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

TL;DR

This paper proves the existence of global viscosity solutions to the eikonal equation on the Abelian cover of a class A Lorentzian 2-torus, exploring related dynamical properties and the differentiability of the stable time separation.

Contribution

It establishes the existence of global viscosity solutions for the eikonal equation on Lorentzian 2-tori and analyzes their dynamical and geometric properties.

Findings

Existence of global viscosity solutions in the interior of the homology cone.

Analysis of dynamical properties related to the solutions.

Differentiability results for the unit sphere of the stable time separation.

Abstract

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.