Hypersurfaces of Low Entropy and Mean Curvature Flow

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This project concerns the geometric calculus of variations, that is the study of the properties of objects which are optimal in some sense for a geometric functional. These variational problems arise in diverse areas of pure and applied mathematics and also in many physical sciences. For example, minimal surfaces arise as minima of the the area functional and provide a mathematical model of soap films. This project focuses on a geometric functional called entropy which has drawn a lot of recent interest as a natural measure of the complexity of a surface. The main tool used to study this functional is the mean curvature flow which is a dynamic process that, roughly speaking, continuously deforms a surface in a manner that decreases its area as quickly as possible. Mean curvature flow was first studied as a model of certain phenomena in materials science and has also found applications in computer graphics and image recognition. Furthermore, as a geometric heat flow, it is closely related to the Ricci flow which was used by Perelman to solve the Poincare conjecture. The mean curvature flow also has promising potential applications to topology - some of which are explored by this project.This project will use 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, entropy, is small. The first goal is to build on work of L. Wang and the PI and 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 be smoothly deformable to the unit sphere. This question is closely related to the smooth 4D Schoenflies conjecture -- an important open problem in low-dimensional topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.