Bellettini, Costante;
Leskas, Konstantinos;
(2025)
Smooth approximations for
constant-mean-curvature hypersurfaces with
isolated singularities.
Advances in Calculus of Variations
10.1515/acv-2024-0132.
(In press).
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Abstract
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.
| Type: | Article |
|---|---|
| Title: | Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1515/acv-2024-0132 |
| Publisher version: | https://doi.org/10.1515/acv-2024-0132 |
| Language: | English |
| Additional information: | Copyright © 2025 Walter de Gruyter GmbH, Berlin/Boston This work is licensed under the Creative Commons Attribution 4.0 International License, https://creativecommons.org/licenses/by/4.0/. |
| Keywords: | Constant-mean-curvature; isolated singularity; Hardt–Simon foliation; smooth approximation |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10214999 |
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