every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. in this talk i will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. this 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, and it also allows to study convergence to singular minimal hyper surfaces. this is a joint result with n. edelen.