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

几何变分演算中的刚性、稳定性、正则性和解析定理

• 批准号: 2247544
• 负责人: Francesco Maggi
• 金额: $ 64.14万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2028-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247544&HistoricalAwards=false
• 关键词: Rigidity; Stability; Regularity; Resolution; Theorems

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.

该项目研究了一套用于物理学领域的数学模型，如广义相对论或表面张力理论。重点是这些模型的方面，需要开发新的数学工具和思想。这种基础研究促进了数学的发展，同时使其扎根于自然科学。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。肥皂膜通常被建模为二维表面，这种方法在第一近似中是正确的，但无法理解它们的一些物理特性，如破裂或膨胀。研究人员最近在毛细理论的背景下开始了对肥皂膜的研究（肥皂膜作为具有正体积的三维区域），该项目将专注于挑战GMT（几何测量理论）和变分法边界的几个新问题，旨在了解体积约束和几何特征（如塌陷和厚度）之间的相互作用。研究人员还打算重新制定的肥皂膜毛细作用模型的扩散界面毛细作用的背景下，从而增加了第二个长度尺度需要进一步调查崩溃的现象。扩散界面毛细作用也被用来提出一个偏微分方程（PDE）的方法来解决长期存在的问题，关于保体积平均曲率流。PDE和GMT方法是该项目的核心部分，研究人员将系统分析一种新的几何规律性定理（中尺度平坦性准则）及其在广义相对论中研究大体积外部等周性和无穷大稳定常数平均曲率叶理的应用。最后，作为项目的一部分，还将利用最优质量传输理论的方法，研究Yamabe和Kazdan-Werner问题在黎曼几何中的自然推广的新方法。该奖项反映了NSF的法定使命，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。