Rigidity, Stability, Regularity, and Resolution Theorems in the Geometric Calculus of Variations

这个项目调查了一组在物理学领域中使用的数学模型，如广义相对论或表面张力理论。重点放在这些模型需要开发新的数学工具和思想的方面。这种基础研究促进了数学的发展，同时又使其立足于自然科学。该项目的一个重要组成部分是研究生和博士后水平的研究培训。SOAP电影被经典地建模为二维表面，这一方法在第一近似值上是正确的，但阻碍了人们对其一些物理属性的理解，如破裂或鼓起。研究人员最近开始在毛细理论(肥皂膜是具有正体积的三维区域)的背景下研究肥皂膜，这个项目将专注于挑战几何测量理论(GMT)和变分的边界的几个新问题，旨在了解体积约束和几何特征(如塌缩和厚度)之间的相互作用。研究人员还打算在扩散界面毛细作用的背景下重新表述肥皂膜毛细模型，从而增加进一步研究坍塌现象所需的第二个长度标尺。利用扩散界面毛细作用，提出了一种偏微分方程组(PDE)方法来求解长期存在的保体积平均曲率流问题。PDE和GMT方法是该项目的核心部分，研究人员将对一类新的几何正则性定理(中尺度平坦度准则)及其在广义相对论中大体积外等周测和无穷远稳定常平均曲率叶的研究中的应用进行系统分析。最后，黎曼几何中的Yamabe和Kazdan-Werner边界问题的自然推广的新方法也将作为项目的一部分，使用最优大众运输理论的方法进行研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates a set of mathematical models used in areas of Physics like General Relativity or surface tension theory. The focus is on aspects of these models that require the development of new mathematical tools and ideas. This kind of basic research advances Mathematics while keeping it grounded in the natural sciences. An important component of the project is research training, both at the graduate and the postdoctoral level.Soap films are classically modeled as two-dimensional surfaces, an approach that is correct in first approximation, but prevents the understanding of some of their physical properties such as bursting or bulging. The investigator has recently initiated the study of soap-films in the context of capillarity theory (soap films as three-dimensional regions with positive volume), and this project will focus on several new problems challenging the boundaries of GMT (Geometric Measure Theory) and the Calculus of Variations and aimed at understanding the interplay between the volume constraint and geometric features like collapsing and thickness. The investigator also intends to reformulate the soap film capillarity model in the context of diffused interface capillarity, thus adding a second length scale needed to further inquiry into collapsing phenomena. Diffused interface capillarity is also used to propose a PDE (Partial Differential Equations) approach to long standing questions concerning the volume-preserving mean curvature flow. PDE and GMT methods are central to the part of the project where the investigator will undertake a systematic analysis of a new kind of geometric regularity theorems (mesoscale flatness criteria) and their applications to the study of large-volume exterior isoperimetry and stable constant mean curvature foliations at infinity in General Relativity. Finally, novel approaches to natural generalizations of the Yamabe and Kazdan–Werner problems with boundary in Riemannian Geometry will also be studied as part of the project using methods from the theory of Optimal Mass Transport.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.