Regularity of minimal surfaces near quadratic cones

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{s}\mathrm{p}\mathrm{t}$ M \cap B_{1/2}$isa$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.