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.

该项目中研究的一个主要主题是平均曲率流。 近一个世纪以来，在材料科学中使用并研究了平均曲率流（MCF），以模拟诸如细胞，谷物和气泡生长等现象。 MCF的计算和理论研究以及其他类似的流量对纯数学和应用数学，理论物理，材料科学和工程的不同领域产生了巨大影响。 该项目的一个特殊重点是了解典型（或通用）表面的演变。 在典型情况下，奇异性发生的奇异性和奇异性发生在不良示例中更简单和更好的集合的结构是否比在不良示例中更简单？该研究项目分为四个部分。 第一部分涉及在平均曲率流下进行超表面的情况。该部分主要涉及流动开始之前的高度或通用位置的高度曲面。 平均曲率流（MCF）是体积的负梯度流，因此任何高表面流经最陡下降的高度下降方向的高度曲面，并最终在有限的时间内灭绝。 该项目进一步发展了通用MCF的理论，将最新结果扩展到其他流。 该项目的第二部分涉及RICCI曲率。以前，PI和NABER已获得了具有应用的估计值，最近PI证明了最近的单调性公式，并与Minicozzi一起使用，以解决有关爱因斯坦指标切线的独特性的长期开放问题。该项目的这一部分讨论了可能的扩展和相关猜想。 该项目的第三部分涉及3个manifolds的Heegaard分裂理论中的长期分类问题：为每个封闭的3个manifold展示一个完整的列表，而无需重复，其所有不可还原的Heegaard拆分，直至同属。该提案的最终和较小部分是关于特征函数的节点集（或零集）的边界。