Global viscosity solutions for eikonal equations on class A Lorentzian 2-tori

On the Abelian cover $(\mathbb{R}^{2},g)$ of a class A Lorentzian 2-torus $(\mathbb{T}^{2},g)$, we showed the existence of global viscosity solutions to the eikonal equation $$g(\nabla u,\nabla u)=-1$$ associated to those homologies in the interior of the homology cone. Some other related dynamical properties are also considered. As an application of the main results, we study the differentiability of the unit sphere of the stable time separation associated to the class A Lorentzian 2-torus.