# The two-dimensional analogue of the Lorentzian catenary and the Dirichlet problem

## TL;DR

This paper extends the concept of the catenary to Lorentz-Minkowski space, solving the Dirichlet problem for certain maximal surfaces and classifying invariant singular maximal surfaces.

## Contribution

It introduces a Lorentzian analogue of the catenary, solves the Dirichlet problem for spacelike boundary data, and classifies invariant singular maximal surfaces in .

## Key findings

- Solved the Dirichlet problem for bounded mean convex domains with spacelike boundary data.
- Classified all invariant singular maximal surfaces under translation and rotation groups.
- Extended classical catenary concepts to Lorentzian geometry.

## Abstract

In this paper we generalize in Lorentz-Minkowski space $\l^3$ the two-dimensional analogue of the catenary of Euclidean space. We solve the Dirichlet problem for bounded mean convex domains and spacelike boundary data that have a spacelike extension to the domain. We also classify all singular maximal surfaces of $\l^3$ invariant by a uniparametric group of translations and rotations.