Comparison Theorems in Lorentzian Geometry

# Comparison Theorems in Lorentzian Geometry and applications to spacelike hypersurfaces

Debora Impera Dipartimento di Matematica, Università degli studi di Milano, via Saldini 50, I-20133 Milano, Italy.
July 26, 2019
###### Abstract.

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.

## 1. Introduction

In general relativity each point of a Lorentzian manifold corresponds to an event. The events that we may experience in the universe are the ones in our chronological future, hence it may be interesting to investigate the geometry of this one. This can be done by means of the analysis of the Lorentzian distance function. Unfortunately this function is not differentiable in any spacetime; precisely, it is not even continuous in general. Nevertheless, in strongly causal spacetimes, the Lorentzian distance function from a point is differentiable at least in a “sufficiently near” chronological future of each point. In this case is possible to analyze the geometry of spacetimes by means of the level sets of the Lorentzian distance function with respect to this point. To do that, the main tools are Hessian and Laplacian comparison theorems for the Lorentzian distance of the spacetime, hence many works have been written in this spirit.
For instance, in a recent paper by F. Erkekoglu, E. García-Rio and D. N. Kupeli ([9]), following the approach of R. E. Greene and H. Wu in [12], the authors obtain Hessian and Laplacian comparison theorems for the Lorentzian distance functions of Lorentzian manifolds comparing their sectional curvatures. Afterwards, in [4], L. J. Alías, A. Hurtado and V. Palmer use these theorems to study the Lorentzian distance function restricted to a spacelike hypersurface immersed into a spacetime . In particular, under suitable conditions, they derive sharp estimates for the mean curvature of spacelike hypersurfaces with bounded image in the ambient spacetime.
In this paper we obtain Hessian and Laplacian comparison theorems for Lorentzian manifolds with sectional curvature of timelike planes bounded by a function of the Lorentzian distance, improving in this way on classical results, and we give some applications to the study of spacelike hypersurfaces.

The paper is organized as follows. In Sections 2 and 3 we present some basic concepts and terminology involving the Lorentzian distance function from a point and we prove our Hessian and Laplacian comparison theorems. To obtain these theorems we use an ‘analytic’ approach inspired by P. Petersen ([13]) avoiding, in this way, the ‘geometric’ approach used by Greene and Wu. In Section 4 we focus on the study of the Lorentzian distance function restricted to spacelike hypersurfaces. Hence, using the Omori-Yau maximum principle, we derive some estimates on the mean curvature that generalize the ones in [4]. Moreover, using a generalized Omori-Yau maximum principle for certain elliptic operators, we also obtain some estimates for the higher order mean curvatures associated to the immersion. Finally, in Section 5, we restrict ourselves to the case when the ambient space has constant sectional curvature and we prove a Bernstein-type theorem for spacelike hypersurfaces with constant -mean curvature that generalizes Corollary 4.6 in [4].

## 2. Preliminaries

Let be an -dimensional spacetime, that is, an -dimensional time-oriented Lorentzian manifold and let . Using the standard terminology and notation in Lorentzian geometry, we say that is in the chronological future of , written , if there exists a future-directed timelike curve from to . Similarly, we say that is in the causal future of , written , if there exists a future-directed causal (that is nonspacelike) curve from to . For a subset , we define the chronological future of as

 I+(S)={q∈M|p≪q for some p∈S},

and the causal future of as

 J+(S)={q∈M|p≤q for some p∈S},

where means that either or . In particular, the chronological and the causal future of a point are, respectively

 I+(p)={q∈M|p≪q},J+(p)={q∈M|p≤q}.

It is well known that is always open, while is neither open nor closed in general. Let . Then the Lorentzian distance is defined as the supremum of the Lorentzian lengths of all the future-directed causal curves from to . If , then by definition. Moreover, if and only if . Given a point one can define the Lorentzian distance function with respect to by

 dp(q)=d(p,q).

