Quantitative Analysis of Rigidity Theorems and Geometric Inequalities

这个研究项目涉及几何和物理兴趣的几个问题的数学，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。大部分的工作集中在稳定性的解决方案的方程模型的几何性质，这样的系统。 该项目的目的是进一步发展数学分析的基础上的现象所支配的表面张力以及其他重要的系统。变分问题的研究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一个尖锐的定量描述的表面几乎恒定的平均曲率，这将导致新的结果的重要性毛细理论。该项目的其他目标是发展一个基于非局部表面能的毛细作用理论，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。在后一个方向的计划将解决定量分析规定的标量曲率方程，Pogorelov定理的表面消失高斯曲率，和Levy-Gromov等周比较定理。相当大的努力将致力于通过对变分法，偏微分方程，几何测量理论和质量运输理论的研究参与学生的培训。

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.