Canonical Coordinates and Natural Equation for Lorentz Surfaces in $\mathbb R^3_1$

TL;DR

This paper introduces canonical isotropic coordinates for Lorentz surfaces in 3D Minkowski space of general type, derives a natural integro-differential equation characterizing them, and proves a fundamental Bonnet-type theorem.

Contribution

It develops a new coordinate system and a natural equation for Lorentz surfaces of general type, extending the understanding of their geometric properties.

Findings

Canonical coordinates simplify the description of Lorentz surfaces.

The natural equation characterizes Lorentz surfaces of general type.

Examples illustrate the theory for special cases like constant mean curvature.

Abstract

We consider Lorentz surfaces in $\mathbb R^3_1$ satisfying the condition $H^2-K\neq 0$, where $K$ and $H$ are the Gauss curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces we introduce special isotropic coordinates, which we call canonical, and show that the coefficient $F$ of the first fundamental form and the mean curvature $H$, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of general type. Using this natural equation we prove a fundamental theorem of Bonnet type for Lorentz surfaces of general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.