Let

 T−1M|p={v∈TpM|v is a future-directed timelike unit vector}

be the fiber of the unit future observer bundle of at . Define the function

 sp:T−1M|p→[0,+∞],sp(v)=sup{t≥0 | dp(γv(t))=t},

where is the future timelike geodesic with , . The future timelike cutlocus of in is defined as

 Γ+(p)={sp(v)v | v∈TpM and 0

and the future timelike cutlocus of in is wherever the exponential map at is defined on .
It is well known that the Lorentzian distance function on arbitrary spacetimes may fail in general to be continuous and finite valued. It is known that this is true for globally hyperbolic spacetimes. We recall that a spacetime is said to be globally hyperbolic if it is strongly causal and it satisfies the condition that is compact for all . Moreover, a Lorentzian manifold is said to be strongly causal at a point if for any neighborhood of there exists no timelike curve that passes through more than once. In general, in order to guarantee the smoothness of this function we need to restrict it on certain special subsets of . Let

 ˜I+(p)={tv | v∈T−1M|p and 0

and let

 I+(p)=exp(int(˜I+(p)))⊂I+(p).

Since

 expp:int(˜I+(p))→I+(p)

is a diffeomorphism, is an open subset of . In the lemma below we summarize the main properties of the Lorentzian distance function.

###### Lemma 1 ([9], Section 3.1).

Let be a spacetime and .

1. If is strongly causal at , then and ,

2. If , then the Lorentzian distance function is smooth on and is a past-directed timelike (geodesic) unit vector field on .

###### Remark 2.

If is a globally hyperbolic spacetime and , then and hence the Lorentzian distance function with respect to is smooth on for each .
We also observe that if is a Lorentzian space form, then it is globally hyperbolic and geodesically complete. Moreover, every timelike geodesic realizes the distance between its points. Hence and we conclude again that the Lorentzian distance function is smooth on for each .

## 3. Hessian and Laplacian Comparison Theorems

This section is devoted to exhibit estimates for the Hessian and the Laplacian of the Lorentzian distance function in Lorentzian manifolds under conditions on the sectional or Ricci curvature. To prove our theorems we will need the following Sturm comparison result.

###### Lemma 3.

Let be a continuous function on and let with be solutions of the problems

 {ϕ′′−Gϕ≤0a.e. in (0,+∞)ϕ(0)=0{ψ′′−Gψ≥0a.e. in (0,+∞)ψ(0)=0, ψ′(0)>0

If for and , then in and

 ϕ′ϕ≤ψ′ψ and ψ≥ϕon (0,T).

For a proof of the lemma see [15]. Using the Sturm comparison result, we obtain a comparison result for solutions of Riccati inequalities with appropriate asymptotic behaviour.

###### Corollary 4.

Let be a continuous function on and let be solutions of the Riccati differentials inequalities

 g′1−g21α+αG≥0, (resp.≤0)g′2+g22α−αG≥0, (resp.≤0)

a.e. in , satisfying the asymptotic conditions

 gi(t)=αt+o(t)ast→0+,

for some . Then (resp. ) and in (resp. in ).

###### Proof.

Since satisfies the conditions in the statement with , without loss on generality we may assume that . Notice that is bounded and integrable in a neighbourhood of . Hence the same is true for the function . Indeed

 −(g1(s)+1s)<−(g1(s)−1s)≤∣∣g1(s)−1s∣∣≤C,

for some constant . Now let be the positive functions defined by

Then , , and

 ϕ′1(t)=−g1(t)ϕ1(t), ϕ′2(t)=g2(t)ϕ2(t)

Hence

 ϕ′′1≤Gϕ1,ϕ′′2≥Gϕ2(resp. ϕ′′1≥Gϕ1,ϕ′′2≤Gϕ2).

Then, it follows by Lemma 3 that (resp. ) and

 −g1(t)=ϕ′1(t)ϕ1(t)≤ϕ′2(t)ϕ2(t)=g2(t)(resp.−g2(t)=ϕ′2(t)ϕ2(t)≤ϕ′1(t)ϕ1(t)=g1(t)).

