Mean curvature flow, minimal submanifolds and Willmore surfaces

在数学中，最小曲面是指三维欧几里得空间或任意维的弯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最小表面。将最小曲面推广到任意维数可得到最小子流形。一类特殊的曲面或子流形，起源于经典力学，由拉格朗日流形组成。在弦理论——粒子物理的最新理论——中，同时是拉格朗日的最小子流形是基本的组成部分。

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.