Explicit Solving of the System of Natural PDEs of Minimal Lorentz Surfaces in $\mathbb R^4_2$

TL;DR

This paper explicitly solves the natural PDE system governing minimal Lorentz surfaces of general type in four-dimensional pseudo-Euclidean space, expressing solutions via four real functions and analyzing transformations under motions.

Contribution

It provides an explicit solution to the PDE system for minimal Lorentz surfaces using Weierstrass-type representations, including transformation formulas under isometries.

Findings

Explicit solutions expressed through four real functions.

Transformation formulas for null curves under motions.

Relation between generating quadruples of functions.

Abstract

A minimal Lorentz surface in $\mathbb R^4_2$ is said to be of general type if its corresponding null curves are non-degenerate. These surfaces admit canonical isothermal and canonical isotropic coordinates. It is known that the Gauss curvature $K$ and the normal curvature $\varkappa$ of such a surface considered as functions of the canonical coordinates satisfy a system of two natural PDEs. Using the Weierstrass type representations of the corresponding null curves, we solve explicitly the system of natural PDEs, expressing any solution by means of four real functions of one variable. We obtain the transformation formulas for the functions in the Weierstrass representation of a null curve under a proper motion in $\mathbb R^4_2$. Using this, we find the relation between two quadruples of real functions generating one and the same solution to the system of natural PDEs.