面，是由对平均曲率的总大小至关重要的曲面组成的，即所谓的Willmore曲面。圆形球体是Willmore的，但不是极小的。从分析上讲，极小曲面和Willmore曲面满足偏微分方程--方程涉及所有空间变量中未知函数的导数。 我们建议研究拉格朗日子流形和其他子流形、极小子流形和Willmore曲面的平均曲率流。

In mathematics, a minimal surface is a surface with zero mean curvature in the 3-dimensional Euclidean space or in a curved space of any dimension. Dipping a closed wire frame into soap solution, the soap film is a minimal surface whose boundary is the wire frame. Generalization of minimal surfaces to arbitrary dimension leads to minimal submanifolds. A distinguished class of surfaces or submanifolds, originated in the classical mechanics, consists of the Lagrangian ones. In string theory - a recent theory of particle physics, minimal submanifolds that are Lagrangian at the same time are basic building blocks. Mathematically, it is very difficult to construct these special objects - being Lagrangian and having least volume. One approach to build them is to use the mean curvature flow which evolves a Lagrangian submanifold by decreasing its volume while keeping it Lagrangian. Minimal surfaces have many nice applications in understanding the geometry and topology of their surrounding region, such as shape and curvature. A larger class of surfaces, including minimal surfaces, is made up by surfaces that are critical to the total magnitude of mean curvature - the so called Willmore surfaces. The round sphere is Willmore but not minimal. Analytically, minimal surfaces and Willmore surfaces satisfy partial differential equations - equations involve derivatives of the unknown functions in all space variables. We propose to study a selected list of topics on the mean curvature flow for Lagrangian submanifolds and other submanifolds, minimal submanifolds and Willmore surfaces.