Singular Structure of Minimal Surfaces

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Minimal surfaces are mathematical models of soap films, or more generally any interface whose energy and area are proportional: they are surfaces that locally minimize area. In addition to having a rich history, minimal surfaces are an important tool in analysis of physical systems and geometric questions. A prominent example is the role stable minimal surfaces play in the study of scalar curvature in general relativity. Just like the soap films they model, minimal surfaces in general have a "singular set" where they are not smooth (for example, consider bubbles in the bathtub: they have singular junctions were multiple bubbles meet). The goal of this project is to work towards better understanding of the singular set of a minimal surface, what structure it possesses, and the behavior of the minimal surface near singularities. In addition to pursuing these research projects, the PI will mentor graduate and undergraduate students and continue outreach work with a local community center's afterschool math program for elementary-, middle-, and high-school students. This project studies higher dimensional minimal hypersurfaces that are area-minimizing, stable, or have finite index. If one precludes, often for natural reasons of orientability, "Y-type" singularities (encountered in the bathtub), then the singular set drops to codimension 7. The project will study the space of 7-dimensional minimal hypersurfaces with isolated singularities and finite index/area in a closed 8-manifold, with the view towards a proving a notion of "bumpiness" for certain singular minimal hypersurfaces. The PI will also investigate non-isolated singularities. The research will also attempt to classify 8-dimensional minimal hypersurfaces in Euclidean space asymptotic to certain cylindrical cones, which would serve to model the geometric degeneration of surfaces towards these cylindrical singularities. The work aims to establish a general isoperimetric-type inequality for the singular set in terms of the minimal hypersurface's boundary data. 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.