Mean Curvature and Complexity of Submanifolds

This project concerns the geometric calculus of variations, that is the study of the properties of objects which are optimal or nearly optimal in various geometric senses. These variational problems arise in various areas of pure and applied mathematics and also in many physical sciences. The project is focused on studying certain measures of complexity for submanifolds and their linkage to interesting geometric partial differential equations such as the minimal surface equation and mean curvature flow. Minimal surfaces are classical geometric objects that naturally arise in the study of surface tension and, in particular, model the shape of soap films. The mean curvature flow is a dynamic process that, roughly speaking, continuously deforms a surface in a manner that decreases its 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. Finally, the project has several educational components and the organization of many workshops and seminars. The project will study several related geometric functionals defined on the space of submanifolds. These functionals share the feature of measuring complexity and are related with the theory of mean curvature. One focus is the Colding-Minicozzi entropy, which is closely linked with the mean curvature flow. Another is the Li-Yau conformal volume which has important applications to minimal surface theory and other geometric problems. The project will investigate these quantities and the ways that they interact with each other and with the theory of minimal submanifolds and mean curvature flow. A major area of emphasis will be on higher codimension submanifolds. Specifically, the PI proposes to study surfaces in certain symmetric four-manifolds. The stability operator is poorly understood when the codimension is more than one and this is a major source of difficulty blocking progress on such topics. The project will start with the simplest settings by systematically utilizing the various available techniques and by developing new ones. 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.