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

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.