Minimal Surfaces in Geometric Variational Problems

微分几何是欧几里得几何的现代版本，它研究任意维数的曲面内部的形状。除了“长度”和“角度”之外，微分几何中的两个关键概念是：“最小曲面”，它概括了直线的概念，以及“曲率”，它衡量了曲面的弯曲程度。微分几何的早期发展部分要归功于物理学。在20世纪早期，爱因斯坦提出了一个几何引力理论，即广义相对论，他断言我们生活在一个弯曲的四维世界（“时空”）中，能量和质量是时空曲率的表现，最小表面是黑洞边界的象征。目前，微分几何和极小曲面是几个物理理论和活跃的数学研究方向的核心。首席研究员（PI）将研究由广义相对论和多组分合金系统的van der Walls- Cahn- Hilliard相变理论驱动的关于最小表面的问题。该项目还将支持提案人通过暑期学校、研讨会、会议以及说明性文章和笔记来促进学生的学习、包容和培训。本项目跨越微分几何最小曲面理论的三个相关的活跃研究领域。首先，PI将研究作为限制最小-最大相变的最小曲面的构造。这种结构最近被证明具有某些理想的稳定性，从而导致PI和Chodosh在三维空间中证明了“多重性1最小-最大猜想”。PI将继续这个项目，以产生不同曲率和不同维度的表面，以及更好地理解稳定相变的几何含义。其次，作为理解非负标量曲率流形极限的一步，PI将通过其嵌入的最小表面的固有弯曲效应来研究具有非负标量曲率的光滑和非光滑三维和四维流形。第三，PI将应用对最小曲面弯曲效应的研究来研究广义相对论中不同质量概念之间的推测关系，其中时间对称初始数据集恰好是具有非负标量曲率的流形。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。

Differential geometry is a modern version of Euclidean geometry that studies shapes inside curved surfaces in any number of dimensions. Two key notions in differential geometry, besides "length" and "angle," are: "minimal surfaces," which generalize the concept of a straight line, and "curvature," which measures how a surface is bent. The early development of differential geometry is partly attributable to physics. In the early 20th century, Einstein formulated a geometric theory of gravitation, general relativity, asserting that we live in a curved four-dimensional world ("spacetime") where energy and mass are manifestations of the curvature of spacetime, and minimal surfaces are indicative of black hole boundaries. Nowadays, differential geometry and minimal surfaces are at the heart of several physical theories and active mathematical research directions. The principal investigator (PI) will study problems regarding minimal surfaces that are motivated by general relativity and by the van der Walls--Cahn--Hilliard theory of phase transitions for multicomponent alloy systems. This project will also support the proposer's efforts to promote student learning, inclusion, and training through summer schools, workshops, and conferences, as well as via expository articles and notes.This project spans three related active research areas of minimal surface theory in differential geometry. First, the PI will investigate the construction of minimal surfaces as limiting min-max phase transitions. This construction has been recently shown to exhibit certain desirable stability properties that led to the proof of the "multiplicity one min-max conjecture" in three dimensions by the PI and Chodosh. The PI will continue this program, to produce surfaces with different curvatures and in different dimensions, as well as to better understand geometric implications of stable phase transitions. Second, as a step toward understanding limits of manifolds with nonnegative scalar curvature, the PI will study smooth and non-smooth three- and four-dimensional manifolds with nonnegative scalar curvature via the inherent bending effects of their embedded minimal surfaces. Third, the PI will apply this study of bending effects of minimal surfaces to investigate conjectured relationships between different mass notions in general relativity, where time-symmetric initial data sets are precisely manifolds with nonnegative scalar curvature.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.