Closed geodesics in stationary manifolds with strictly convex boundary 1

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1 Differential Geometry and its Applications 13 (2) North-Holland Closed geodesics in stationary manifolds with strictly convex boundary 1 A.M. Candela and A. Salvatore Dipartimento di Matematica, via E. Orabona 4, 7125 Bari, Italy Communicated by M. Willem Received 2 May 1999 Abstract: Let (M,, L ) be a non-compact Lorentzian manifold. If the metric, L is stationary and M has a strictly space-convex boundary, then variational tools allow to prove the existence of at least one closed spacelike geodesic in it. Keywords: Stationary Lorentzian manifolds, closed geodesics, Ljusternik Schnirelmann category, spaceconvex boundary. MS classification: 53C5, 58E5, 58E1. 1. Introduction The aim of this paper is to look for closed geodesics in a special class of Lorentzian manifolds with boundary. Let us recall that if (M,, L ) is a Lorentzian manifold, a smooth curve z :[a, b] M is a geodesic if D s ż(s) = for all s [a, b], where ż is the tangent field along z and D s ż is the covariant derivative of ż along z induced by the Levi-Civita connection of, L. It is well known that, if z is a geodesic, then there exists a constant E z, named energy of z, such that E z = ż(s), ż(s) L for all s [a, b] ; hence z is called spacelike, respectively lightlike or timelike, if E z is positive, respectively null or negative. This classification is called causal character of geodesics and comes from General Relativity (cf. [12, 15]). We say that a geodesic z :[a, b] M is closed if z(a) = z(b), ż(a) =ż(b). The research of closed geodesics with prescribed causal character is useful above all in order to have more information about the geometry of a Lorentzian manifold. Some existence results 1 Supported by M.U.R.S.T. (research funds ex 4% and 6%) and E.E.C., Program Human Capital Mobility (Contract ERBCHRXCT 94494) //$ see frontmatter c 2 Published by Elsevier Science B.V. All rights reserved PII S ()33-4

2 252 A.M. Candela, A. Salvatore for closed non-spacelike geodesics have been stated if M is compact (cf. [8, 9, 18]) while the existence of a closed spacelike geodesic holds, for example, if M = M S 1 is compact and of splitting type (cf. [1]). Anyway in [9] an example of a compact Lorentzian manifold without spacelike closed geodesics makes interesting to investigate more about the existence of spacelike closed geodesics in non-compact Lorentzian manifolds. To this aim, let us introduce the following definition. Definition 1.1. A Lorentzian manifold (M,, L ) is stationary if there exists a smooth connected finite-dimensional Riemannian manifold (M,, ) such that M = M R while, L has the following form: ζ,ζ L = ξ,ξ + δ(x ),ξ τ + δ(x ),ξ τ β(x ) ττ, (1.1) for any z = (x, t) M = M R and ζ = (ξ, τ), ζ = (ξ,τ ) T z M T x M R, where δ : M T M and β : M ], + [ are smooth. In particular, the metric (1.1) is called static if δ(x). It is easy to see that a closed geodesic on a stationary Lorentz manifold can never be timelike while it is lightlike only if it is reduced to a single point. On the other hand, the existence of closed spacelike geodesics has been proved in a stationary Lorentzian manifold if M is compact (cf. [13]). Here, we want to extend this result to stationary manifolds with strictly space-convex boundary (cf. Definition 1.5). First of all let us recall the following definition (cf. [14]). Definition 1.2. Let M be an open connected subset of a Lorentzian manifold M. We say that M has a strictly space-convex boundary M if any spacelike geodesic z :[a, b] M M is such that z(]a, b[) M. Remark 1.3. Let M be strictly space-convex and z :[a, b] M M be a closed spacelike geodesic. It is easy to see that, by extending z to a bigger interval by periodicity, there results z([a, b]) M. Remark 1.4. If M has a smooth boundary M, there exists a smooth function : M R such that (z) = z M, (z) > z M, (1.2) L (z) if z M (such a function can be defined by using the distance from the boundary). It is known that, if M is strictly space-convex too, then there results H L (z)[ζ, ζ] (1.3) for any z M and ζ T z M spacelike, where HL (z) : T z M T zm R denotes the Hessian of at the point z (cf. [14, Proposition 4.1.3]). Vice versa if H L (z)[ζ, ζ] < for all z M and ζ T z M spacelike, (1.4)

