some results about the level sets of lorentzian busemann function

In the early 1980s yau posed the problem of establishing the rigidity of the hawking-penrose singularity theorems. approaches to this problem have involved the introduction of lorentzian busemann functions and the study of the geometry of their levelsets. also they have been used to prove bartnik’s splitting conjecture. in this paper, we haveshown several properties of the level sets of lorentzian busemann function.