A Liouville-type theorem for cylindrical cones

Abstract

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

arXiv comment21 pages; fixed a mistake