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

提出的研究的第一部分是关于在平均曲率流下的超曲面会发生什么。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，因此任何超曲面都沿着体积下降最陡的方向流过超曲面，并最终在有限时间内消失。在它灭绝之前，当它经过奇点时，拓扑结构会发生变化。因此，在某种意义上，拓扑被编码在奇异点中。拟议的项目进一步发展了PI与Minicozzi发起的一般平均曲率流理论。我们已经对所有的一般奇点进行了分类，现在的主要任务之一是完全理解流本身，并证明一般超曲面确实只经过一般奇点。第一个重点是证明一个稳定流形定理，即证明在一个不稳定自收缩器的邻域中稳定流形包含在超图中。我们预计这些结果将有许多应用，并计划继续研究。第二部分讨论了流形和具有里奇曲率边界的空间，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，特别是对爱因斯坦度量的应用。第三部分是将开流形的等长群和基本群表示为有限维向量空间的一般线性群。最后一个较小的部分是关于特征函数的节点集（或零集）的边界。当一个表面随着时间的推移，局部地向面积减少最快的方向移动时，它被称为平均曲率流。从数学上讲，这导致了一个非线性偏微分方程，它在形式上类似于物理学中控制热量流动的方程。移动界面出现在广泛的科学和工程应用中。平均曲率流和其他几何流因其内在的美以及它们在其他领域的潜在应用而发展起来，例如期权定价，退火金属中晶粒的运动和晶体生长。虽然已经获得了关键的基础结果，但几个最基本的问题仍未得到解答。期望在数学内外都有许多应用。

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.