Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

TL;DR

This paper classifies helicoidal surfaces in Minkowski space with constant mean and Gauss curvature, focusing on those generated by polynomial graphs or Lorentzian circles, revealing specific geometric constraints.

Contribution

It provides a complete classification of such surfaces, showing polynomial degree restrictions and geometric configurations for Lorentzian circle generators.

Findings

Polynomial generating curves are of degree 0 or 1 and produce ruled surfaces.

Lorentzian circle generators require a spacelike axis with the circle centered on the axis.

The study characterizes the geometric structure of these helicoidal surfaces.

Abstract

In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is $0$ or $1$ and that the surface is ruled. If the generating curve is a Lorentzian circle, we show that the only possibility is that the axis is spacelike and the center of the circle lies in the axis.