Generic Flows, Ricci Curvature, Heegaard Splittings, and Nodal Sets

本项目研究的一个主要课题是平均曲率流。平均曲率流在材料科学中已经使用和研究了近一个世纪，用来模拟细胞、颗粒和气泡生长等现象。MCF的计算和理论研究，以及其他类似流动的研究，在纯数学和应用数学、理论物理、材料科学和工程的不同领域产生了巨大的影响。这个项目的一个特别重点是了解典型(或通用)表面的演变。流的奇点和奇点出现的集合的结构在典型情况下比在坏的例子中更简单和更好吗？本研究项目分为四个部分。第一部分是关于在平均曲率流作用下超曲面的情况。这部分主要涉及在流动开始之前处于一般或一般位置的超曲面。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下降的方向流经超曲面，最终在有限时间内消失。该项目进一步发展了一般MCF的理论，将最近的结果推广到其他流动中。该项目的第二部分与Ricci Curvature有关。以前，PI和Naber已经得到了估计及其应用，最近PI证明了单调性公式，并与Minicozzi一起用于解决爱因斯坦度规中关于切锥唯一性的长期悬而未决的问题。项目的这一部分讨论了可能的扩展和相关的猜想。项目的第三部分涉及三维流形的Heegaard分裂理论中长期存在的分类问题：为每个封闭的三维流形展示其所有不可约Heegaard分裂的完整列表，没有重复，直到等距。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。

A major topic under study in this project is mean curvature flow. Mean curvature flow (MCF) has been used and studied in materials science for almost a century to model phenomena such as cell, grain, and bubble growth. MCF's computational and theoretical study, as well as that of other similar flows, have had enormous impact in diverse areas of pure and applied mathematics, theoretical physics, materials science, and engineering. A particular emphasis of this project is to understand the evolution of a typical (or generic) surface. Are the singularities of flow and the structure of the set where the singularities occur simpler and nicer in the typical case than in bad examples?This research project is divided into four parts. The first part concerns what happens to a hypersurface under the mean curvature flow. This part is mostly concerned with hypersurfaces that are in general or generic position before the flow starts. The mean curvature flow (MCF) 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. The project further develops the theory of generic MCF, extending recent results to other flows. A second part of the project concerns Ricci curvature. Previously estimates with applications have been obtained by the PI and Naber, and recently monotonicity formulas have been proved by the PI and used with Minicozzi to settle a long standing open problem about uniqueness of tangent cones of Einstein metrics. This part of the project discusses possible extensions and related conjectures. The third part of the project concerns the longstanding classification problem in the theory of Heegaard splittings of 3-manifolds: to exhibit for each closed 3-manifold a complete list, without duplication, of all its irreducible Heegaard splittings, up to isotopy. The final and smaller part of the proposal is about bounds for nodal sets (or zero sets) of eigenfunctions.