This research project studies two important geometric objects: minimal surfaces and mean curvature flows. A minimal surface mathematically models the shape of a soap film; the energy of such a film is proportional to its surface area, and so stable configurations are those with least area, that is, minimal surfaces. The theory of minimal surfaces directly connects to problems arising in physics, chemistry, biology, and materials science. More broadly, minimal surfaces are an important model for many geometric variational problems -- that is, problems where one seeks to find and study the properties of geometric objects that are optimal in some sense. In addition to being a fundamental principle in the physical sciences, variational problems arise in diverse areas of pure and applied mathematics. In contrast to minimal surfaces, which are static, the mean curvature flow is a dynamic process. Roughly speaking, mean curvature flow continuously deforms a surface in a manner that decreases area as quickly as possible. It was first studied as a model of certain phenomena in materials science and has also found applications in computer graphics and image recognition. Furthermore, it is closely related to the Ricci flow that was employed in the solution of the Poincaré conjecture. As such, the mean curvature flow has promising potential applications to topology, several of which are explored by this project.
This project will use the mean curvature flow to investigate hypersurfaces in n-dimensional Euclidean space of low entropy, that is, hypersurfaces for which a natural measure of geometric complexity, the entropy, is small. It also studies properties of minimal surfaces in Euclidean three-space using a variety of techniques. The first goal is to better understand properties, especially topological ones, of hypersurfaces of low entropy. This requires the investigation of the structure of non-compact self-similar (both shrinking and expanding) solutions to the mean curvature flow. The overarching objective is to see if hypersurfaces of low entropy in Euclidean four-space must smoothly bound a closed ball. This question is closely related to the smooth four-dimensional Schoenflies conjecture, an important open problem in low-dimensional topology. The project also studies several problems connected to the theory of minimal surfaces. Chief among these is the question raised by Calabi, refined by Yau, and partially answered by Colding-Minicozzi, asking whether a complete, embedded minimal surface is properly embedded. In addition, the project explores the relationship between ideas in projective differential geometry and minimal surface theory. This includes studying an analog of the Korteweg-de Vries equation and investigating free-boundary minimal surfaces in the ball.
