Abstract

The well-known Simons cone suggests that singularities may exist in a stable minimal hypersurface in Riemannian manifolds of dimension greater than 7, locally modeled on minimal hypercones. It was conjectured that generically they can be perturbed away. In this talk, we shall present a way to resolve these singularities by perturbing metric in an 8-manifold and hence obtain smoothness under a generic metric. We shall also talk about certain generalizations of this generic smoothness of minimal submanifold in other dimensions and codimensions as well as their applications.