Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold
Given a Riemannian manifold and an embedded hypersurface in , a result by R. L. Bishop states that infinitesimal convexity on a neighborhood of a point in implies local convexity. Such result was extended very recently to Finsler manifolds by the author et al. . We show in this note that the techniques in , unlike the ones in Bishop’s paper, can be used to prove the same result when is semi-Riemannian. We make some remarks for the case when only timelike, null or spacelike geodesics are involved. The notion of geometric convexity is also reviewed and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.
Key words and phrases:Semi-Riemannian manifolds, convex hypersurface, geodesics
2000 Mathematics Subject Classification:53C60, 53C22, 58E10
Let be an open of subset of and be a differentiable function in . We say that is locally convex at a point if there exists a neighborhood of such that the graph of the restriction of to is nowhere below the tangent plane at or, equivalently, all the straight lines through the point are nowhere above the graph of . If is twice differentiable then the above condition is satisfied if the Hessian of is positive semidefinite in a neighborhood of , while the positive semi-definiteness of at the single point is necessary but not sufficient, as the function at shows.
Let be the hypersurface in which is the graph of the function ; the following quadratic form associated to the Hessian of at a point , i.e.
is the second fundamental form of at . Thus, the convexity of a twice differentiable function in a neighborhood of is equivalent to the fact that the is positive semidefinite in the same neighborhood.
Now assume that is a smooth embedded hypersurface in a Riemannian manifold . The natural generalizations of the above notions are the following ones:
the first one becomes the requirement that there exists a neighborhood in of such that the intersections of the images of the geodesics through with velocity vector at tangent to with are contained on the closure of one of the two connected component or, equivalently, there exists a neighborhood of and a neighborhood of such that the is contained , where is one of the two connected component of ; this condition is called local convexity at ;
the second one becomes the requirement that there exists an open neighborhood of such that the second fundamental form of , with respect to a smooth unit normal vector field on , is positive semidefinite111According to the case when is the graph of a function and is the smooth unit vector which lies on the same side of the graph as the canonic unit vector defining the axis of the values of the function, from , we know that the geodesics issuing from and tangent to are locally contained in the closure of the component of a tubular neighborhood of individuated by . As the sign of changes if one changes with , it is clear that what is really important in this definition is the fact that is semidefinite in , either positive or negative. Clearly in the case when is negative semidefinite the geodesics tangent at to are locally contained in the component individuated by itself. for each , i.e.
where is the Levi-Civita connection of and is a vector field on extending ; this condition is called infinitesimal convexity in .
The global versions of these definitions require that the embedded hypersurface is orientable, i.e. must admits a smooth unit normal vector field . In such a case, is said infinitesimally convex if , for each , where is the second fundamental form of with respect to .
As in the case of the graph of a function, infinitesimal convexity at a neighborhood of implies local convexity. Surprisingly enough, this property was proved only relatively recently for a hypersurface in a Riemannian manifold having constant sectional curvature in  and in the general case in . Besides, the proof in  is rather involved and it seems to depend drastically on the Riemannian setting. In Section 2, we will describe which are the obstacles one encounters trying to extend Bishop’s proof when the ambient manifold is a Finsler or a semi-Riemannian one. The implication in the Finsler setting was obtained recently in . In Section 3 we will show that the techniques in  also give the implication in a semi-Riemannian manifold and in Section 4 we will consider the same problem in relation to the causal character of the geodesics and to another notion of convexity called geometric convexity. Finally, we will see some applications to geodesic connectedness of an open subset of a Lorentzian manifold by means of causal geodesics.
2. A review of Bishop’s proof
The proof in  that infinitesimal convexity in a neighborhood of a point implies local convexity at is based on a reduction of the general problem to the case of a Riemannian manifold of dimension two. Such a reduction is quite natural and it is based on the following idea. For any geodesic starting from with initial velocity tangent to at , consider the surface ruled by the geodesics orthogonal to and passing through . Two such surfaces , , , intersect along the geodesic which is orthogonal to at (see Fig. 1).
Then, for each , define the unit vector field which is the geodesic field (with respect to the Riemannian metric induced on by ) such that is orthogonal to for all . The surface can be parametrized by the coordinates and which are, respectively, the affine parameters of and (thus is represented on by the equation ). The unit vector field in which is tangent to can be written as , where is the smooth unit vector field in orthogonal to (oriented in such way that ). Using the infinitesimal convexity assumption, it is possible to obtain a differential inequality, satisfied by the function , implying that is non positive on . Then local convexity at follows from the observation that .
