Comparison theory of Lorentzian distance with applications to spacelike hypersurfaces

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 [1], where complete details and further related results may be found.


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 I + (p) = {q ∈ M : p q}, and J + (p) = {q ∈ M : p ≤ q}.
As is well-known, I + (p) is always open, while J + (p) is neither open nor closed in general.
If q ∈ J + (p), then the Lorentzian distance d(p, q) is the supremum of the Lorentzian lengths of all the future-directed causal curves from p to q (possibly, d(p, q) = +∞).If q / ∈ J + (p), then the Lorentzian distance d(p, q) = 0 by definition.In particular, d(p, q) > 0 if and only is q ∈ I + (p).Let us recall that the Lorentzian distance function d : M × M → [0, +∞] for an arbitrary spacetime may fail to be continuous in general, and may also fail to be finite valued; globally hyperbolic spacetimes turn out to be the natural class of spacetimes for which the Lorentzian distance function is finite-valued and continuous (see, for instance, [3] and [2]).
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 T −1 M| p = {v ∈ T p M : 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
For every c ∈ R, let us define It is worth pointing out that the index of a Jacobi field J c along a timelike geodesic γ c in a Lorentzian space form of constant curvature c is given by where J c (0) = 0 and J c (s) = x ⊥ γ c (s).On the other hand, when c .Our first result assumes that the sectional curvatures of the timelike planes of M are bounded from above by a constant c and reads as follows.
Lemma 2 [1, Lemma 3.1] Let M be a spacetime such that K M (Π) ≤ c, c ∈ R, for all timelike planes in M. Assume that there exists a point p ∈ M such that I + (p) = / 0, and let q ∈ I + (p) (with d p (q) < π/ √ −c when c < 0).Then for every spacelike vector x ∈ T q M orthogonal to ∇d p (q) it holds that where ∇ 2 stands for the Hessian operator on M.
Observe that if K M (Π) ≤ c for all timelike planes in an (n + 1)-dimensional spacetime M, then for every unit timelike vector Z ∈ T M one gets that Ric M (Z, Z) ≥ −nc.Our next result holds under this weaker hypothesis on the Ricci curvature of M. When c = 0 this is nothing but the so called timelike convergence condition.
Lemma 3 [1, Lemma 3.3] Let M be an (n + 1)-dimensional spacetime such that Ric M (Z, Z) ≥ −nc, c ∈ R, for every unit timelike vector Z. Assume that there exists a point p ∈ M such that I + (p) = / 0, and let q 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.
timelike planes in M. Assume that there exists a point p ∈ M such that I + (p) = / 0, and let q ∈ I + (p) (with d p (q) < π/ √ −c when c < 0).Then for every spacelike vector x ∈ T q M orthogonal to ∇d p (q) it holds that 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 ψ(Σ) ⊂ I + (p).Let r = d p denote the Lorentzian distance function with respect to p, and let u = r • ψ : Σ→(0, ∞) be the function r along the hypersurface, which is a smooth function on Σ.Our first objective is to compute the Laplacian of u.To do that, observe that ∇r = ∇u − ∇r, N N along Σ, where ∇u stands for the gradient of u on Σ.In particular, Moreover, 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 that for every unit tangent vector field X ∈ T Σ, and therefore by (1) Tracing this inequality, one gets the following inequality for the Laplacian of where H = −(1/n)tr(A) is the mean curvature of Σ.Similarly, under the assumption Ric M (Z, Z) ≥ −nc, c ∈ R, for every unit timelike vector Z, we know from the Laplacian comparison result given in Lemma 3 that ∆r ≥ −n f c (r) along the hypersurface.Therefore, we conclude that For further details, see [1, Section 3].

