A Note on Helicoidal Singular Minimal Surfaces

TL;DR

This paper characterizes helicoidal singular minimal surfaces in Euclidean space, proving they are circular cylinders with specific parameters and axes orthogonal to a given vector.

Contribution

It provides a classification of helicoidal singular minimal surfaces, showing they must be circular cylinders with the axis orthogonal to the vector involved in the mean curvature condition.

Findings

Helicoidal singular minimal surfaces have axes orthogonal to the vector v.

Such surfaces are necessarily circular right cylinders.

The parameter alpha must be -1 for these 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.