Global Viscosity Solutions Associated To Geodesic Flows On Lorentzian Manifolds

In the 80s of last century,to study positive definite time-periodic Hamiltonian systems(which is not nearly integrable)with one degree of freedom,S.Aubry and J.Mather independently developed a global variational method under different backgrounds,then proved the existence of some invariant sets with complicated dynamic structure(see[1],[6],[36],[37]).In his paper[42],J.Mather extended the above method to higher dimensional Tonelli Lagrangian system and then defined a series of minimal(in the sense of variation)invariant sets,which formed the so-called Aubry-Mather theory.Later,A.Fathi found another approach to those minimal invariant sets by using some weak solutions to the corresponding Hamilton-Jacobi equation.Today we call his approach the weak KAM method.In Tonelli Lagrangian systems,Fathi established the existence of the weak KAM solutions and proved that the notion of weak KAM solutions are identical with the notion of global viscosity solutions,see[23],[24].This inspires us to study the global dynamics of the corresponding system by establishing the existence of the global viscosity solutions in more extensive Hamiltonian systems.Aubry-Mather theory and weak KAM theory are all established for positive definite Hamiltonian systems,but with the rapid developments in modern physics,the investigation of non positive definite systems becomes more and more important.Among a variety of non positive definite systems,Lorentzian geodesic flow systems which involve the general theory of relativity form a very typical case,not only because of its importance in theoretic physics,but also from the non positive definiteness assumption itself.In this case,the non positive definiteness assumption leads to essential difficulties,namely that restricted to every cotangent space,the corresponding Hamiltonian function is not even coercive.It was until recently that S.Suhr succeeded to establish a Aubry-Mather theory for the appropriate spacetime model(which he called class A),see[51],[52].Based on the parallelism of weak KAM theory and Aubry-Mather theory,we begin to consider how to establish weak KAM theory on a appropriate spacetime model.According to the close relation between weak KAM solution and global viscosity solution,we believe that the existence of the global viscosity solution must be a first test.In joint works with Xiaojun Cui,we prove the following results:Let(M,g)be a spacetime,in this paper,we consider the Lorentzian eikonal equation g((?)u,(?)u)=-1.For the non-compact case,we show that for a spacetime(M,g)with regular cosmological time function ?,-? is a viscosity solution to the Lorentzian eikonal equation and shares some analogous properties with the so-called weak KAM solutions.For the compact case,we showed the existence of global viscosity solutions to the Lorentzian eikonal equation on the Abelian cover(R2,g)of a class A Lorentzian 2-torus(T2,g)for every asymptotic direction in the interior of the homology cone.Some other related dynamical properties are also considered.As an application of the above results,we also study the differentiability of the unit sphere of the stable time separation associated to the class A Lorentzian 2-torus.