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

TL;DR

This paper proves that certain constant-mean-curvature hypersurfaces with isolated singularities can be approximated by smooth hypersurfaces with the same mean curvature, extending known results and providing new approximation techniques.

Contribution

It introduces a method to approximate CMC hypersurfaces with isolated singularities by smooth ones, even in higher dimensions, generalizing the Hardt--Simon theorem.

Findings

Existence of smooth CMC hypersurfaces converging to the singular one

Approximation holds in ambient dimension 8 without cone restrictions

Extension of Hardt--Simon approximation to non-zero mean curvature

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.)