Minimal Surfaces in Geometric Variational Problems

微分几何是欧几里德几何的现代版本，它研究任意维数的曲面内部的形状。微分几何中的两个关键概念，除了“长度”和“角度”，是：“最小曲面”，它概括了直线的概念，和“曲率”，它测量曲面如何弯曲。微分几何的早期发展部分归因于物理学。在世纪早期，爱因斯坦提出了一个引力的几何理论，即广义相对论，声称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指示。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。主要研究者（PI）将研究由广义相对论和货车der Walls-Cahn-Hilliard多组分合金系统相变理论激发的最小表面问题。该项目还将通过暑期学校、研讨会和会议以及临时文章和笔记支持提议者促进学生学习、包容和培训的努力。该项目跨越微分几何中最小曲面理论的三个相关的活跃研究领域。首先，PI将研究最小表面的构造作为限制最小-最大相变。这种结构最近已被证明表现出某些理想的稳定性，导致证明的“多重性一个最小-最大猜想”在三维的PI和Chodosh。PI将继续这一计划，以产生不同曲率和不同尺寸的表面，以及更好地理解稳定相变的几何含义。其次，作为理解具有非负标量曲率的流形极限的一步，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将应用最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关系，其中时间对称的初始数据集是具有非负标量曲率的精确流形。该奖项反映了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而被认为值得支持。

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.