Regularity of minimal surfaces near quadratic cones

TL;DR

This paper demonstrates that minimal surfaces near quadratic cones are smooth perturbations of the cone or its foliation, revealing local structure and singularity behavior for such surfaces.

Contribution

It proves regularity and structural results for minimal surfaces close to quadratic cones, extending understanding of their local geometry and singularities.

Findings

Minimal surfaces near quadratic cones are $C^{1,eta}$ perturbations.

Singularities modeled on quadratic cones determine local surface structure.

Characterization of minimal surfaces asymptotic to quadratic cones as cones or foliation leaves.

Abstract

Hardt-Simon proved that every area-minimizing hypercone $\mathbf{C}$ having only an isolated singularity fits into a foliation of $\mathbb{R}^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf{C}$. In this paper we prove that if a stationary $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb{R}^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone $\mathbf{C}^{3,3}$), then $\mathrm{spt} M \cap B_{1/2}$ is a $C^{1,\alpha}$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of $M$, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.