Duality of zero mean curvature surfaces in the Lorentzian Heisenberg group

TL;DR

This paper explores the duality of zero mean curvature surfaces in the Lorentzian Heisenberg group, showing that the associated transformation surface also exhibits zero mean curvature and deriving the Sym formula for dual surfaces.

Contribution

It introduces a duality transformation for zero mean curvature surfaces in the Lorentzian Heisenberg group and derives the Sym formula for the dual surfaces.

Findings

The transformation surface of a zero mean curvature surface also has zero mean curvature.

A Sym formula for the dual surface is derived in both metric cases.

The duality preserves the zero mean curvature property in the transformation surface.

Abstract

We study a transformation surface associated with a zero mean curvature surface in the three-dimensional Heisenberg group with respect to two left-invariant semi-Riemannian metrics. We investigate the duality and prove that the transformation surface also has zero mean curvature. Furthermore, we derive the Sym formula for the dual surface in both metric cases.