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.

该研究项目涉及几个与自然界系统所呈现的形状有关的几何和物理问题的数学问题。正在研究的数学模型与界面形成和表面形状的物理描述有关，例如基板上和容器中的液滴。大部分工作都集中在对此类系统的几何特性进行建模的方程解的稳定性上。 该项目旨在进一步发展由表面张力以及其他重要系统控制的现象的数学分析。所研究的变分问题涉及常平均曲率曲面、规定曲率方程、曲率流和等周比较定理。该项目的主要目标是获得平均曲率几乎恒定的表面的清晰定量描述，这将导致毛细管理论中重要的新结果。该项目的其他目标是开发基于非局部表面能的毛细管理论，并以定量形式解决涉及超曲面的内在或外在曲率的各种刚性定理。在后一个方向上，该程序将解决规定标量曲率方程、高斯曲率消失表面上的 Pogorelov 定理以及 Levy-Gromov 等周比较定理的定量分析。将投入大量精力通过变分法、偏微分方程、几何测度论和公共交通理论的研究来培养学生。