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Submanifolds and Foliations in Riemannian Manifolds

黎曼流形中的子流形和叶状结构

基本信息

批准号：
1810913
负责人：
Marco Radeschi
金额：
$ 15.48万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810913&HistoricalAwards=false
关键词：
Submanifolds Foliations Riemannian Manifolds

项目摘要

Differential geometry is the study of the geometry of spaces. Sometimes these spaces exist as physical objects such as the surface of planet or a cell, or the universe itself; sometimes they appear as abstract spaces in applied situations, such as the set of all possible states of a physical system, the set of possible configurations of a machine, or the set of strategies of a group of players in the stock market. In these situations, it is often important to find and study the properties of objects inside these spaces that are optimal in some sense: for example, the shortest way to move between two configurations, or the surface that minimizes area or surface tension. Sometimes these geometric sub-objects come in families (e.g., foliations that slice the space into thin pieces), and it is important to study not just each of these pieces by itself, but the whole family as a unique entity. This project will study several instances of these important and very natural objects, and draw new bridges between these geometric concepts and other more abstract areas of mathematics.The main goal of the project is to study certain structures of Riemannian manifolds such as singular Riemannian foliations and closed minimal submanifolds, and relate them to the geometry and topology of the ambient space. In the study of singular Riemannian foliations, the main focus will be on laying new theoretical ground (e.g., continuing the development of an invariant theory in this context) and applying these concepts in different areas, such as producing new manifolds with positive and non-negative curvature, or exhibiting new immersed minimal hypersurfaces in round spheres and projective spaces. In the study of closed minimal submanifolds, the main focus will be on the index of such objects, especially in the two extreme cases of closed geodesics and minimal hypersurfaces. The PI will continue working on a conjecture of Berger on manifolds with all geodesics closed, and on a conjecture of Marques-Neves-Schoen on minimal hypersurfaces in manifolds with positive Ricci curvature. In the first case, the plan is to improve the understanding of the equivariant Morse theory on the free loop space of spaces with all geodesics closed. In the second case, the PI will continue to develop tools such as virtual immersions, recently developed by the PI, which could have independent interest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

微分几何是研究空间几何的学科。有时这些空间以物理对象的形式存在，如行星或细胞的表面，或宇宙本身；有时它们在应用情况中表现为抽象空间，例如物理系统的所有可能状态的集合，机器的可能配置的集合，或者股票市场中一组参与者的策略集合。在这些情况下，从某种意义上来说，寻找和研究这些空间内物体的最优属性通常是很重要的：例如，在两种构型之间移动的最短路径，或者使面积或表面张力最小化的表面。有时，这些几何子对象以家族为单位出现（例如，将空间切成薄片的叶状结构），重要的是不仅要单独研究这些小块，而且要将整个家族作为一个独特的实体来研究。这个项目将研究这些重要的和非常自然的物体的几个实例，并在这些几何概念和其他更抽象的数学领域之间建立新的桥梁。本课题的主要目标是研究黎曼流形的某些结构，如奇异黎曼叶形和闭极小子流形，并将它们与周围空间的几何和拓扑联系起来。在奇异黎曼叶理的研究中，主要的焦点将是奠定新的理论基础（例如，在此背景下继续发展不变理论）并将这些概念应用于不同的领域，例如产生具有正曲率和非负曲率的新流形，或者在圆球体和投影空间中展示新的浸入式极小超曲面。在闭极小子流形的研究中，主要关注的是闭极小子流形的指标问题，特别是在闭测地线和极小超曲面两种极端情况下的指标问题。PI将继续研究Berger关于所有测地闭合流形的猜想，以及Marques-Neves-Schoen关于正Ricci曲率流形极小超曲面的猜想。在第一种情况下，计划是提高对所有测地线闭合的空间的自由环空间的等变莫尔斯理论的理解。在第二种情况下，PI将继续开发诸如PI最近开发的虚拟沉浸等工具，这些工具可能具有独立的利益。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。