Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow

Northwestern University, Evanston IL

Investigators

Abstract

Singularities arise naturally in many areas of science and mathematics. Sometimes they are technical obstacles to overcome, while in other cases they encode essential features of the problem being considered. In this research the focus is on singularities of minimal submanifolds and of the Lagrangian mean curvature flow. Minimal submanifolds are higher dimensional generalizations of minimal surfaces, which model soap films. Their study is very classical, but some basic questions remain unanswered about their behavior near singularities. Special Lagrangian submanifolds are a particular kind of minimal submanifolds, which have received a great deal of attention recently due to their appearance in string theory. The Lagrangian mean curvature flow is a natural evolution process by which we can attempt to find special Lagrangian submanifolds, however once again the appearance of singularities is the basic difficulty. This research project aims to understand some common features of singularities which appear in families, in contrast with most previous research that focused on isolated singularities. Progress on this problem will have applications to many other related questions in geometry and analysis. The project also includes several educational activities aimed at increasing interest and success in STEM fields at all levels. Specifically, the educational activities include: a week long summer math circle aimed at students in grades three to five; support for undergraduate research; the continuation of a summer undergraduate workshop in geometry and topology; and the continuation of a bridge program for beginning graduate students to ease transitioning to graduate school. The most basic information that can be obtained from a singularity in many geometric problems is its tangent cone or tangent flow. The question of the uniqueness of such "tangent objects" is one of the most basic problems in geometric analysis and it is only well understood when singularities are isolated. The PI will study non-isolated singularities in two related settings: minimal hypersurfaces and the Lagrangian mean curvature flow. The tools developed for understanding the uniqueness of tangent cones and flows in these settings will also have applications to important geometric problems: the classification of minimal hypersurfaces with prescribed behavior at infinity; the generic smoothness of minimal hypersurfaces; and the possibility of surgeries at singularities along the Lagrangian mean curvature flow in connection with the Thomas-Yau conjecture. In addition to these specific applications, the PI expects that the new methods introduced in connection with the above problems will have applications in other areas of geometric analysis. 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.