Regularity of minimal surfaces near quadratic cones

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 integral $n$-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.