Harmonic vector fields on compact Lorentz surfaces

We study harmonic vector fields on a Lorentzian torus $T^2$ i.e. critical points of the total bending functional $\mathcal{B} :\mathcal{ E} \to\mathbb{R}$ were $\mathcal{E}$ consists of all unit timelike vector fields on $T^2$. We derive the first variation formula for $\mathcal{B}$ in terms of the Lorentz angle function associated to each $X\in \mathcal{E}$ and give applications on flat Lorentzian tori.