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.
