Weierstrass Representations of Lorentzian Minimal Surfaces in $\mathbb R^4_2$

Abstract

The minimal Lorentzian surfaces in $\mathbb{R}^4_2$ whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy $K^2-\varkappa^2 >0$ are called minimal Lorentzian surfaces of general type. These surfaces admit canonical parameters and with respect to such parameters are determined uniquely up to a motion in $\mathbb{R}^4_2$ by the curvatures $K$ and $\varkappa$ satisfying a system of two natural PDEs. In the present paper we study minimal Lorentzian surfaces in $\mathbb{R}^4_2$ and find a Weierstrass representation with respect to isothermal parameters of any minimal surface with two-dimensional first normal space. We also obtain a Weierstrass representation with respect to canonical parameters of any minimal Lorentzian surface of general type and solve explicitly the system of natural PDEs expressing any solution to this system by means of four real functions of one variable.