A Liouville-type theorem for cylindrical cones

Suppose that C0n subset of Rn+1$\mathbf {C}_0<^>n \subset \mathbb {R}<^>{n+1}$ is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), l >= 0$l \ge 0$, and M$M$ a complete embedded minimal hypersurface of Rn+1+l$\mathbb {R}<^>{n+1+l}$ lying to one side of C=C0xRl$\mathbf {C}= \mathbf {C}_0 \times \mathbb {R}<^>l$. If the density at infinity of M$M$ is less than twice the density of C$\mathbf {C}$, then we show that M=H(lambda)xRl$M = H(\lambda) \times \mathbb {R}<^>l$, where {H(lambda)}lambda$\lbrace H(\lambda)\rbrace _\lambda$ is the Hardt-Simon foliation of C0$\mathbf {C}_0$. This extends a result of L. Simon, where an additional smallness assumption is required for the normal vector of M$M$.