The delicate point in this argument is that could not be a well defined vector field in , for any convex neighborhood of in .
By convex neighborhood of a point in a Riemannian or semi-Riemannian manifold, here and hereafter, we mean an open neighborhood which has the property that any two point in are connected by a unique geodesic whose support is contained in (it is well known than any point on a Riemannian or semi-Riemannian manifold has a convex neighborhood, see for example [27, Ch. 5, Prop. 7]).
Bishop rules out this pathological case by using an estimate on the distance in of the focal points of in . Such estimate is based on the one hand on the fact that, the Gauss curvatures of the surfaces are uniformly bounded from above (in a neighborhood of ) by the sectional curvature of the ambient manifold (provided that the metric is at least ), on the other one on the fact that the geodesics in having as their velocity vector fields, leave orthogonally and then the the distance from (in ) of their points can be evenly controlled.
Such type of arguments have no straightforward extension when the ambient manifold is semi-Riemannian. Indeed, as shown in , if the sectional curvature is bounded for all timelike or spacelike planes at a point then it is constant at . Moreover the Morse index of spacelike geodesics is always . Thus, Rauch comparison theorems (see  for the case of geodesics starting orthogonally to a given geodesic) do not generally hold in semi-Riemannian geometry. Some results are available in literature, but they concern either causal geodesics (cf. [5, §11.2] for geodesics connecting two given points and [24, 16] for more general boundary conditions) or Jacobi fields along a given single geodesic of any causal character satisfying two different types of boundary conditions .
In , Bishop’s result was extended to a hypersurface in a Finsler manifold by using a different approach from that in . In the Finslerian setting, the problem of directly extending the proof in  is not related to the lack of satisfactory comparison results (cf. [1, §9.1]) but rather on the fact that the differential inequality satisfied by the function above (that comes from the evaluation of the second fundamental form of the curve , in the two dimensional manifold , which assumes by construction the same values of that of the hypersurface ) is obtained by using, in an essential way, the metric compatibility of the Levi-Civita connection. For a Finsler manifold, the role of the Levi-Civita connection can be played by the Chern connection which is metric compatible (and torsion free) if and only the Finsler structure is Riemannian (see, e.g. [1, Exercise 2.4.2]).222In some cases, the Chern connection which is a linear connection on the vector bundle over , the natural projection, reduces to a linear connection on , even if the Finsler metric is not a Riemannian one. When this happens the Finsler metric is said of Berwald type. Since from a theorem of Szabó (cf. [1, §10.1]), given a Berwald metric on , there exists a Riemannian metric such that its Levi-Civita connection coincides with the Chern connection of , we can state, as already observed in , that Bishop’s proof is also valid in any Berwald space.
The approach followed in  will be briefly described in the next section. Here, we would like to emphasize that when the Finsler metric comes from a Riemannian one (i.e. ) the result in  improves the one in  with respect to the smoothness assumption on the metric and on the hypersurface. Indeed from , it is enough that the metric is differentiable with locally Lipschitz differential (and we will assume the same on the semi-Riemannian metric ) while in  it assumed that it is .
In regarding to the smoothness of the hypersurface, we slightly improve the requirement in  that the hypersurface is an embedded submanifold locally described as the regular level of a twice differentiable function having Lipschitz second differential. Since at each point of an embedded submanifold there is a coordinate system adapted to (see e.g. [27, Ch.1, Prop. 28]), it will be clear in the proof of Lemma 3.5 that it is enough to assume that the hypersurface is a embedded submanifold of (of course, also must have a differentiable structure of class at least ).
3. Infinitesimal convexity implies local convexity
Let be a semi-Riemannian manifold of dimension and be an embedded non-degenerate hypersurface in . Let be a function defined in a neighborhood of such that and its gradient at any has the same orientation of the smooth unit normal unit vector field in with respect to which the second fundamental form of is defined. Let be the Hessian of , where is the Levi-Civita connection of . In a coordinate system of , using the Einstein summation convention, we get
for any and any , where are the Christoffel symbols of the metric . From (2), it’s easy to see that if is a geodesic then
The following lemma, concerning the equivalence between the second fundamental form of and the Hessian of a function as above, is well known. We give here a proof for the sake of completeness.