SPACELIKE HYPERSURFACES BOUNDED BY A LEVEL SET OF THE LORENTZIAN DISTANCE
For the applications of our comparison results to the estimate of the mean curvature of spacelike hypersurfaces, we will make use of a generalized version of the well-known Omori-Yau maximum principle.Following the terminology introduced by Pigola, Rigoli and Setti [5, Definition 1.10], the Omori-Yau maximum principle is said to hold on an n-dimensional Riemannian manifold Σ n if, for any smooth function u ∈ C ∞ (Σ) with u * = sup Σ u < +∞ there exists a sequence of points {p k } k∈N in Σ with the properties In this sense, the classical maximum principle given by Omori [6] and Yau [7] states that the Omori-Yau maximum principle holds on every complete Riemannian manifold with Ricci curvature bounded from below.More generally, as shown by Pigola, Rigoli and Setti [5, Example 1.13], a sufficiently controlled decay of the radial Ricci curvature suffices to imply the validity of the Omori-Yau maximum principle.Now we are ready to give our main results.If the Omori-Yau maximum principle holds on Σ, then its future mean curvature H satisfies inf where u denotes the Lorentzian distance d p along the hypersurface.
For a sketch of the proof, observe that since Ric M (Z, Z) ≥ −nc, we have that Applying the Omori-Yau maximum principle to the function u, we get that On the other hand, where u denotes the Lorentzian distance d p along the hypersurface.In particular, if As a direct application of Theorem 6 we get the following.For a sketch of the proof of Theorem 6, observe that since K M (Π) ≥ c, we know that Applying the Omori-Yau maximum principle to the positive function u, we get that It follows from here that sup Therefore, making k → ∞ here we get the result.The last assertion follows from the fact that lim s→0 f c (s) = +∞.On the other hand, for a proof of Corollary 7, simply observe that sup Σ H < +∞ implies that inf Σ u > 0. For further details, see [1,Section 4].
In particular, when the ambient spacetime is Lorentzian space form, Theorems 5 and 6 yield the following consequences.For a proof simply observe that the Ricci curvature of a spacelike hypersurface Σ in an arbitrary spacetime M is given by Ric In particular, if M n+1 c is a Lorentzian space form of constant sectional curvature c then Ric Σ (X, X) ≥ (n − 1)c − n 2 H 2 /4 |X| 2 .Thus, every spacelike hypersurface Σ with bounded mean curvature in M n+1 c has Ricci curvature bounded from below.Hence, if complete, it satisfies the Omori-Yau maximum principle.
As observed in [1, Remark 1], our last results have specially simple and illustrative consequences when the ambient is the Lorentz-Minkowski spacetime.For instance, we can state the following improvement of Theorem 2 in [8].
Corollary 10 [1, Corllary 4.7] The only complete spacelike hypersurfaces with constant mean curvature in the Minkowski space L n+1 which are contained in I + (p) (for some fixed p ∈ L n+1 ) and bounded from above by a hyperbolic space centered at p are precisely the hyperbolic spaces centered at p.

THE LORENTZIAN DISTANCE FUNCTION FROM AN ACHRONAL HYPERSURFACE
Given S ⊂ M n+1 an achronal spacelike hypersurface, one can define the Lorentzian distance function from S by d S (q) = sup{d(p, q) : p ∈ S}.As in the previous case of the Lorentzian distance from a point, to guarantee the smoothness of Lemma 11 Let S be an achronal spacelike hypersurface in a spacetime M.
1.If S is compact and M is globally hyperbolic, then s(p) > 0 for all p ∈ S and I + (S) = / 0. 2. If I + (S) = / 0, then d S is smooth on I + (S) and its gradient ∇d S is a past-directed timelike (geodesic) unit vector field on I + (S).
Doing now a similar analysis of the Lorentzian distance function to an achronal hypersurface, one can derive also sharp estimates for the mean curvature of spacelike hypersurfaces which contained in its chronological future.Further details about this may be found in [1].
d S , one needs to restrict this function on certain special subsets M. Let η be the future-directed Gauss map of S, and let s : S → [0, +∞] the function given by s(p) = sup{t ≥ 0 : d S (γ p (t)) = t}, where γ p : [0, a) → M is the future inextendible geodesic starting at p with initial velocity η p .Then, one can define Ĩ + (S) = {tη p : for all p ∈ S and 0 < t < s(p)} and consider the subset I + (S) = exp S (int( Ĩ + (S))) ⊂ I + (S), where exp S denotes the exponential map with respect to the hypersurface S. Below we collect some interesting properties about d S (see [4, Section 3.2]).
is the future inextendible geodesic starting at p with initial velocity v.Then, one can define the subset Ĩ + (p) ⊂ T p M as Ĩ + (p) = {tv : for all v ∈ T −1 M| p and 0 < t < s p (v)} and consider the subset I + (p) = exp p (int( Ĩ + (p))) ⊂ I + (p).Observe that exp p : int( Ĩ + (p)) → I + (p) is a diffeomorphism and I + (p) ⊂ M is an open subset (possible empty).In the following result we summarize the main properties about the Lorentzian distance function (see [4, Section 3.1]).