A Liouville-type theorem for cylindrical cones

Abstract

Suppose that $C_0^n\subset R^{n+1}$ is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), $l\ge 0$, and $M$ a complete embedded minimal hypersurface of $R^{n+1+l}$ lying to one side of $C=C_0\times R^l$. If the density at infinity of $M$ is less than twice the density of $C$, then we show that $M=H\left(\lambda \right)\times R^l$, where ${\left\{H\left(\lambda \right)\right\}}_{\lambda }$ is the Hardt–Simon foliation of $C_0$. This extends a result of L. Simon, where an additional smallness assumption is required for the normal vector of $M$.