Existence of maximal surface containing given curve and special singularity

TL;DR

This paper introduces a new formulation for maximal surfaces in Lorentz-Minkowski space, proves a singular Björling problem for closed null curves, and demonstrates the existence of maximal surfaces with prescribed curves and singularities.

Contribution

It provides a novel approach to describing maximal surfaces and establishes existence results for surfaces with specific boundary curves and singularities.

Findings

New formulation for maximal surfaces in Lorentz-Minkowski space

Proof of singular Björling problem for closed null curves

Existence of maximal surfaces containing given curves with special singularities

Abstract

We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, $\mathbb{L}^3$, using the identification of $\mathbb L^3$ with $\mathbb C\times \mathbb R$. Further we give a different proof for the singular Bj\"orling problem for the case of closed real analytic null curve. As an application, we show the existence of maximal surface which contains a given curve and has a special singularity.