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。这些流的特征是几何对象（例如度量、映射和子流形）通过曲率等几何量的变形，并由抛物线类型的偏微分方程组成。几何流出现在许多现实世界的应用中。例如，流体和材料中移动界面的表面张力与平均曲率成正比；平均曲率流和仿射平均曲率流对于图像处理非常有用。然而，由于非线性和奇点的可能发展，尤其是拓扑变化，研究几何方程可能具有挑战性。理解这些奇点的一种方法是放大并理解解决方案在接近奇点时间时的样子，之后平滑的解决方案不再存在。在这个限制过程中，我们得到了几何方程的特殊解，称为古代解，它们在过去已经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初级研究人员的指导。该项目的目的是对非线性几何流的古代解进行分类，例如里奇流和平均曲率流。该项目将结合偏微分方程技术和几何估计来研究这些流动的古老解决方案。目标是将古代封闭非塌缩解分类为更高维的 Ricci 流（已解决 n = 2、3 的情况），假设解随着时间接近负无穷大而变为渐近圆柱。这种分类的一个动机来自于展示 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里奇流中进行手术，而无需最初假设全局曲率条件。作为与合作者完成的项目的延续，PI 将研究平均曲率流的非塌缩古代解，除了众所周知的圆柱体案例之外，这些解对于其他广义圆柱体是渐近的。该奖项反映了 NSF 的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。