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

本课题研究的主要课题是平均曲率流。平均曲率流（MCF）已经在材料科学中应用和研究了近一个世纪，用于模拟细胞、颗粒和气泡生长等现象。MCF的计算和理论研究，以及其他类似的流动，在纯数学和应用数学、理论物理、材料科学和工程等各个领域产生了巨大的影响。这个项目特别强调的是理解一个典型（或通用）表面的演变。流的奇异点和奇异点所在的集合的结构在典型情况下比在坏例子中更简单更好吗？本研究项目分为四个部分。第一部分关注的是平均曲率流下的超曲面。这部分主要关注流开始前处于一般或一般位置的超曲面。平均曲率流（MCF）是体积的负梯度流，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。该项目进一步发展了一般MCF理论，将最近的结果扩展到其他流动。该项目的第二部分涉及利玛窦曲率。以前的估计是由PI和Naber得到的，最近的单调性公式由PI证明，并与Minicozzi一起解决了一个长期存在的关于爱因斯坦度量的切锥唯一性的开放问题。项目的这一部分讨论了可能的扩展和相关的猜想。该项目的第三部分涉及3流形heegard分裂理论中长期存在的分类问题：为每个封闭的3流形展示其所有不可约heegard分裂的完整列表，没有重复，直至同位素。该建议的最后和较小的部分是关于特征函数的节点集（或零集）的边界。

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.