Approximation and Interpolation Theorems for Maximal Surfaces with Singularities

Abstract

In this paper, we prove an approximation and interpolation theorem for maxfaces in the Lorentz--Minkowski $3$-space $\mathbb{L}^3$. Alarcón, Forstnerič, and López established approximation and interpolation theorems for conformal minimal surfaces using the Enneper--Weierstrass representation formula. We survey their methods and apply them to maxfaces. Furthermore, by incorporating singularity criteria based on the Weierstrass data of maxfaces into the approximation and interpolation theorem, we demonstrate the existence of a maxface with prescribed singularities at specified points, as well as the existence of a maxface whose singular set has a dense image in $\mathbb{L}^3$.