Björling problem for zero mean curvature surfaces in the three-dimensional light cone

TL;DR

This work develops a Björling framework for zero mean curvature surfaces in the 3D light cone $\mathbb{Q}^3_+$ by using spacelike analytic Björling data $(\gamma,\mathcal{L})$ with a conformality and orientability constraint, leading to a Weierstrass-type representation $X=F f_3 F^*$ and a precise existence/uniqueness theorem. It then applies the representation to classify all rotational ZMC surfaces in $\mathbb{Q}^3_+$, producing explicit elliptic, hyperbolic, and parabolic catenoids with concrete Weierstrass data, and provides an analytic extension example across a lightlike circle. The results extend the Björling approach to light-cone geometry, yielding explicit parametrizations and a complete rotational classification, with potential implications for the study of ZMC surfaces in Lorentzian and isotropic settings. Overall, the paper combines a novel data formulation, a Weierstrass-type construction, and a complete rotational taxonomy to advance understanding of ZMC surfaces in the light cone.

Abstract

We solve the Björling problem for zero mean curvature surfaces in the three-dimensional light cone. As an application, we construct and classify all rotational zero mean curvature surfaces.

Figures (5)

• Figure 1: Elliptic catenoids in $3$-dimensional light cone with $a = \frac{3}{2}$ on the left, and $a = 4$ on the right, where the initial given curve is highlighted.
• Figure 2: Hyperbolic catenoid in the $3$-dimensional light cone with parameter $b = \frac{3}{2}$, where the initial given curve is highlighted.
• Figure 3: Parabolic catenoid in $3$-dimensional light cone with parameter $b = \frac{1}{2}$, where the initial given curve is highlighted.
• Figure 4: A parabolic catenoid and a non-rotational zero mean curvature surface sharing the same initial curve, where the initial given curve is highlighted (on the left); the non-rotational zero mean curvature surface drawn over bigger domain (on the right).
• Figure 5: Analytic extension of the non-rotational zero mean curvature surface across a lightlike circle (on the top); closer look at the analytic extension near the lightlike circle (on the bottom). On all figures, the lightlike circle is highlighted.

Theorems & Definitions (13)

• Remark 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Remark 3.6
• Remark 3.7