A Note on Helicoidal Singular Minimal Surfaces

Abstract

Let $\alpha\in\r$ and let $\vec{v}\in\r^3$ be a unit vector. A singular minimal surface $\Sigma$ in Euclidean space is a surface $\Sigma$ whose mean curvature $H$ satisfies $H=\alpha\frac{\langle N,\vec{v}\rangle}{\langle p,\vec{v}\rangle}$, where $N$ is the unit normal vector of $\Sigma$. In this short note we study singular minimal surfaces which are invariant by a one-parameter group of helicoidal motions. We prove that if $\Sigma$ is a helicoidal singular minimal surface, then the axis of the helicoidal motion is orthogonal to $\vec{v}$, $\alpha=-1$ and $\Sigma$ is a circular right cylinder.