Minimal surfaces in a certain 3-dimensional homogeneous spacetime

TL;DR

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

Contribution

It introduces a generalized integral representation formula unifying existing formulas for minimal timelike surfaces in these manifolds.

Findings

Unified representation formula for minimal timelike surfaces

Analysis of the harmonicity of the normal Gauss map

Application to various homogeneous Lorentzian 3-manifolds

Abstract

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group 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 unificaton of representation formulas for minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gau{\ss} map of minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discussed.