Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.08822 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908967296303104 |
|---|---|
| author | Cheng, Zehua |
| author_facet | Cheng, Zehua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generic Metrics on $S^{n+1}$ Preclude Linearly Stable Singular Tangent Cones of Area-Minimizing Boundaries Cheng, Zehua Differential Geometry 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. |
| title | Generic Metrics on $S^{n+1}$ Preclude Linearly Stable Singular Tangent Cones of Area-Minimizing Boundaries |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2604.08822 |