We are now ready to prove the Hessian and Laplacian comparison theorems. In both cases we will follow the proofs given by S. Pigola, M. Rigoli and A. G. Setti in [15] of the corresponding theorems in the Riemannian setting.
We will denote by and respectively the Levi-Civita connection and the Laplacian on the spacetime . Moreover, for a given function , we denote by the symmetric operator given by for every , and by the metrically equivalent bilinear form given by

 ¯¯¯¯¯¯¯¯¯¯¯Hessf(X,Y)=⟨¯¯¯¯¯¯¯¯¯¯hessf(X),Y⟩.
###### Theorem 5 (Hessian Comparison Theorem).

Let be an -dimensional spacetime. Assume that there exists a point such that and let be the Lorentzian distance function from . Given a smooth even function on , let be a solution of the Cauchy problem

 {h′′−Gh=0h(0)=0, h′(0)=1

and let be the maximal interval where is positive and , where

 B+(p,r0)={q∈I+(p)|dp(q)

If

 (1) KM(Π)≤G(r)

for all timelike planes , then

 ¯¯¯¯¯¯¯¯¯¯¯Hessr(X,X)≥−h′h(r)⟨X,X⟩

for every spacelike which is orthogonal to . Analogously, if

 (2) KM(Π)≥G(r)

for all timelike planes , then

 ¯¯¯¯¯¯¯¯¯¯¯Hessr(X,X)≤−h′h(r)⟨X,X⟩

for every spacelike which is orthogonal to .

###### Proof.

Let and let , , be the radial future directed unit timelike geodesic with , , . Recall that and . Since satisfies the timelike eikonal inequality, is diagonalizable (see [11] Chapter 6 or [10] for more details) and has an orthonormal basis consisting of eigenvectors of . Let us denote by and respectively its greatest and smallest eigenvalues in the orthogonal complement of . Notice that the theorem is proved once one shows that

• if (1) holds, then

 λmin(q)≥−h′h(r(q)).
• if (2) holds, then

 λmax(q)≤−h′h(r(q)).

Let us prove claim first. We claim that if (1) holds, then satisfies

 (3) ⎧⎨⎩ddt(λmin∘γ)−(λmin∘γ)2≥−Gfor a.e. t>0λmin∘γ=1t+o(t)as t→0+

Namely, by the definition of covariant derivative

 (¯¯¯¯¯∇X¯¯¯¯¯¯¯¯¯¯hessu)(Y)=¯¯¯¯¯∇X(¯¯¯¯¯¯¯¯¯¯hessu(Y))−¯¯¯¯¯¯¯¯¯¯hessu(¯¯¯¯¯∇XY).

Hence, recalling the definition of the curvature tensor we find

 (¯¯¯¯¯∇Y¯¯¯¯¯¯¯¯¯¯hessu)(X)−(¯¯¯¯¯∇X¯¯¯¯¯¯¯¯¯¯hessu)(Y)=¯¯¯¯R(X,Y)¯¯¯¯¯∇u.

Choose , . For every spacelike unit vector , is orthogonal to and we can define a vector field orthogonal to by parallel translation along . Then

 ¯¯¯¯¯∇γ′(s)(¯¯¯¯¯¯¯¯¯¯hessr(Y))= (¯¯¯¯¯∇γ′(s)¯¯¯¯¯¯¯¯¯¯hessr)(Y)+¯¯¯¯¯¯¯¯¯¯hessr(¯¯¯¯¯∇γ′(s)Y) = −(¯¯¯¯¯∇¯¯¯¯∇r¯¯¯¯¯¯¯¯¯¯hessr)(Y) = −(¯¯¯¯¯∇Y¯¯¯¯¯¯¯¯¯¯hessr)(¯∇r)+¯¯¯¯R(¯¯¯¯¯∇r,Y)¯¯¯¯¯∇r = ¯¯¯¯¯¯¯¯¯¯hessr(¯¯¯¯¯∇Y¯¯¯¯¯∇r)+¯¯¯¯R(¯¯¯¯¯∇r,Y)¯¯¯¯¯∇r.

