The primary focus of this proposal is to study the local structure of weak limits of (smooth as well as singular) immersed stable minimal hypersurfaces of arbitrary dimension, with the ultimate goal of extending the partial regularity theory of embedded stable minimal hypersurfaces, developed in 1981 by R. Schoen and L. Simon. Relaxing the embeddedness hypothesis allows the presence of additional singularities, for instance points where the tangent cones are hyperplanes with multiplicity greater than one. It is proposed to investigate the nature of these singularities, with the aim of obtaining information concerning the size and the structure of these singularities, possible uniqueness of their tangent cones as well as the local nature of the hypersurface near the singularities. The second project proposed is a study of the effect of certain topological and analytic constraints (on the target manifold) on the regularity propeties of the singular sets of energy minimizing harmonic maps between Riemannian manifolds.
The regularity theory of minimal surfaces and harmonic maps has a very rich history and the results and ideas developed in these areas have profoundly influenced several other fields of contemporary research in mathematics and physics, such as Yang-Mills fields, free boundary problems, curvature flows and general relativity. Obtaining a good structural description near singularities turns out to be highly desirable in many problems in pure and applied mathematics, and any new advances in minimal surface theory in this regard will likely have an impact on these other fields.
