Ancient Solutions and Singularities in Geometric Flows

该项目的重点是研究各种几何流动中的奇点形成。这些流的特征是几何对象的变形，如度量、映射和子流形，由几何量（如曲率）组成，并由抛物线型的偏微分方程组成。几何流出现在许多实际应用中。例如，流体和材料中沿运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在图像处理中非常有用。然而，由于非线性和奇点的可能发展，特别是拓扑变化，研究几何方程是具有挑战性的。理解这些奇点的一种方法是放大并理解解在接近奇点时的样子，之后光滑解不再存在。在这个极限过程中，我们得到一个几何方程的特殊解，这些解被称为古解，它们在过去已经存在了无限长的时间。理解这些解决方案有助于获得更多关于几何对象的拓扑和几何信息。该项目还将包括培训学生和指导初级研究人员。该项目的目的是对非线性几何流，如Ricci流和平均曲率流的古老解进行分类。该项目将结合PDE技术和几何估计来研究这些流动的古老解决方案。目标是对高维Ricci流的古老闭合非坍缩解进行分类（情况n = 2,3已被解决），假设解随着时间趋于负无穷渐近圆柱形。这种分类的一个动机来自于里奇流的平均凸邻域定理的类比。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。作为与合作者已完成的项目的延续，PI将研究平均曲率流的非坍缩古解，这些解除熟知的圆圆柱体外，还对其他广义圆圆柱体渐近。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.