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arXiv · 2410.05566

Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities

Costante Bellettini·Konstantinos Leskas
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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.)

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Costante Bellettini, Konstantinos Leskas. 2025-10-08. Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities. https://doi.org/10.1515/acv-2024-0132

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