Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities

TL;DR

This paper investigates how isolated singularities in stable minimal hypersurfaces influence their local behavior, stability, and convergence properties, especially under metric perturbations and in the context of singularities in higher dimensions.

Contribution

It introduces new results on the existence, smoothness, and uniqueness of minimal hypersurfaces near singularities under metric perturbations and convergence scenarios.

Findings

Existence of nearby minimal hypersurfaces under small metric perturbations.

Smoothness of minimal hypersurfaces for generic perturbation directions.

Uniqueness of homologically minimizing hypersurfaces near 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.