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

Costante Bellettini, Konstantinos Leskas

TL;DR

The paper develops local smoothing results for constant-mean-curvature (CMC) hypersurfaces with isolated singularities under a regular tangent cone. Using the prescribed-mean-curvature functional $J_ extlambda$ on finite-perimeter sets, it proves the existence of smooth CMC hypersurfaces $T_j$ with mean curvature $ extlambda$ that converge to the original boundary $ar{igtriangleup}^* E$ in multiple senses, with $T_j=ar{igtriangleup}^* E_j$ and $E_j o E$. In ambient dimension $8$ the cone regularity is automatic, while for higher dimensions a regular tangent cone is assumed; a singular maximum principle for CMC hypersurfaces and a detailed Jacobi-operator analysis on cones underpin the approximation. The work extends Hardt--Simon-type results to the CMC Plateau problem and provides a robust framework for smoothing singularities in geometric variational problems, with implications for geometric constructions in high dimensions. The methods combine variational minimisation, blow-up analysis, cone stability, and foliation techniques to achieve strong convergence and regularity results.

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.)

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

TL;DR

The paper develops local smoothing results for constant-mean-curvature (CMC) hypersurfaces with isolated singularities under a regular tangent cone. Using the prescribed-mean-curvature functional on finite-perimeter sets, it proves the existence of smooth CMC hypersurfaces with mean curvature that converge to the original boundary in multiple senses, with and . In ambient dimension the cone regularity is automatic, while for higher dimensions a regular tangent cone is assumed; a singular maximum principle for CMC hypersurfaces and a detailed Jacobi-operator analysis on cones underpin the approximation. The work extends Hardt--Simon-type results to the CMC Plateau problem and provides a robust framework for smoothing singularities in geometric variational problems, with implications for geometric constructions in high dimensions. The methods combine variational minimisation, blow-up analysis, cone stability, and foliation techniques to achieve strong convergence and regularity results.

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 the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt--Simon approximation theorem.)
Paper Structure (6 sections, 16 theorems, 111 equations)

This paper contains 6 sections, 16 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Prescribed CMC Plateau problem
  3. Regular minimal cones, graphs, Jacobi operator
  4. A singular maximum principle
  5. Approximation
  6. Auxiliary results

Key Result

Theorem 1

Let $E$ be a set with locally finite perimeter in an open set $U\subset \mathbb{R}^8$, and assume that $E$ minimises $J_\lambda$ in a ball $\hat{B} \subset \subset U$, for a given $\lambda \in \mathbb{R}$. There exists a ball $B \subset \hat{B}$, with the same centre, and a sequence of hypersurfaces

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 3
  • Lemma 2.1
  • proof
  • ...and 36 more