On the other hand, since is parallel

Hence

 ddt¯¯¯¯¯¯¯¯¯¯¯Hessr(γ)(Y,Y)−⟨¯hessr(γ)(Y),¯¯¯¯¯¯¯¯¯¯hessr(γ)(Y)⟩=−KMγ(Y∧γ′)

Notice that

 ¯¯¯¯¯¯¯¯¯¯¯Hessr(X,X)≥λmin

for every spacelike unit vector field . Let us choose so that at

 ¯¯¯¯¯¯¯¯¯¯¯Hessr(γ)(Y,Y)=λmin(γ(s)).

Then, the function attains its minimum at . Hence

 ddt¯¯¯¯¯¯¯¯¯¯¯Hessr(γ)(Y,Y)∣∣s=ddt(λmin∘γ)∣∣s

and we have proved that satisfies the first equation in (3), since . The asymptotic behaviour follows from the expression

 (4) ¯¯¯¯¯¯¯¯¯¯¯Hessr=1r(⟨,⟩+dr⊗dr)+o(1)

that can be proved using normal coordinates around . Now, if we set , we find that satisfies

 {ϕ′+ϕ2=Gon (0,r0)ϕ=1t+o(t)as t→0+

Then, using Corollary 4 with , and we conclude that

 λmin(q)≥−h′h(r(q))

and this concludes the proof of .
Finally, for what concerns claim , we observe that reasoning as in the proof of claim and choosing so that at

 ¯¯¯¯¯¯¯¯¯¯¯Hessr(γ)(Y,Y)=λmax(γ(s))

we can prove that, if (2) holds, satisfies

 ⎧⎨⎩ddt(λmax∘γ)−(λmax∘γ)2≤−Gfor a.e. t>0λmax∘γ=1t+o(t)as t→0+

In this case, setting again , we find that satisfies

 {ϕ′+ϕ2=Gon (0,r0)ϕ=1t+o(t)as t→0+

Then, we can conclude again using Corollary 4 with , and . ∎

###### Theorem 6 (Laplacian Comparison Theorem).

Let be an - dimensional spacetime. Assume that there exists a point such that and let . Let be the Lorentzian distance function from . Given a smooth even function on , let be a solution of the Cauchy problem

 {h′′−Gh≥0h(0)=0, h′(0)=1

and let be the maximal interval where is positive. If

 (5) RicM(¯¯¯¯¯∇r,¯¯¯¯¯∇r)≥−nG(r),

then

 ¯¯¯¯¯Δr≥−nh′h(r)

holds pointwise on .

###### Proof.

Let and let , , be the radial future directed unit timelike geodesic with , , . Recall that and . Define

 φ(t)=¯¯¯¯¯Δr∘γ(t),t∈(0,s].

Then tracing Equation (4)

 φ(t)=nt+o(t)as t→0+.

Recall that given the following Bochner formula holds

 12¯¯¯¯¯Δ⟨¯¯¯¯¯∇f,¯¯¯¯¯∇f⟩=∥∥¯¯¯¯¯¯¯¯¯¯hessf∥∥2+RicM(¯¯¯¯¯∇f,¯¯¯¯¯∇f)+⟨¯¯¯¯¯∇¯¯¯¯¯Δf,¯¯¯¯¯∇f⟩.

See [11] for more details. Since , it follows that

 0=∥∥¯¯¯¯¯¯¯¯¯¯hessr∥∥2+RicM(¯¯¯¯¯∇r,¯¯¯¯¯∇r)+⟨¯¯¯¯¯∇¯¯¯¯¯Δr,¯¯¯¯¯∇r⟩.

Since and , we have

 1n(¯¯¯¯¯Δr)2+⟨¯¯¯¯¯∇¯¯¯¯¯Δr,¯¯¯¯¯∇r⟩≤nG(r).

Computing along

 φ′(t)=ddt(¯¯¯¯¯Δr(γ(t)))∣∣s=⟨¯¯¯¯¯∇¯¯¯¯¯Δr(γ(t)),γ′(t)⟩∣∣s=−⟨¯¯¯¯¯∇¯¯¯¯¯Δr,¯¯¯¯¯∇r⟩.

