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

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 centre...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Bellettini, Costante, Leskas, Konstantinos
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 07.10.2024
Subjects
Approximation
Curvature
Hyperspaces
Singularity (mathematics)
Online AccessGet full text

Cover

More Information
Summary: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.)
ISSN:2331-8422