Geometry of Surfaces and Four-Dimensional Manifolds

曲面几何和四维流形

• 批准号: 2104988
• 负责人: Hung Tran
• 金额: $ 21.19万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104988&HistoricalAwards=false
• 关键词: Geometry; Surfaces; Four; Dimensional; Manifolds

The goal of this project is to explore critical points of natural geometric functionals, particularly in the fields of minimal surfaces with boundaries and four-dimensional manifolds. These concepts arise naturally such as when one dips a wireframe into a soap solution, forming a soap film, which is an example of a minimal surface. Similarly, the world we are living in can be modeled as a four-dimensional manifold with three spatial and one time directions. Thus, advancements in this area will have applications in physics, biology, and applied sciences. As a consequence, the project's objective is to advance knowledge by exploring the following research directions. The project also includes mentoring of students (at both the undergraduate and graduate levels) and the organization of a mini-school aimed at broadening participation of under-represented minorities in STEM fields.The first research direction in this project aims to study stationary surfaces with boundaries, examining the first and second variations to draw out classification and uniqueness results. In particular, motivated by Lawson and Willmore's conjectures for minimal surfaces in a sphere, the PI will conduct a parallel study for free boundary minimal surfaces in a ball. By introducing creative ideas (Jacobi-Steklov eigenvalue, interpreting constraints as linear functionals), the PI will develop an approach that gives a framework for further investigations. The second thrust studies the connection between the geometry and topology of a four-dimensional manifold. The PI aims to classify these manifolds under natural conditions, particularly resolving a differentiable sphere conjecture of the second kind and providing insights on one of Hopf's conjectures and a folklore conjecture about Einstein metrics. The guiding principle is that the Hodge star operator gives rise to several elliptic identities in dimension four. The project also includes several mentoring activities as well as impactful activities aimed at broadening participation of under-represented minorities in STEM fields.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.

该项目的目的是探索自然几何功能的临界点，尤其是在具有边界和四维流形的最小表面领域。这些概念自然出现，例如当一个人将线框浸入肥皂溶液中时，形成了肥皂膜，这是最小表面的一个例子。同样，我们所生活的世界也可以建模为具有三个空间和一个时间方向的四维流形。因此，该领域的进步将在物理，生物学和应用科学中应用。 结果，该项目的目标是通过探索以下研究方向提高知识。 该项目还包括对学生的指导（在本科和研究生级别）以及旨在扩大代表性不足的少数群体参与STEM领域的小学生的组织。该项目的第一个研究方向旨在研究边界的固定表面，以研究第一和第二个变体，以吸引分类和独特性和独特性结果。特别是，由劳森（Lawson）和威尔莫尔（Willmore）对球体中最小表面的猜想进行的，PI将对球中的自由边界最小表面进行平行研究。通过引入创意（Jacobi-Steklov特征值，将约束为线性功能解释），PI将开发一种方法，为进一步研究提供框架。 第二个推力研究了四维流形的几何形状和拓扑之间的联系。 PI的目的是在自然条件下对这些歧管进行分类，尤其是解决第二类的可区分领域，并提供有关Hopf的一种猜想和关于爱因斯坦指标的民俗猜想的见解。指导原则是，霍奇恒星操作员在第四维中产生了几个椭圆形身份。 该项目还包括几项指导活动以及旨在扩大代表性不足少数民族在STEM领域的参与的有影响力的活动。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估评估的评估来审查标准的。

项目成果

On the Morse index with constraints: An abstract formulation

关于带有约束的莫尔斯指数：一个抽象的表述

• DOI: 10.1016/j.jmaa.2023.127317
• 发表时间: 2023
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Tran, Hung;Zhou, Detang
• 通讯作者: Zhou, Detang

Curvature of the second kind and a conjecture of Nishikawa

第二类曲率与西川猜想

• DOI: 10.4171/cmh/545
• 发表时间: 2023
• 期刊: Commentarii Mathematici Helvetici
• 影响因子: 0.9
• 作者: Cao, Xiaodong;Gursky, Matthew;Tran, Hung
• 通讯作者: Tran, Hung

Stationary surfaces with boundaries

有边界的静止表面

• DOI: 10.1007/s10455-022-09850-4
• 发表时间: 2022
• 期刊: Annals of Global Analysis and Geometry
• 影响因子: 0.7
• 作者: Gruber, Anthony;Toda, Magdalena;Tran, Hung
• 通讯作者: Tran, Hung

On the Morse Index with Constraints for Capillary Surfaces

• DOI: 10.1007/s12220-022-01135-3
• 发表时间: 2023-02
• 期刊: The Journal of Geometric Analysis
• 作者: Hung Tran;Detang Zhou

FIRST STABILITY EIGENVALUE OF SINGULAR HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPHERES

球内平均曲率恒定的奇异超曲面的第一稳定特征值

• 发表时间: 2022
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Juncheol, Pyo;Nguyen, Dung;Tran, Hung
• 通讯作者: Tran, Hung