A Liouville-type theorem for cylindrical cones

TL;DR

This paper proves a Liouville-type theorem for minimal hypersurfaces lying on one side of a cylindrical cone, showing they must be part of a known foliation if their density at infinity is sufficiently small.

Contribution

It extends L. Simon's result by removing the smallness assumption on the normal vector, establishing a new rigidity theorem for minimal hypersurfaces near cylindrical cones.

Findings

Minimal hypersurfaces with low density at infinity are classified as foliation members.

The result applies to strictly stable, strictly minimizing minimal hypercones.

The theorem generalizes previous work by relaxing geometric assumptions.

Abstract

Suppose that $\mathbf{C}_0^n \subset \mathbb{R}^{n+1}$ is a smooth strictly minimizing and strictly stable minimal hypercone, $l \geq 0$, 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 $M$.