Singularity in Minimal Submanifolds and Geometric Flows

Minimal hypersurfaces are hypersurfaces that locally minimize area, while mean curvature flow evolves a hypersurface in the direction that decreases its area as fast as possible. This flow often converges to a minimal hypersurface over time. Both concepts—minimal hypersurfaces and mean curvature flow—play an important role in a wide range of scientific disciplines, including physics, materials science, and computer vision. The proposed project aims to deepen our understanding of the singularities that arise in these settings, which are central to current research in the field. In addition to advancing scientific knowledge, the project incorporates an appropriate educational component through teaching, mentoring, and the organization of seminars and conferences. This project has two main parts. The first part focuses on leveraging the theory of self-expanders for mean curvature flow to establish sharp lower bounds for the densities of minimal cones that are topologically nontrivial in specific senses. This work builds on and extends previous results by the PI and Bernstein to higher dimensions. The second part addresses a fundamental question in the study of mean curvature flow: the uniqueness of blow-up limits at singularities. It also explores the behavior of the flow as it emerges from a conical singularity. 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.