Mean Curvature Flow, Manifolds with Ricci curvature bounds, Representations of Isometry groups, and Eigenfunctions

平均曲率流、具有 Ricci 曲率界限的流形、等距群的表示以及本征函数

• 批准号: 1104392
• 负责人: Tobias Colding
• 金额: $ 38.9万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104392&HistoricalAwards=false
• 关键词: Mean; Curvature; Flow; Manifolds; Ricci

The first part of the proposed research concerns what happens to a hypersurface under the mean curvature flow. We are particularly interested in hypersurfaces that are in general or generic position before the flow start. The mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. Thus, in some sense, the topology is encoded in the singularities. The proposed project further develops the theory of generic mean curvature flow that the PI has initiated with Minicozzi. We have already classified all generic singularities and one of the main task now is to completely understand the flow itself and show that indeed a generic hypersurface only go through generic singularities. The first key point is to prove a stable manifold theorem, that is, to prove that in a neighborhood of an unstable self-shrinker the stable manifold is contained in a hyper-graph. We expect that these results will have a number of applications and plan to pursue them. A second part concerns manifolds and spaces with Ricci curvature bounds and give some new estimates for these manifolds and various applications of these estimates, in particular to Einstein metrics. A third part concerns representations of isometry groups and fundamental groups of open manifolds into general linear groups of finite dimensional vector spaces. The final and smaller part is about bounds for nodal sets (or zero sets) of eigenfunctions. When a surface evolves over time by locally moving it in the direction where the area decreases the fastest it is said to be moving by mean curvature flow. Mathematically, this leads to a nonlinear partial differential equation which is formally similar to the equation that governs the flow of heat in physics. Moving interfaces occur in a wide range of scientific and engineering applications. Mean curvature flow and other geometric flows were developed for their intrinsic beauty as well as their potential applications to other fields to model, for instance, option pricing, motion of grains in annealing metals, and crystal growth. While key foundational results have been obtained, several of the most basic questions remain unanswered. Many applications are expected both within and outside mathematics.

研究的第一部分是关于平均曲率流下超曲面的变化。我们对流开始前处于一般或一般位置的超曲面特别感兴趣。平均曲率流是体积的负梯度流，因此任何超曲面都以体积最陡峭的下降方向流过超曲面，最终在有限时间内消失。在它灭绝之前，当它经历奇点时，拓扑可能会发生变化。因此，在某种意义上，拓扑被编码在奇点中。拟议的项目进一步发展了PI与Minicozzi共同发起的一般平均曲率流理论。我们已经对所有的泛型奇点进行了分类，现在的主要任务之一是完全理解流本身，并证明泛型超曲面确实只通过泛型奇点。第一个关键点是证明一个稳定流形定理，即证明在一个不稳定自缩器的邻域中，稳定流形包含在一个超图中。我们预计这些成果将有许多应用，并计划继续下去。第二部分讨论了具有Ricci曲率界的流形和空间，并给出了这些流形的一些新的估计以及这些估计的各种应用，特别是对于爱因斯坦度量。第三部分是关于开流形的等距群和基本群到有限维向量空间的一般线性群的表示。最后也是较小的部分是关于特征函数的节点集(或零集)的界。当一个曲面随着时间的推移而沿着面积减小最快的方向局部移动时，它就被称为按平均曲率流移动。在数学上，这导致了一个形式上类似于物理中控制热流的方程的非线性偏微分方程式。移动界面存在于广泛的科学和工程应用中。平均曲率流和其他几何流动是因为它们的内在美以及它们在其他领域的潜在应用而被发展起来的，例如期权定价、退火金属中的颗粒运动和晶体生长。虽然已经取得了关键的基础性成果，但仍有几个最基本的问题没有得到回答。预计在数学内外都会有很多应用程序。