Hardt-Simon and Wang showed that any area-minimizing hypercone fits into a foliation of area-minimizing hypersurfaces which, apart from the cone itself, are all smooth radial graphs asymptotic to the cone.  For cones having a smooth link Hardt-Simon additionally showed the foliation is unique, in the sense that any complete area-minimizing hypersurface lying to one side of the cone must be a leaf of the foliation.  In this talk we prove a similar uniqueness for minimizing cylindrical hypercones of the form $C = C_0 \times R^k$ when $C_0$ is a Simons cone (or more generally is smooth, strictly-minimizing, and strictly stable). 

This is joint work with Gábor Székelyhidi.