# Minimal Lorentz Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

## TL;DR

This paper classifies a special class of minimal Lorentz surfaces in pseudo-Euclidean 4-space with neutral metric, introducing canonical parameters and invariant functions to characterize them through PDEs.

## Contribution

It introduces a new class of minimal Lorentz surfaces of general type and provides a method to classify and construct them using invariant functions and PDEs.

## Key findings

- Canonical parameters are introduced for these surfaces.
- Surfaces are characterized by two invariant functions satisfying PDEs.
- An explicit example of such a surface is constructed.

## Abstract

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that any minimal Lorentz surface of general type is determined (up to a rigid motion) by two invariant functions satisfying a system of two natural partial differential equations. Using a concrete solution to this system we construct an example of a minimal Lorentz surface of general type.