Minimal Surfaces in Geometric Variational Problems

微分几何是欧几里得几何的现代版本，它研究任意维度的曲面内部的形状。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”，它推广了直线的概念，以及“曲率”，它测量曲面的弯曲程度。微分几何的早期发展在一定程度上要归功于物理学。在20世纪初，爱因斯坦提出了引力的几何理论，广义相对论，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，最小表面指示黑洞边界。如今，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元合金系统的van der walls-Cahn-Hilliard相变理论驱动的极小曲面问题。该项目还将通过暑期学校、研讨会和会议，以及通过说明性文章和笔记来支持提出者促进学生学习、包容和培训的努力。该项目横跨微分几何中极小曲面理论的三个相关活跃研究领域。首先，PI将研究作为限制最小-最大相变的极小曲面的构造。这种结构最近被证明具有某些理想的稳定性性质，这些性质导致PI和Chodosh证明了三维空间中的“重数一极小极大猜想”。PI将继续这一计划，以生产具有不同曲率和不同维度的曲面，以及更好地理解稳定相变的几何含义。其次，作为理解非负数量曲率流形极限的一步，PI将通过嵌入极小曲面的固有弯曲效应来研究光滑和非光滑的具有非负数量曲率的三维和四维流形。第三，PI将应用这项关于极小曲面弯曲效应的研究来调查广义相对论中不同质量概念之间的猜想关系，其中时间对称的初始数据集恰恰是具有非负标量曲率的流形。这一奖项反映了NSF的法定使命，并通过使用基金会的智力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.