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

平均曲率流下完全嵌入自相似曲面的构造

• 批准号: 0908835
• 负责人: Xuan Hien Nguyen
• 金额: $ 2.24万
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-11-16 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908835&HistoricalAwards=false
• 关键词: Construction; Complete; Embedded; Self; Similar

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.

该项目的主要目标是在平均曲率流（表面区域的梯度流）下构建完整嵌入自相似表面的新示例。构造的策略是通过调整方法N来对两个已知示例的交集进行去奇异化。Kapouleas用于构建嵌入式极小曲面;首席研究员将使用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。该项目是非常几何的性质，但它涉及到许多技术，从非线性偏微分方程。自相似曲面模拟了在某些条件下奇点附近的平均曲率流的行为;因此，这种曲面的示例的可用性对于理解奇点附近的流及其在奇点之后的延续是很重要的。不幸的是，目前只有四个已知的例子，完全自相似表面在二维嵌入在欧几里得空间。一般的分类是无望的;因此，必须找到成功的方法来构建新的例子。这个项目的更广泛的目标是更好地理解平均曲率流的奇点的形成。