Framed null curves and timelike surfaces via Lorentzian harmonic maps into de-Sitter 2-space

We construct a class of Lorentzian harmonic maps into the de-Sitter $2$-space satisfying the eigenvalue equation $\mathrm{\square }N=2H^{2}N$ for the d’Alambert operator $\mathrm{\square }$ and a non-zero constant $H$ from framed null curves. We also investigate two classes of timelike surfaces associated with these Lorentzian harmonic maps: the first one is timelike surfaces with constant mean curvature $H$ in Lorentz-Minkowski $3$-space and the second one is timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group ${\mathrm{Nil}}_3(H)$. In particular, we characterize some properties of singularities on timelike minimal surfaces in ${\mathrm{Nil}}_3(H)$ via an invariant of framed null curves.