For all ,
By definition, , where is any vector field on extending and is the gradient of . Using the metric compatibility of the Levi-Civita connection we get . As and are orthogonal at any point of , we get on
hence on . As and we get the thesis. ∎∎
As already observed in [4, Section 3], locally defining the hypersurface as a regular level set of a function allows to extend the notion of infinitesimal convexity to hypersurfaces having points where the tensor induced by the ambient metric is degenerate. At these points, the notion of second fundamental form is meaningless. In what follows, unless differently specified, we will assume that is an embedded hypersurface in , non necessarily non-degenerate.
Let be a semi-Riemannian manifold and be a embedded hypersurface in . We say that is infinitesimally convex in a neighborhood of a point if there exists a neighborhood of in and a function such that , is a regular value of , and is semi-definite (either negative or positive), for all .
From Lemma 3.1, it is clear that this definition is independent of the function if is a non-degenerate hypersurface. A posteiori, from Theorem 3, the same is true also in the degenerate case. Indeed if there exists a function with respect to which is infinitesimally convex in , by Theorem 3, is locally convex in and then it is infinitesimally convex with respect to any other function. Moreover the “local convex side” of (i.e the closure of the connected component of where the geodesics starting at the points of with velocity vector tangent to are locally contained) is , if , and , if .
A very natural approach in trying to prove that infinitesimal convexity in a neighborhood of a point of a hypersurface implies local convexity is to evaluate along any geodesic arc starting at with initial velocity vector tangent to . The infinitesimal convexity assumption can be used to get a differential inequality satisfied by the function , at least when the image of is contained on the side of which is the candidate convex one (recall Remark 3.4). The differential inequality and the initial conditions satisfied by imply that is actually constant and equal to , that is the geodesic is contained in . This is an intermediate fundamental step to prove local convexity at . The remaining part of the proof is a quite trivial consequence of the fact that any point in a semi-Riemannian manifold has a convex neighborhood.
Assume that is infinitesimally convex in a neighborhood of . Let be a neighborhood of in and such that is a regular value of and , and for each . Let be a geodesic satisfying the initial conditions , and such that , then .
Since is an embedded submanifold, we can assume, without loss of generality, that is the domain of a coordinate system, , centered at and that , for each , and the function is equal to . Then the map , , is well defined. Let be the projection on of the geodesic , , where , are the components of the geodesic in the coordinate system (see Fig. 2).
Since is infinitesimally convex in ,
Thus we can estimate as follows
For each , let us call the symmetric bilinear operator on defined by the Christoffel symbols at . The first term in the summand in (4) is bounded above by , where is the norm of the bilinear operators on and is the Euclidean norm of the vector . Since is smooth on and the Christoffel symbols are Lipschitz on , this last quantity is bounded above by , where is a positive constant depending on and is the Euclidean norm of , hence it is equal to . By continuity, the second term in (4) is bounded above by , where is a positive constant depending on . Therefore, satisfies the differential inequality
and the initial condition and from [2, Lemma 3.1], it must be equal to on (i.e. ). ∎∎
Let be a semi-Riemannian manifold and be an embedded hypersurface in . Let and be a neighborhood of in , then is infinitesimally convex in if and only if it is locally convex in .
Let be an open subset of and be a function such that is a regular value, and .
Assume that is locally convex at any point , i.e there exists a neighborhood of such that is contained in the closure one of the connected component of , say . Let and consider the affinely parametrized geodesic such that , . Thus there exists such that the function is well defined and has a maximum point at . Therefore .
Assume now that is infinitesimally convex in , with for all . Let ; we are going to show that for any open convex neighborhood of in the set is also convex, i.e any two points in are joined by a unique geodesic whose support is contained in . Let be the subset of given by the couple of points that can be connected by a unique geodesic with support in . As is a connected subset of , it is enough to show that is non-empty and it is an open and closed subset in . Clearly, each couple can be connected by a constant geodesic, i.e. . If and is the unique geodesic connecting them and whose inner points are in , by smooth dependence of geodesics by boundary conditions in a convex neighborhood (cf. [27, Ch. 5, Lemma 9]), we can consider two small enough neighborhoods and of and, respectively, in , such that the unique geodesic in connecting to lies in a small neighborhood of , hence its points are contained in and is open. Now let and consider a sequence converging to . The sequence of geodesics , parametrized on and connecting to in converges in the topology to the geodesic connecting to . Thus lies in . As the points and are in , from Lemma 3.5, cannot be tangent to at any of its inner points.
