Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow

最小超曲面的奇异性和拉格朗日平均曲率流

• 批准号: 2306233
• 负责人: Gabor Szekelyhidi
• 金额: $ 33.74万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-01-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306233&HistoricalAwards=false
• 关键词: Singularities; Minimal; Hypersurfaces; Lagrangian; Mean

Singularities arise naturally in many areas of science and mathematics. Sometimes they are technical obstacles to overcome, while in other cases they encode essential features of the problem being considered. In this research the focus is on singularities of minimal submanifolds and of the Lagrangian mean curvature flow. Minimal submanifolds are higher dimensional generalizations of minimal surfaces, which model soap films. Their study is very classical, but some basic questions remain unanswered about their behavior near singularities. Special Lagrangian submanifolds are a particular kind of minimal submanifolds, which have received a great deal of attention recently due to their appearance in string theory. The Lagrangian mean curvature flow is a natural evolution process by which we can attempt to find special Lagrangian submanifolds, however once again the appearance of singularities is the basic difficulty. This research project aims to understand some common features of singularities which appear in families, in contrast with most previous research that focused on isolated singularities. Progress on this problem will have applications to many other related questions in geometry and analysis. The project also includes several educational activities aimed at increasing interest and success in STEM fields at all levels. Specifically, the educational activities include: a week long summer math circle aimed at students in grades three to five; support for undergraduate research; the continuation of a summer undergraduate workshop in geometry and topology; and the continuation of a bridge program for beginning graduate students to ease transitioning to graduate school.The most basic information that can be obtained from a singularity in many geometric problems is its tangent cone or tangent flow. The question of the uniqueness of such "tangent objects" is one of the most basic problems in geometric analysis and it is only well understood when singularities are isolated. The PI will study non-isolated singularities in two related settings: minimal hypersurfaces and the Lagrangian mean curvature flow. The tools developed for understanding the uniqueness of tangent cones and flows in these settings will also have applications to important geometric problems: the classification of minimal hypersurfaces with prescribed behavior at infinity; the generic smoothness of minimal hypersurfaces; and the possibility of surgeries at singularities along the Lagrangian mean curvature flow in connection with the Thomas-Yau conjecture. In addition to these specific applications, the PI expects that the new methods introduced in connection with the above problems will have applications in other areas of geometric analysis.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.

奇点自然出现在科学和数学的许多领域。有时它们是需要克服的技术障碍，而在其他情况下它们编码了正在考虑的问题的基本特征。本研究的重点是最小子流形和拉格朗日平均曲率流的奇点。最小子流形是最小表面的高维概括，用于模拟肥皂膜。他们的研究非常经典，但关于其在奇点附近的行为的一些基本问题仍未得到解答。特殊拉格朗日子流形是一种特殊的最小子流形，由于其在弦理论中的出现而近年来受到了广泛的关注。拉格朗日平均曲率流是一个自然演化过程，通过它我们可以尝试找到特殊的拉格朗日子流形，然而奇点的出现再次成为基本困难。该研究项目旨在了解家族中出现的奇点的一些共同特征，这与之前大多数关注孤立奇点的研究形成鲜明对比。这个问题的进展将应用于几何和分析中的许多其他相关问题。该项目还包括多项教育活动，旨在提高各级 STEM 领域的兴趣并取得成功。具体来说，教育活动包括：针对三至五年级学生的为期一周的夏季数学圈；支持本科生研究；继续举办几何和拓扑学夏季本科生研讨会；在许多几何问题中，从奇点可以获得的最基本信息是其切锥或切流。这种“相切物体”的唯一性问题是几何分析中最基本的问题之一，只有当奇点被隔离时才能很好地理解。 PI 将研究两个相关设置中的非孤立奇点：最小超曲面和拉格朗日平均曲率流。为理解这些设置中的相切锥体和流动的独特性而开发的工具也将应用于重要的几何问题：在无穷远处具有规定行为的最小超曲面的分类；最小超曲面的一般平滑度；以及与托马斯-丘猜想相关的拉格朗日平均曲率流奇点手术的可能性。除了这些具体应用之外，PI 预计针对上述问题引入的新方法将在几何分析的其他领域得到应用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。