Quantitative Analysis of Rigidity Theorems and Geometric Inequalities

刚性定理和几何不等式的定量分析

• 批准号: 1565354
• 负责人: Francesco Maggi
• 金额: $ 18万
• 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: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1565354&HistoricalAwards=false
• 关键词: Quantitative; Analysis; Rigidity; Theorems; Geometric

This research project concerns the mathematics of several questions of geometric and physical interest having to do with the shapes taken on by systems in nature. The mathematical models under study are related to the physical description of interface formation and the shapes of surfaces, such as liquid droplets on substrates and in containers. Much of the work is centered on the stability of solutions to the equations that model geometric properties of such systems. The project aims to further develop the mathematical analysis underlying phenomena governed by surface tension as well as other important systems.The variational problems under study concern constant mean curvature surfaces, prescribed curvature equations, curvature flows, and isoperimetric comparison theorems. A main goal of the project is obtaining a sharp quantitative description of surfaces with almost constant mean curvature, which would lead to new results of importance in capillarity theory. Other goals of the project are developing a capillarity theory based on nonlocal surface energies, and addressing in quantitative form various rigidity theorems involving intrinsic or extrinsic curvatures of hypersurfaces. In the latter direction the program will address the quantitative analysis of prescribed scalar curvature equations, of the Pogorelov theorem on surfaces with vanishing Gauss curvature, and of the Levy-Gromov isoperimetric comparison theorem. Considerable effort will be devoted to the training of students through involvement in research on the calculus of variations, partial differential equations, geometric measure theory, and mass transportation theory.

该研究项目涉及几种几何和物理兴趣问题的数学，与系统中系统所采取的形状有关。所研究的数学模型与界面形成的物理描述和表面的形状，例如底物和容器中的液滴。大部分工作集中在解决方案对此类系统的几何特性的方程式的稳定性。 该项目旨在进一步发展受表面张力和其他重要系统控制的基本现象的基本现象。该项目的一个主要目标是获得几乎持续平均曲率的表面的尖锐定量描述，这将导致毛细血管理论中重要的新成果。该项目的其他目标是开发基于非局部表面能的毛细血性理论，并以定量形式解决各种刚性定理，涉及超出表面的内在或外在曲率。在后一个方向上，该程序将解决针对规定的标量曲率方程，PogoreLov定理在表面上具有消失的高斯曲率以及Levy-Gromov等速度比较定理的定量分析。通过研究变异，部分微分方程，几何测量理论和大众运输理论的研究，将致力于培训学生的培训。