Having proved that is a convex neighborhood in , the rest of the proof follows by contradiction. Indeed, assume that there exists and a sequence of vectors such that and . Let be the geodesic in connecting the first point of the sequence to . As converges to the sequence converges uniformly (actually in the topology) to the geodesic connecting to in . By uniform convergence, the image of must be contained in . Since and , from Lemma 3.5, . But . ∎∎
4. Some remarks and applications
4.1. Convexity with respect to the geodesics having the same causal character
In a semi-Riemannian manifold, the set of geodesics through a point can be divided into three disjoint subsets according to the causal character of the initial velocity vector at . It is natural to ask if the equivalence in Theorem 3.6 holds restricting the geodesics, or equivalently the tangent vectors to at any , involved in the definitions of local, or infinitesimal convexity, to one of these subsets.
To be more precise, let us give the definition of local and infinitesimal convexity, taking into account causality. Let be a non-degenerate embedded hypersurface of , and be the set of the timelike (resp. lightlike, spacelike) vectors in . Let be a neighborhood of in and be a unit normal smooth vector field on . We say that is time- (resp. null-, space-) locally convex at if there exists a neighborhood of such that is contained in the closure of the connected component of a tubular neighborhood of individuated by . Analogously, we say that a non-degenerate embedded hypersurface is time- (resp. null-, space-) infinitesimally convex in a neighborhood of if its second fundamental form with respect to is positive semidefinite on timelike (resp. null, spacelike) vector field on , i.e , for each and for all vectors . By Lemma 3.1, this last condition is equivalent to , for all , where is a function as in Definition 3.3. As in Remark 3.2, defining time- (resp. null-, space-) infinitesimal convexity in this last way allows one to consider also degenerate hypersurface.
Since the null cone at a point is the boundary of both the subsets of spacelike and timelike vectors at , if is time- or space-infinitesimally convex at a point then, by continuity, it is also null-infinitesimally convex.
We observe that Lemma 3.5 continues to hold for time- and space-infinitesimal convexities, up to taking a smaller as the upper limit of the interval of definition of the geodesic . Indeed the projection map is smooth and therefore, since is timelike (resp. spacelike), the curve remains timelike (resp. spacelike) on a right neighborhood of , thus inequality (4) is true for in such neighborhood (clearly, for null-infinitesimal convexity, this argument is invalid).
On the contrary, under the weaker hypothesis that is only time- (resp. space-) infinitesimally convex in , the proof of the “only if” part of Theorem 3.6 becomes wrong because the set , now defined as the set of the couples of points in that can be joined by a timelike (resp. spacelike) geodesic contained in is no longer closed, in fact the limit of a sequence of timelike or spacelike geodesics might be a null geodesic, Remark 4.1. Moreover, allowing the limit be a null geodesic leads to change the definition of including points that can be connected by a null geodesic but then becomes non-open. Thus we leave the following as an open problem:
Prove or disprove that for any smooth embedded hypersurface in a semi-Riemannian manifold (or, at least, in a Lorentzian manifold) time-, null- or space- infinitesimal convexity in a neighborhood of a point implies the same type (time-, null- or space-) of local convexity at .
A positive answer to this problem has been given in [12, Cor. 3.5 and Rem. 3.8] for null/time-infinitesimal and local convexities of a hypersurface of the type in a standard stationary Lorentzian manifold , by reducing the problem to a Finslerian one (cf. [13, 14] or ) and using the above mentioned result .
4.2. Geometric convexity
Let be an open subset of the semi-Riemannian manifold with differentiable boundary . Let be a function such that is a regular value, and , for each .
In what follows, infinitesimal convexity of will be always considered as defined in terms of the function globally defining as a regular level set.
Observe that under these assumptions is a a embededd and oriented hypersuface in (the orientation is given by the transversal vector field , where is the gradient operator with respect to any auxiliary Riemannian metric on ).
From Lemma 3.5, it immediately follows that if is infinitesimally convex at any of its points then the following condition, called (e.g. in [29, 4, 3, 2]) geometric convexity of or, equivalently, of , holds:
for any two points and for any geodesic arc from to , if is contained in then, actually, it is contained in .
(observe that the same also happens for respectively time/space-infinitesimal convexity of and timelike and spacelike geodesics).