Hence the function satisfies

 ⎧⎨⎩φ′(t)−φ2(t)n≥−nGφ(t)=nt+o(t)as t→0+

Set . Then satisfies

 ⎧⎨⎩ϕ′(t)+ϕ2(t)n≥nGon (0,r0)ϕ(t)=nt+o(t)as t→0+

Then we conclude again using Corollary 4. ∎

## 4. Applications to spacelike hypersurfaces

Let be a spacelike hypersurface isometrically immersed into the spacetime . Since is time-orientable, there exists a unique future-directed timelike unit normal field globally defined on . We will refer to that normal field as the future-pointing Gauss map of the hypersurface. We let denote the second fundamental form of the immersion. Its eigenvalues are the principal curvatures of the hypersurface. Their elementary symmetric functions

 Sk= ∑i1<...

define the -mean curvatures of the immersion via the formula

 (nk)Hk=(−1)kSk.

Thus is the mean curvature of and , where and are, respectively, the scalar curvature of and and is the Ricci tensor of . Even more, when is even, it follows from the Gauss equation that is a geometric quantity which is related to the intrinsic curvature of .

The classical Newton transformations associated to the immersion are defined inductively by

 P0=I,Pk=(nk)HkI+APk−1,

for every .

###### Proposition 7.

The following formulas hold:

1. ,

2. ,

3. ,

where .

We refer the reader to [3] for the proof of the last proposition and for further details on the Newton transformations (see also [16] and [17] for others details on the Newton transformations in the Riemannian setting). Let be the Levi-Civita connection of . We define the second order linear differential operator associated to by

 Lkf=Tr(Pk∘hessf).

It follows by the definition that the operator is elliptic if and only if is positive definite. Let us state two useful lemmas in which geometric conditions are given in order to guarantee the ellipticity of when (Recall that is always elliptic).

###### Lemma 8.

Let be a spacelike hypersurface immersed into a spacetime. If on , then is an elliptic operator (for an appropriate choice of the Gauss map ).

For a proof of Lemma 8 see Lemma 3.10 in [8]. The next Lemma is a consequence of Proposition 3.2 in [7].

###### Lemma 9.

Let be a spacelike hypersurface immersed into a -dimensional spacetime. If there exists an elliptic point of , with respect to an appropriate choice of the Gauss map , and on , , then for all the operator is elliptic.

We recall here that given a spacelike hypersurface , a point is said to be elliptic if the second fundamental form of the immersion is negative definite at .

Now consider and assume that there exists a point such that and that . Let be the Lorentzian distance function from and let be the function along the hypersurface, which is a smooth function on . Let us calculate the Hessian of on . Notice that

 ¯¯¯¯¯∇r=∇u−⟨¯¯¯¯¯∇r,ν⟩ν.

Hence, since and , we have

Hence

 ¯¯¯¯¯∇r=∇u−ν√1+∥∇u∥2

Moreover

 ¯¯¯¯¯∇X¯¯¯¯¯∇r=∇X∇u+√1+∥∇u∥2AX+⟨AX,∇u⟩ν−X(√1+∥∇u∥2)ν

for every spacelike . Thus

 Hessu(X,PkX)=¯¯¯¯¯¯¯¯¯¯¯Hessr(X,PkX)−√1+∥∇u∥2⟨PkAX,X⟩

On the other hand, we have the following decompositions

 X= X∗−⟨X,∇u⟩¯¯¯¯¯∇r PkX= (PkX)∗−⟨X,Pk∇u⟩¯¯¯¯¯∇r,

where , are respectively the components of , orthogonal to . Then

and, taking into account that

 ¯¯¯¯¯∇¯¯¯¯∇r¯¯¯¯¯∇r=0

we find

 ¯¯¯¯¯¯¯¯¯¯¯Hessr(X,PkX)=¯¯¯¯¯¯¯¯¯¯¯Hessr(X∗,(PkX)∗).

