Ancient Solutions and Singularities in Geometric Flows

几何流中的古代解和奇点

• 批准号: 2105508
• 负责人: Natasa Sesum
• 金额: $ 30.57万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105508&HistoricalAwards=false
• 关键词: Ancient; Solutions; Singularities; Geometric; Flows

The focus of the project is to study singularity formation in various geometric flows. These flows are characterized by the deformation of geometric objects such as metrics, mappings, and submanifolds by geometric quantities such as curvature and consist of partial differential equations of parabolic type. Geometric flows appear in many real world applications. For example, surface tension along moving interfaces in fluids and materials is proportional to mean curvature; mean curvature flow and affine mean curvature flow are useful for image processing. However, studying geometric equations can be challenging due to nonlinearities and the possible development of singularities, especially topological changes. One way to understand those singularities is to zoom in and understand how solutions look as they approach the singular time after which a smooth solution no longer exists. During this limiting process we get special solutions to a geometric equation that are called ancient solutions, which have existed for an infinite amount of time in the past. Understanding those solutions could be useful in obtaining more topological and geometric information about a geometric object. The project will also include training of students and the mentoring of junior researchers.The aim of the project is the classification of ancient solutions to nonlinear geometric flows, such as, the Ricci flow and the mean curvature flow. This project will combine the PDE techniques and geometric estimates to study ancient solutions of these flows. The goal is to classify ancient closed noncollapsed solutions to higher dimensional Ricci flow (cases n = 2, 3 have been solved), under the assumption that solution becomes asymptotically cylindrical as time approaches minus infinity. One motivation for this classification comes from showing an analogue of the Mean Convex Neighborhood Theorem for the Ricci flow. This could potentially enable us to perform surgery in the Ricci flow in higher dimensions without assuming global curvature conditions initially. As a continuation of a completed project with collaborators, PI will investigate noncollapsed ancient solutions to the mean curvature flow that are asymptotic to other generalized round cylinders besides the well understood case of a round cylinder.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的重点是研究各种几何流中奇点的形成。这些流动的特点是几何对象的变形，如度量，映射和子流形的几何量，如曲率和组成的抛物型偏微分方程。几何流出现在许多真实的应用中。例如，流体和材料中沿运动界面的表面张力沿着与平均曲率成比例;平均曲率流和仿射平均曲率流对于图像处理是有用的。然而，由于非线性和奇点的可能发展，特别是拓扑变化，研究几何方程可能具有挑战性。理解这些奇异性的一种方法是放大并理解解在接近奇异时间时的样子，在奇异时间之后，光滑解不再存在。在这个极限过程中，我们得到了一个几何方程的特殊解，这些解被称为古代解，它们在过去存在了无限长的时间。了解这些解决方案可能有助于获得有关几何对象的更多拓扑和几何信息。该项目还将包括对学生的培训和对初级研究人员的指导，其目的是对非线性几何流（如Ricci流和平均曲率流）的古代解进行分类。本计画将联合收割机结合偏微分方程技术与几何估计来研究这些流动的古解。我们的目标是分类古代封闭的noncollapsed解决方案，高维Ricci流（情况n = 2，3已解决），假设解决方案成为渐近圆柱时间接近负无穷大。这种分类的一个动机来自于显示Ricci流的平均凸邻域定理的类似物。这可能使我们能够在更高维度的Ricci流中进行手术，而无需最初假设全局曲率条件。作为一个与合作者完成的项目的延续，PI将调查noncollapsed古代的平均曲率流的解决方案，是渐近的其他广义圆柱体除了一个圆cylinder.This奖项的情况下，充分理解反映了NSF的法定使命，并已被认为是值得通过使用该基金会的智力价值和更广泛的影响审查标准进行评估的支持。