Harmonic vector fields on compact Lorentz surfaces

摘要

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.