Geometry of Surfaces and Four-Dimensional Manifolds

该项目的目标是探索自然几何泛函的临界点，特别是在具有边界的最小曲面和四维流形领域。这些概念自然产生，例如当一个人将线框浸入肥皂溶液中，形成肥皂膜，这是一个最小表面的例子。同样，我们生活的世界可以被建模为具有三个空间和一个时间方向的四维流形。因此，这一领域的进步将在物理学、生物学和应用科学中得到应用。因此，该项目的目标是通过探索以下研究方向来推进知识。该项目还包括对学生（本科生和研究生）进行指导，并组织一所小型学校，旨在扩大代表性不足的少数民族在STEM领域的参与。本课题的第一个研究方向是研究有边界的静止曲面，考察第一和第二种变化，得出分类和唯一性结果。特别地，在Lawson和Willmore关于球面最小曲面的猜想的激励下，PI将对球的自由边界最小曲面进行平行研究。通过引入创造性的想法（雅可比-斯特克洛夫特征值，将约束解释为线性泛函），PI将开发一种方法，为进一步的研究提供框架。第二个推力研究四维流形的几何和拓扑之间的联系。PI的目标是在自然条件下对这些流形进行分类，特别是解决第二类可微球猜想，并提供对Hopf猜想之一的见解和关于爱因斯坦度量的民间猜想。指导原则是Hodge星算子在四维空间中产生几个椭圆恒等式。该项目还包括几项指导活动以及旨在扩大代表性不足的少数民族在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。

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.