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 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.)