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（即表面积的梯度流）下构建完全嵌入自相似曲面的新实例。构造的策略是通过采用N. Kapouleas用于构造嵌入最小曲面的方法来使两个已知例子的交集去具体化；首席研究员将使用Scherk单周期极小曲面得到近似解，然后对其进行扰动得到完全自相似曲面。这个项目本质上是非常几何的，但它涉及到许多非线性偏微分方程的技术。在一定条件下，自相似曲面模拟了平均曲率流在奇异点附近的行为；因此，这些曲面的可用性对于理解在奇点附近的流动及其越过奇点的延续是很重要的。不幸的是，目前在欧几里得空间中只有四个已知的二维完全自相似曲面的例子。笼统的分类是没有希望的；因此，找到构建新实例的成功方法是至关重要的。这个项目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.