Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911198006476800 |
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| author | Bellettini, Costante Leskas, Konstantinos |
| author_facet | Bellettini, Costante Leskas, Konstantinos |
| contents | We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension $8$ the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt--Simon approximation theorem.) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities Bellettini, Costante Leskas, Konstantinos Differential Geometry Analysis of PDEs 53A10, 49Q20, 35J93 We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension $8$ the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt--Simon approximation theorem.) |
| title | Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities |
| topic | Differential Geometry Analysis of PDEs 53A10, 49Q20, 35J93 |
| url | https://arxiv.org/abs/2410.05566 |