Maximal spacelike surfaces in a certain homogeneous Lorentzian 3-manifold

TL;DR

This paper develops a unified integral representation for maximal spacelike surfaces in a family of homogeneous Lorentzian 3-manifolds, including Minkowski and anti-de Sitter spaces, and explores their geometric properties.

Contribution

It introduces a generalized integral representation formula unifying previous formulas for these surfaces in various homogeneous Lorentzian 3-manifolds.

Findings

Unified integral representation formula derived.

Normal Gauss map harmonicity analyzed.

Includes Minkowski and anti-de Sitter spaces as special cases.

Abstract

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula which is the unification of representation formulas for maximal spacelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gau{\ss} map of maximal spacelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discussed.