Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities

Abstract

Locally stable minimal hypersurface could have singularities in dimension $\geq 7$ in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this paper, we present different aspects of how these singularities may affect the local behavior of minimal hypersurfaces. First, 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 entire smoothness of the resulting minimal hypersurface. Second, given a strictly stable minimal hypersurface $\Sigma$ with strictly minimizing tangent cone at each singular point, we show that $\Sigma$ is uniquely homologically minimizing in its neighborhood. Lastly, given a family of stable minimal hypersurfaces multiplicity $1$ converges to a stable minimal hypersurface $\Sigma$ in an eight manifold, we show that there exists a non-trivial Jacobi field on $\Sigma$ induced by these converging hypersurfaces, which generalizes a result by Ben Sharp in smooth case; Moreover, positivity of this Jacobi field implies smoothness of the converging sequence.