Generic Metrics on $S^{n+1}$ Preclude Linearly Stable Singular Tangent Cones of Area-Minimizing Boundaries

TL;DR

This paper shows that for a generic set of Riemannian metrics on the sphere, area-minimizing boundaries cannot have singular tangent cones that are linearly stable, using perturbation and Baire category methods.

Contribution

It introduces a perturbation theorem to eliminate stable tangent cones and demonstrates that such metrics form a residual set, advancing understanding of singularities in minimal surface theory.

Findings

Residual set of metrics with no stable singular tangent cones

Constructed explicit metric perturbations destroying tangent cone stability

Proved openness of metrics admitting no prescribed tangent cone

Abstract

We prove that for a residual (and hence dense) subset $\mathcal{G}$ of Riemannian metrics on $S^{n+1}$ in the $C^{3}$ topology, no area-minimizing integral $n$-current that is a boundary admits a singular tangent cone which is linearly stable in the Euclidean sense. The proof proceeds in three stages. First, we develop a perturbation theorem: given any area-minimizer possessing an isolated singularity whose unique tangent cone $C$ is linearly stable, we construct an explicit $C^{3}$-small metric perturbation that destroys the compatibility conditions required for $C$ to persist as a tangent cone. The construction rests on the Hardt--Simon asymptotic expansion near isolated singularities, the spectral theory of the Jacobi operator on the cross-section of $C$, and a surjectivity argument showing that the map from compactly supported metric variations to forcing terms in the linearised minimal-surface equation on $C$ has dense range. Second, we establish that the set of metrics admitting no area-minimizer with a prescribed cone type as tangent cone is open, using compactness of integral currents and upper-semicontinuity of the density function. Third, we assemble these ingredients via a Baire category argument, intersecting countably many open dense sets to obtain the residual set $\mathcal{G}$. An extension to non-isolated singularities is outlined using Federer--Almgren dimension reduction.