Mean curvature flow in higher co-dimensions

在这个研究项目中，我们打算研究黎曼流形或Kähler子流形的高维余维的平均曲率流(MCF)的解析性质及其对子流形的拓扑和几何的影响。具体地说，我们的主要目标之一是使用MCF来理解黎曼流形或Kählerian流形之间的光滑映射的拓扑和几何。该方案可以通过在乘积空间中用平均曲率流来演化相应的图形子流形来实现。研究图的平均曲率流的一个主要动机是Gromov提出的一个问题，即给定的黎曼流形之间的映射的雅可比如何影响映射的同伦型。第二个动机是Yau提出的一个一般性问题，即如何将一个辛同构变形为一个同态映射。我们想提的是，在过去的十年里，这些问题引起了许多数学家的注意。另一个目的是了解MCF的奇异性。特别是，我们感兴趣的是详细研究平均曲率流的平移孤子。马尔可夫随机场奇异性分析的主要问题之一是对这类对象的分类。我们解决这个问题的方法是将椭圆型偏微分方程组的分析与极小子流形理论的技巧结合起来。

In this research project we intend to investigate analytic properties of the mean curvature flow (MCF) in higher co-dimension of submanifolds of a Riemannian or Kählerian manifold and its implications on the topology and geometry of submanifolds. Specifically, one of our main objectives is to use MCF to understand the topology and geometry of smooth maps between Riemannian or Kählerian manifolds. This plan can be achieved by evolving the corresponding graphical submanifold in the product space by the mean curvature flow. A prime motivation for the study of the graphical mean curvature flow is a question posed by Gromov of how the Jacobians of a map between given Riemannian manifolds affects the homotopy type of the map. A second motivation is a general question of Yau of how to deform a symplectomorphism into a homolorphic map. We would like to mention that these problems have attracted the attention of many mathematicians during the last decade. Another aim is to understand the singularities of the MCF. In particular, we are interested to study in detail translating solitons of the mean curvature flow. One of the major problems of singular analysis of MCF is the classification of such objects. Our approach to attack this problem is to combine the analysis of elliptic systems of PDEs with techniques arising from minimal submanifold theory.