Abstract. In this note we summarize some comparison results for the Lorentzian distance function in spacetimes, with applications to the study of the geometric analysis of the Lorentzian distance function on spacelike hypersurfaces. In particular, we will consider spacelike hypersufaces whose image under the immersion is bounded in the ambient spacetime and derive sharp estimates for the mean curvature of such hypersurfaces under appropriate hypotheses on the curvature of the ambient spacetime. The results in this note are part of our recent paper Keywords: Lorenzian distance function, Hessian and Laplacian comparison results, spacelike hypersurface, mean curvature, Omori-Yau maximum principle. PACS: 04.20.Cv,02.40.Vh THE LORENTZIAN DISTANCE FUNCTION Consider M n+1 an (n + 1)-dimensional spacetime, and let p, q be points in M. Using the standard terminology and notation from Lorentzian geometry, one says that q is in the chronological future of p, written p q, if there exists a future-directed timelike curve from p to q. Similarly, q is in the causal future of p, written p < q, if there exists a future-directed causal (i.e., nonspacelike) curve from p to q. Obviously, p q implies p < q. As usual, p ≤ q means that either p < q or p = q. For a subset S ⊂ M, one defines the chronological future of S as I + (S) = {q ∈ M : p q for some p ∈ S}, and the causal future of S as J + (S) = {q ∈ M : p ≤ q for some p ∈ S}. Thus S ∪ I + (S) ⊂ J + (S). In particular, the chronological future I + (p) and the causal future J + (p) of a point p ∈ M are Given a point p ∈ M, one can define the Lorentzian distance function from p by d p (q) = d (p, q). In order to guarantee the smoothness of d p as a function on M, one needs to restrict this function on certain special subsets of M. Consider v is a future-directed timelike unit vector} the fiber of the unit future observer bundle of M at p, and set s p : COMPARISON RESULTS FOR THE LORENTZIAN DISTANCE FROM A POINT where ∇ 2 stands for the Hessian operator on M. Observe where ∆ stands for the (Lorentzian) Laplacian operator on M. On the other hand, under the assumption that the sectional curvatures of the timelike planes of M are bounded from below by a constant c, we get the following result. where ∇ 2 stands for the Hessian operator on M. The proofs of Lemma 2, Lemma 3 and Lemma 4 follow from the fact that where γ is the radial future directed unit timelike geodesic from p to q and J is the Jacobi field along γ with J(0) = 0 and J(s) = x, and it is strongly based on the maximality of the index of Jacobi fields. For the details, see [1, Section 3]. SPACELIKE HYPERSURFACES CONTAINED IN I + (p) Consider ψ : Σ n → M n+1 a spacelike hypersurface immersed into a spacetime M. Since M is time-oriented, there exists a unique future-directed timelike unit normal field N globally defined on Σ. Let A stand for the shape operator of Σ with respect to N. We will assume that there exists a point p ∈ M such that I + (p) = / 0 and that for every tangent vector field X ∈ T Σ, where ∇ 2 r and ∇ 2 u stand for the Hessian of r and u in M and Σ, respectively. Assume now that K M (Π) ≤ c (resp. K M (Π) ≥ c) for all timelike planes in M, and that u < π/ √ −c on Σ when c < 0. Then by the Hessian comparison results for r given in Lemma 2 (resp. Lemma 4), one gets tha