Hence, if we assume that for all timelike planes , then

 ¯¯¯¯¯¯¯¯¯¯¯Hessr(X,PkX)= ¯¯¯¯¯¯¯¯¯¯¯Hessr(X∗,(PkX)∗)≥−h′h(u)⟨X∗,(PkX)∗⟩ =

where is a solution of the problem

 {h′′−Gh=0h(0)=0, h′(0)=1

Therefore

 Hessu(X,PkX)≥ −√1+∥∇u∥2⟨PkAX,X⟩.

Tracing

 Lku≥−h′h(u)(ckHk+⟨∇u,Pk∇u⟩)+√1+∥∇u∥2ckHk+1.

Summarizing, we have proved the following

###### Proposition 10.

Let be an -dimensional spacetime. Assume that there exists a point such that and let be the Lorentzian distance function from . Given a smooth even function on , let be a solution of the Cauchy problem

 {h′′−Gh=0h(0)=0, h′(0)=1

and let be the maximal interval where is positive. Let be a spacelike hypersurface such that . If

 (6) KM(Π)≤G(r)

for all timelike planes , then

 (7) Lku≥−h′h(u)(ckHk+⟨∇u,Pk∇u⟩)+√1+∥∇u∥2ckHk+1.

On the other hand, if we assume that for all timelike planes in , the same computations yield the following

###### Proposition 11.

Let be an -dimensional spacetime. Assume that there exists a point such that and let be the Lorentzian distance function from . Given a smooth even function on , let be a solution of the Cauchy problem

 {h′′−Gh=0h(0)=0, h′(0)=1

and let be the maximal interval where is positive. Let be a spacelike hypersurface such that . If

 (8) KM(Π)≥G(r)

for all timelike planes , then

 (9) Lku≤−h′h(u)(ckHk+⟨∇u,Pk∇u⟩)+√1+∥∇u∥2ckHk+1.

In the following, under suitable bounds on the sectional curvature of the ambient spacetime, we will find some lower and upper bounds for the mean cuevature and the higher order mean curvatures associated to the immersion. In order to do it we will use the Omori-Yau maximum principle for the Laplacian and for more general elliptic operators (for more details and others applications of this technique see [5], [6]). Namely, if , where is a symmetric operator with trace bounded above, using the terminology introduced by S. Pigola, M. Rigoli and A. G. Setti in [14],we say that the Omori-Yau maximum principle holds on for if for any smooth function with there exists a sequence of points such that

 (10) (i) u(pi)>u∗−1i, (ii) ∥∇u(pi)∥<1i, (iii) Lu(pi)<1i.

Equivalently if , we can find a sequence such that

 (11) (i) u(qi)>u∗−1i, (ii) ∥∇u(qi)∥<1i, (iii) Lu(qi)>−1i.

Clearly the Laplacian belong to this class of operators. In this case, S. Pigola, M. Rigoli and A. G. Setti showed in [14] that a condition of the form

 (12) Ric(∇ρ,∇ρ)≥−C2G(ρ),

where is the distance function on to a fixed point and is a smooth function satisfying

 (13) (i) G(0)>0,(ii) G′(t)≥0on [0,+∞),(iii) G(t)−12∉L1(+∞),(iv) limsupt→∞tG(√t)G(t)<+∞.

is sufficient to guarantee the validity of the Omori-Yau maximum principle for the Laplacian on . Analogously, in [5], L. J. Alias, M. Rigoli and the author showed that the condition

 (14) K(∇ρ,X)≥−G(ρ),

where is any vector field tangent to and satisfies (13), is sufficient to guarantee the validity of the Omori-Yau maximum principle on for operators with the properties described above.
Applying the Omori-Yau maximum principle we find the following estimates for the mean curvature. The proof of the following theorems is essentially the same as that of Theorems 4.1 and 4.2 in [4].

###### Theorem 12.

Let be an -dimensional spacetime. Assume that there exists a point such that and let be the Lorentzian distance function from . Given a smooth even function on , let be a solution of the Cauchy problem

 {h′′−Gh=0h(0)=0, h′(0)=1

and let be the maximal interval where is positive. Let