# Enneper representation of minimal surfaces in the three-dimensional Lorentz-Minkowski space

## TL;DR

This paper develops an Enneper-type representation for spacelike and timelike minimal surfaces in Lorentz-Minkowski space using complex and paracomplex analysis, providing new explicit examples and connecting to existing Weierstrass representations.

## Contribution

It introduces an Enneper representation for minimal surfaces in Lorentz-Minkowski space, linking complex and paracomplex analysis with existing Weierstrass formulas.

## Key findings

- Derived Enneper-type formulas for spacelike and timelike minimal surfaces.
- Constructed explicit examples of minimal surfaces in Lorentz-Minkowski space.
- Established equivalence with known Weierstrass representations.

## Abstract

In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz-Minkowski space $L^{3}$, using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples of minimal surfaces in $L^{3}$ constructed via the Enneper representation formula, that it is equivalent to the Weierstrass representation obtained by Kobayashi (for spacelike immersions) and by Konderak (for the timelike ones).