Hardt-Simon proved that every area-minimizing hypercone $C$ having only an isolated singularity fits into a foliation of $R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $C$. We prove that if a minimal hypersurface $M$ in the unit ball $B_1 \subset R^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), then $M \cap B_{1/2}$ 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 in $R^{n+1}$ asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation. This is joint work with Luca Spolaor.
