Regularity of minimal surfaces near quadratic cones

Hardt-Simon proved that every area-minimizing hypercone C having only an isolated singularity fits into a foliation of Rn+1 by smooth, areaminimizing hypersurfaces asymptotic to C. In this paper we prove that if a stationary integral n-varifold M in the unit ball B-1 subset of Rn+1 lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone C-3,C-3), then spt M boolean AND B-1/2 is a C-1,C- 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.