Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow

该项目的主要目标是在平均曲率流下构造完全嵌入的自相似曲面的新实例，该平均曲率流是表面积的梯度流。构造的策略是通过采用N.Kapouleas构造嵌入极小曲面的方法来去奇化两个已知例子的交点；主要研究者将使用Scherk的单周期极小曲面来获得近似解，然后对其进行扰动以获得完整的自相似曲面。该项目本质上是非常几何的，但它涉及到许多来自非线性偏微分方程组的技术。自相似曲面模拟了奇点附近的平均曲率流在一定条件下的行为；因此，这种曲面的例子的可用性对于理解奇点附近的流动以及它在奇点之后的延续是重要的。不幸的是，目前只有四个已知的二维完全自相似曲面嵌入到欧几里德空间中。一般的分类是没有希望的；因此，找到成功的方法来构建新的例子是至关重要的。这个项目的更广泛的目标是更好地理解平均曲率流奇点的形成。

The main goal of the project is the construction of new examples of complete embedded self-similar surfaces under the mean curvature flow, which is the gradient flow of the surface area. The strategy for the construction is to desingularize the intersection of two known examples by adapting the method N. Kapouleas used to construct embedded minimal surfaces; the Principal Investigator will use a Scherk's singly periodic minimal surface to get an approximate solution which is then perturbed to obtain a complete self-similar surface. The project is very geometric in nature, but it involves many techniques from nonlinear partial differential equations. Self-similar surfaces model the behavior of the mean curvature flow near singularities under certain conditions; therefore, the availability of examples of such surfaces is important to the understanding of the flow near its singularities and its continuation past the singularities. Unfortunately, there are currently only four know examples of complete self-similar surfaces in dimension two embedded in the Euclidean space. A general classification is hopeless; therefore, it is essential to find successful methods to construct new examples. The broader goal of this project is to obtain a better understanding of the formation of singularities for the mean curvature flow.