Singularities on timelike minimal surfaces in Lorentzian Heisenberg group

TL;DR

This paper studies timelike minimal surfaces in the Lorentzian Heisenberg group, linking their construction to harmonic maps into de-Sitter space, and classifies their singularities with explicit examples.

Contribution

It introduces criteria for various singularities on these surfaces and provides explicit examples, advancing understanding of their geometric properties.

Findings

Criteria for cuspidal edges, swallowtails, and cuspidal cross caps.

Explicit examples illustrating singularities.

Connection between harmonic maps and minimal surface singularities.

Abstract

Timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group are shown to be constructed from Lorentzian harmonic maps into the de-Sitter two-sphere, and they naturally admit singular points. In particular, we provide criteria for cuspidal edges, swallowtails, and cuspidal cross caps, and present several explicit examples.