Ruled zero mean curvature surfaces in the three-dimensional light cone

TL;DR

This work provides a complete classification of ruled zero mean curvature surfaces in the three-dimensional light cone $\mathbb{Q}^3_+$. It develops the Weierstrass representation for ZMC surfaces in $\mathbb{Q}^3_+$, analyzes geodesics and screw motions to construct helicoids, and connects helicoids with catenoids via the associated family, including a Lawson-type correspondence with isotropic space. The main result shows that every ruled ZMC surface in $\mathbb{Q}^3_+$ is, up to isometries and homotheties, either a helicoid or a parabolic catenoid, with a detailed account of when helicoids appear in the associated family of a given catenoid. The paper also illuminates dualities of lightlike Gauss maps and clarifies how these ruled surfaces fit into broader correspondences across space forms. Overall, it extends the taxonomy of ZMC geometry to a degenerate metric setting and links it to classical constructions in Euclidean and isotropic geometries.

Abstract

We obtain a complete classification of ruled zero mean curvature surfaces in the three-dimensional light cone. En route, we examine geodesics and screw motions in the space form, allowing us to discover helicoids. We also consider their relationship to catenoids using Weierstrass representations of zero mean curvature surfaces in the three-dimensional light cone.

Figures (6)

• Figure 1: A helicoid (left) and a parabolic catenoid (right) in the ball model of $\mathbb{Q}^3_+$ (cf. CKLLY1). Blue curves represent geodesics.
• Figure 2: Types of surface given by the Weierstrass data $(g,\omega) = (w, \delta \frac{\dif{w}}{w^2})$, depending on the value of $\delta \in \mathbb{C}$. The red line represents elliptic catenoids, black point represents parabolic catenoid, green line represents hyperbolic catenoids, and blue cardioid represents helicoids.
• Figure 3: Associated families of catenoid given by the Weierstrass data $(g,\omega) = (w, \frac{1}{2\sqrt{5}} \frac{\dif{w}}{w^2})$.
• Figure 4: Associated families of catenoid given by the Weierstrass data $(g,\omega) = (w, \frac{1}{4} \frac{\dif{w}}{w^2})$.
• Figure 5: Associated families of catenoid given by the Weierstrass data $(g,\omega) = (w, \frac{1}{\sqrt{13}} \frac{\dif{w}}{w^2})$.

Theorems & Definitions (28)

• Example 2.1
• Definition 2.2
• Remark 2.3
• Example 2.4: A family of CMC surfaces in $\mathbb{Q}^3_+$
• Lemma 2.5: Liu2
• Remark 2.6
• Definition 3.1
• Lemma 3.2
• Definition 3.3
• Theorem 3.4