This fact was already observed in [18, Th. 6] for a domain in a complete Riemannian manifold. We emphasize that, from a technical point of view, local convexity at each point of (which clearly implies geometric convexity), is harder to prove than the other convexity notions except in the case when strongly (i.e. (1) is satisfied with the strict inequality) infinitesimal convexity holds (cf. [29, Section 1], where strongly infinitesimal convexity is called strictly infinitesimal convexity); for example, in trying to prove that infinitesimal convexity in a neighborhood of a point implies local convexity at , we must encompass the possibility of a geodesic oscillating between and , as , with . In other words, the fact that is nonnegative is essential in differential inequality (5). Anyway, if geometric convexity holds we immediately get, as in the first part of the proof of Theorem 3.6, that is infinitesimal convex at any of its points. Thus we can state:
For any open subset of a semi-Riemannian manifold local, infinitesimal and geometric convexity of are equivalent; moreover time/space-infinitesimal convexity are respectively equivalent to time/space-geometric convexity.
We recall that the first chain of equivalence holds also for a open subset of a Finsler manifold (see [2, Cor. 1.2] and recall Remark 2.1), while both the equivalences between full infinitesimal and geometric convexity and time/space-infinitesimal convexity and time/space-geometric convexity were obtained in [4, Ths. 4.3, 4.4 and Appendix A] for a open subset of the type in a standard stationary Lorentzian manifold . The equivalence in the null case, already obtained in  for a static standard region, was proved in [3, Th. 2.5]. We observe that this last equivalence also follows by the results in  as in Remark 4.4.
Space-, null- and time-geometric convexity, in a Lorentzian setting, were introduced in  and were used there, together with variational methods, to prove existence and multiplicity results about the number of spacelike and timelike geodesics connecting a couple of points (for timelike ones, only certain chronologically related points are to be considered) in an open subset having space or time-geometrically convex boundary of a standard static Lorentzian manifold . These results were extended to standard stationary Lorentzian manifolds in . Remarkably, null- and time-geometric convexity of some open subsets of this type, contained in the outer Schwarzschild, Reissner-Nordström and Kerr spacetimes, have been proved in [25, §7] (see also [7, 17, 12]). We stress that these results were obtained by showing that the boundaries of such open subsets are strongly infinitesimally convex and indeed the advantage of the notion of infinitesimal convexity with respect to the other ones relies, of course, in its direct computability.
4.3. Some applications to geodesic connectedness
It is worth to observe that geometric convexity extends the classical notion of convexity of a subset of . For example, assume that is a smooth complete Riemannian manifold and is a smooth, connected open subset of having geometrically convex boundary ; then there exists a (non necessarily unique) geodesic connecting to , contained in and having length equal to the distance in between and . A way to prove that is to apply both a minimization and a penalization argument to the energy functional of the Riemannian manifold. We refer to [25, Cor. 4.4.7] for a proof, using these variational methods, of the existence of a geodesic connecting to ; the minimizing property of such a geodesic can be proved as in [21, Remark at p.448] while the case when is not differentiable and/or is not complete has been studied in .
This result also holds in a forward or backward complete Finsler manifold [2, Th. 1.3]. A more general result is obtained replacing the completeness of the Finsler manifold with the assumption that the closure in of any ball, with respect to the symmetric distance associated to the the pseudo-distance induced by on , is compact [2, Th. 1.3] and [12, Rem. 4.2].
Another way to prove such results is by using a shortening argument to the length functional. In an open subset of a Lorentzian manifold satisfying good causality properties, similar and in some aspects dual (cf. [27, p. 409]) arguments are applicable. We refer to [27, Ch. 14] for more details on the the definitions and notations about causality that we are going to use.
Let be a time-oriented Lorentzian manifold and be a causal curve on (i.e., assuming for simplicity that is piecewise smooth, ). The Lorentzian length of is defined as . Let be a subset of and , two points in . We will denote by the set of the future pointing causal curves such that and . We say that is causally related to in and we write , if . We define the time separation in , , as , if otherwise .
Let be a time-oriented Lorentzian manifold and be a , open subset such that is compact and strongly causal. Assume that is convex (recall the first part of Corollary 4.5), then for any with there is a future pointing causal geodesic such that (hence is finite).
