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

通用流、Ricci 曲率、Heegaard 分裂和节点集

• 批准号: 1404540
• 负责人: Tobias Colding
• 金额: $ 46.5万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404540&HistoricalAwards=false
• 关键词: Generic; Flows; Ricci; Curvature; Heegaard

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.

在这个项目中研究的一个主要课题是平均曲率流。 平均曲率流（MCF）在材料科学中的应用和研究已有近世纪的历史，用于模拟细胞、晶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，材料科学和工程的各个领域产生了巨大的影响。 这个项目的一个特别的重点是了解一个典型的（或通用的）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例子中更简单和更好吗？本研究项目共分为四个部分。 第一部分是关于超曲面在平均曲率流下会发生什么。这一部分主要关注在流开始之前处于一般或通用位置的超曲面。 平均曲率流（MCF）是体积的负梯度流，因此任何超曲面都沿体积的最陡下降方向流过超曲面，并最终在有限时间内灭绝。 该项目进一步发展了通用MCF的理论，将最近的结果扩展到其他流。 该项目的第二部分涉及里奇曲率。以前的估计与应用程序已获得的PI和Naber，最近的单调性公式已被证明的PI和Minicozzi解决一个长期存在的公开问题的唯一性切锥的爱因斯坦度量。项目的这一部分讨论了可能的扩展和相关的配置。 该项目的第三部分涉及3-流形的Heegaard分裂理论中长期存在的分类问题：为每个封闭的3-流形展示一个完整的列表，没有重复，所有不可约的Heegaard分裂，直到合痕。最后一个较小的部分是关于本征函数的节点集（或零点集）的界限。