Minimal surfaces near Hardt-Simon foliations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916930397405184 |
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| author | Nandakumaran, Vishnu |
| author_facet | Nandakumaran, Vishnu |
| contents | Caffarelli-Hardt-Simon used the minimal surface equation on the Simons cone $C(S^3\times S^3)$ to generate newer examples of minimal hypersurfaces with isolated singularities. Hardt-Simon proved that every area-minimizing quadratic cone $\mathcal{C}$ having only an isolated singularity can be approximated by a unique foliation of $\mathbb R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathcal C$. This paper uses methods similar to Caffareli-Hardt-Simon to solve the minimal surface equation for the Hardt-Simon surfaces in the sphere for some boundary values. We use gluing methods to construct minimal surfaces over Hardt-Simon surfaces and near quadratic cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22795 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal surfaces near Hardt-Simon foliations Nandakumaran, Vishnu Differential Geometry Analysis of PDEs 49Q05, 53A10 Caffarelli-Hardt-Simon used the minimal surface equation on the Simons cone $C(S^3\times S^3)$ to generate newer examples of minimal hypersurfaces with isolated singularities. Hardt-Simon proved that every area-minimizing quadratic cone $\mathcal{C}$ having only an isolated singularity can be approximated by a unique foliation of $\mathbb R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathcal C$. This paper uses methods similar to Caffareli-Hardt-Simon to solve the minimal surface equation for the Hardt-Simon surfaces in the sphere for some boundary values. We use gluing methods to construct minimal surfaces over Hardt-Simon surfaces and near quadratic cones. |
| title | Minimal surfaces near Hardt-Simon foliations |
| topic | Differential Geometry Analysis of PDEs 49Q05, 53A10 |
| url | https://arxiv.org/abs/2410.22795 |