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

Abstract

In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz–Minkowski space $${\mathbb {L}}^{3}$$ , using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples of minimal surfaces in $${\mathbb {L}}^{3}$$ 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).