Quantitative Analysis of Rigidity Theorems and Geometric Inequalities

这项研究项目涉及几个几何和物理问题的数学，这些问题与自然界中系统所呈现的形状有关。正在研究的数学模型涉及界面形成和表面形状的物理描述，例如衬底和容器中的液滴。许多工作都集中在对此类系统的几何特性进行建模的方程的解的稳定性上。该项目旨在进一步发展由表面张力和其他重要系统控制的现象背后的数学分析。所研究的变分问题涉及常平均曲率曲面、规定的曲率方程、曲率流和等周比较定理。该项目的一个主要目标是获得具有几乎恒定平均曲率的曲面的精确定量描述，这将导致毛细理论中具有重要意义的新结果。该项目的其他目标是发展基于非局部表面能量的毛细理论，并以定量的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，该程序将解决规定的标量曲率方程的定量分析，具有零高斯曲率的曲面上的Pogorelov定理，以及Levy-Gromov等周比较定理。将致力于通过参与变分、偏微分方程、几何测量理论和质量运输理论的研究来培养学生。

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.