平均曲率流中非紧Self-shrinkers的结构

During the last thirty years, curvature flows, such as ricci flow and mean curvature flow, are considered to be a very powerful tool to study the geometry and topology of the given manifolds. The mean curvature flow generally meets a singularity in finite time, as the result of the surface disappearing in a point, by thin neck formations and many other more complicated possible behaviors.So one of the most important problems in mean curvature flow is to understand the possible singularities that the flow goes through. Self-shrinkers play an important role, which describe all possible blow ups at a given singularity of a mean curvature flow. There are many results about the classification and rigidity of self-shrinkers. But a classical expectation in the field is the following: Any smooth complete embedded self-shrinker in R3 with at most quadratic area growth has a finite number of infinite ends, each of which are either asymptotic to a smooth cone or a cylinder. In this project, we will study the structure of complete noncompact embedded self-shrinkers. Furthermore, we will estimate the number of ends and the asymptotic behavior of ends of self-shrinker under some condition.

近三十年来，Ricci流、平均曲率流等曲率流一直被认为是研究给定流形几何与拓扑的强有力的工具。平均曲率流一般会在有限时间能产生奇点，曲面可能会收缩为一个点，或者曲面的一部分会形成细长的neck型奇点，或者是其他更为复杂的行为。所以平均曲率流中最重要的研究问题之一就是了解其产生的奇点。在奇点分析中self-shrinker起到了至关重要的作用，因为它描述了各种奇点爆破的情形。目前已经有很多关于self-shrinkers分类，刚性和gap定理等结果，很多结果都表明下面T.Ilmanen的猜测很有可能成立： R3中光滑完备嵌入的面积具有至多平方增长的self-shrinkers，具有有限个末端，每个末端渐近为光滑锥或者柱面。 在本项目中，我们将研究完备非紧self-shinkers的结构，具体的我们将估计在一定条件下self-shrinker末端的个数，研究末端的渐近行为。