Geometric Analysis Seminar
Minimal hypersurface has been frequently studied in recent years in dimension ≤ 7; In higher dimensions, singularities are known to exist in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this talk, we present different aspects of how these singularities may affect the local behavior of minimal hypersurfaces. In particular, given a non-degenerate minimal hypersurface with strictly stable and strictly minimizing tangent cone at each singular point, under any small perturbation of the metric, we show the existence of a nearby minimal hypersurface under new metric. For a generic choice of perturbation direction, we show the entirely smoothness of the resulting minimal hypersurface.