Generic Regularity for Minimal Hypersurfaces in 8-Manifolds

 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 stable minimal hypercones. It was conjectured that generically they can be perturbed away. In this talk, we present a way to eliminate these singularities by perturbing metric in an 8-manifold. In particular, this proves the existence of a closed embedded smooth minimal hypersurface in an 8-manifold with generic metric, which generalizes the result by N. Smale in minimizing setting and by Chodosh-Liokumovich-Spolaor in Ric>0 case.