Timelike minimal surfaces in the three-dimensional Heisenberg group

TL;DR

This paper studies timelike minimal surfaces in the 3D Heisenberg group with semi-Riemannian metrics, characterizing them via Lorentz harmonic maps and establishing a Weierstrass representation using loop groups.

Contribution

It introduces a novel characterization of non-vertical timelike minimal surfaces through Lorentz harmonic maps and develops a generalized Weierstrass representation for these surfaces.

Findings

Characterization of non-vertical timelike minimal surfaces via Lorentz harmonic maps

Development of a generalized Weierstrass type representation

Application of loop group decompositions to surface representation

Abstract

Timelike surfaces in the three-dimensional Heisenberg group with left invariant semi-Riemannian metric are studied. In particular, non-vertical timelike minimal surfaces are characterized by the non-conformal Lorentz harmonic maps into the de Sitter two phere. On the basis of the characterization, the generalized Weierstrass type representation will be established through the loop group decompositions.