A Note on Helicoidal Singular Minimal Surfaces

Abstract

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