Minimal surfaces near Hardt-Simon foliations

TL;DR

This paper extends techniques used on the Simons cone to construct minimal surfaces near Hardt-Simon foliations, providing new examples of minimal hypersurfaces with controlled singularities.

Contribution

It introduces a gluing method to solve the minimal surface equation near Hardt-Simon surfaces and quadratic cones, expanding the class of known minimal hypersurfaces.

Findings

Constructed new minimal surfaces near Hardt-Simon foliations.

Demonstrated existence of solutions with specific boundary conditions.

Extended methods from Simons cone to more general singularities.

Abstract

Caffarelli-Hardt-Simon used the minimal surface equation on the Simons cone $C(S^3\times S^3)$ to generate newer examples of minimal hypersurfaces with isolated singularities. Hardt-Simon proved that every area-minimizing quadratic cone $\mathcal{C}$ having only an isolated singularity can be approximated by a unique foliation of $\mathbb R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathcal C$. This paper uses methods similar to Caffareli-Hardt-Simon to solve the minimal surface equation for the Hardt-Simon surfaces in the sphere for some boundary values. We use gluing methods to construct minimal surfaces over Hardt-Simon surfaces and near quadratic cones.