Let be a sequence of curves, , such that , and . From [27, Ch. 14, Prop. 8] there is a limit sequence (see [27, Ch. 14, Def. 7]) for . Arguing as in the proof of [27, Ch. 14, Lemma 14] the limit sequence must be finite, and the last point has to be equal to . From the definition of a limit sequence, there exist a subsequence and sequences , , such that and then ; moreover there are convex neighborhoods associated to the limit sequence such that , for each . From the second part of the proof of Theorem 3.6, we know that the sets are convex neighborhoods. Consider the future pointing causal broken geodesic from to , with vertices , having one segment in each convex set (such broken geodesic is a quasi-limit of , [27, p. 406]). Consider also the future pointing causal geodesics connecting with in333Observe that such geodesics must be causal and future pointing as the points and are causally related, see [27, Ch. 14, Lemma 2]. . By the smooth dependence of geodesics by boundary conditions in a convex neighborhood ([27, Ch.5, Lemma 9]) the geodesics converge in the topology to and . Hence is contained in and, by the length maximizing property of each causal geodesic segment in (see [27, Ch. 5, Prop 34]), we get , where is the broken geodesics in with vertices and segments . Since belongs to , the portion of given by the first segment plus a small part starting at and contained in cannot be entirely contained in and then, by Lemma 3.5, it cannot be a pregeodesic. Therefore there exists a causal future pointing geodesic having length strictly greater than , connecting to the end point of . By choosing a sequence of points , such that , the future pointing causal geodesics defined by the points and and having support in converge, in the topology, to . Hence by Lemma 3.5, does not intersect in any of its inner points. As the end point belongs to , in an analogous way, replacing with , we can construct a broken geodesic segment between and , longer than plus the segment of from and , which does not intersect in any of its inner point. Thus in steps, we construct a future pointing causal broken geodesic connecting to , entirely contained in and such that . This contradiction comes from the fact that we have assumed that is not a geodesic. Using again Lemma 3.5, we conclude that must be contained in and . ∎∎
Since a globally hyperbolic Lorentzian manifold is strongly causal and the intersection of the causal future of with the causal past of is compact, as in Proposition 4.6 we get the following.
Let be a globally hyperbolic Lorentzian manifold and be a open subset. Assume that is convex, then for any with there is a future pointing causal geodesic such that .
We don’t know if we can replace the assumption of (infinitesimal or, equivalently, local) convexity of in Proposition 4.6 with time-infinitesimal convexity (or even time and null local convexity). In fact, in that case, we cannot state that the sets associated to the limit sequence in the proof of Proposition 4.6 are convex neighborhoods (recall Remark 4.2).
In some globally hyperbolic stationary Lorentzian manifold we can obtain the full geodesic connectedness of an open connected subset having convex boundary. Indeed the following proposition holds.
Let be a Lorentzian manifold endowed with a complete timelike Killing vector field and admitting a smooth, spacelike, complete Cauchy hypersurface. Let be a open connected subset of having convex boundary and which is invariant for the flow of . Then for each and in there is a geodesic connecting to whose support is in . Moreover, if is also non contractible, a sequence of spacelike geodesics in connecting to exists such that and, if is a flow line of with , then the number of timelike future pointing geodesics in connecting each to diverges as .
The above proposition extends the results in , where is also assumed to be static (i.e. its orthogonal distribution is integrable). It is based on the fact that the existence of a smooth, spacelike, Cauchy hypersurface and the completeness of imply that is isometric by the map , where is the flow of and the Cauchy hypersurface, to a standard stationary spacetime (cf. [11, Th. 2.3]). As is invariant for the flow of , then , with and since is complete, is complete. Then, as in the case without boundary in , one can prove that the projections on of the curves in a sublevel of the restriction of the energy functional of the standard stationary region to the manifold of the curves connecting to and such that is constant a.e., are contained in a compact subset of . Then, using a penalization argument as in [25, Section 4.2], the result follows as in [25, Sections 4.4 and 4.5].
We point out that, due to the particular variational setting, last part of the statement of the above Proposition follows as well assuming that is only time-infinitesimally convex (recall Problem 4.8). For a standard stationary region , a result of this type, without assuming global hyperbolicity of the ambient spacetime , has been obtained in [12, Th. 4.6].
I would like to thank M. Sánchez for suggesting to investigate the topic of this note and for several useful comments. Moreover, I thank the local organizing committee of the “VI International Meeting on Lorentzian Geometry, Granada 2011” for the financial support and the hospitality during the workshop.
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