3 Closed geodesics in stationary manifolds 253 then M is strictly space-convex. Indeed, suppose that z :[a, b] M M is a spacelike geodesic such that there exists s ]a, b[ with z(s ) M. Then s is a minimum point of the C 2 function z; hence, d2 ds z(s 2 ) = H L (z(s ))[ż(s ), ż(s )] in contradiction with (1.4). From now on, let M be a stationary Lorentzian manifold. Definition 1.5. We say that (M,, L ), M = M R, is a stationary manifold with strictly space-convex boundary if there exists (M, g ) stationary Lorentzian manifold, M = M R, such that M is an open connected subset of M and g M =, L. Moreover (H 1 ) M is a C 2 submanifold of M ; (H 2 ) M M is complete with respect to the Riemannian structure on M (i.e., any geodesic x :]a, b[ M can be extended to a continuous curve x :[a, b] M M ); (H 3 ) M = M R is strictly space-convex. Remark 1.6. As M is a C 2 submanifold of M then there exists a C2 function φ : M R such that M = {x M : φ(x ) > }, M = {x M : φ(x ) = } (1.5) and φ(x ) for any x M, where φ denotes the gradient of φ with respect to the Riemannian structure on M.Ifweset there results (z) = φ(x ) for any z = (x, t ) M, (1.6) L (z) = ( φ(x ), ), where L denotes the gradient of with respect to the Lorentzian structure on M. It is easy to prove that satisfies (1.2); moreover by (H 3 ) it follows that satisfies (1.3), too. Theorem 1.7. Let (M,, L ), M = M R, be a stationary manifold with strictly spaceconvex boundary such that M is not contractible in itself and its fundamental group π 1 (M ) is finite or it has infinitely many conjugacy classes. Suppose that there exist ν, N > such that ν β(x ) N for all x M. (1.7) Assume that there exist x M, U C 2 (M, R + ) and some positive constants R, ρ, λ, such that x M, d(x, x ) R H U R (x )[ξ,ξ] λ ξ,ξ for all ξ T x M (1.8) and x M, d(x, x ) R φ(x ) ρ, (1.9)

4 254 A.M. Candela, A. Salvatore where d(, ) is the distance in M and HR U (x)[ξ,ξ] denotes the Hessian of the function U at x in the direction ξ both induced by the Riemannian structure, on M.Hereφ defines the boundary M as in (1.5). Furthermore, let us suppose that lim δ(x ),δ(x) =, (1.1) d(x,x ) + sup x M U(x ) δ (x ) < +, sup x M U(x ) β(x ) < +, (1.11) where 2 =, and δ (x) = sup{ δ (x)[v] : v T x M, v =1}. Then there exists at least one spacelike closed geodesic in M. Remark 1.8. If δ(x), Theorem 1.7 implies the existence of closed geodesics in static manifolds with strictly space-convex boundary. Let us point out that z = (x, t) is a closed geodesic in a static manifold if and only if t is constant and x is a closed geodesic in M (see, e.g., [13, Remark 2.9]). Consequently, the study of closed geodesics in static Lorentzian manifolds can be reduced to the research of closed geodesics in Riemannian manifolds; in particular, by [6, Theorem 1.2] it follows the existence of closed geodesics in static manifolds with non-smooth convex boundary. Remark 1.9. It is easy to see that if z = (x, t) is a closed geodesic, then for any c R z c = (x, t + c) is a closed geodesic, too. 2. Variational setting In this section we want to introduce a suitable functional whose critical points can be led back to closed geodesics. From now on, let (M,, L ) be a stationary Lorentzian manifold such that M = M R while its metric, L is given in (1.1). Assume [a, b] = [, 1] = I. For all n N, define the Hilbert space { H 1 (S 1, R n ) = γ L 2 ( I, R n ) : γ is absolutely continuous, } γ 2 ds < +, γ() = γ(1) equipped with the norm γ 2 = γ γ 2 2 = γ (s) 2 ds + γ (s) 2 ds, where 2 is the usual norm in L 2 (I, R n ). By Nash Embedding Theorem we can assume that M is a submanifold of an Euclidean space R N and its Riemannian metric, is just the Euclidean metric on R N ; hence, the following definition can be stated: 1 = 1 (M ) = { x H 1 (S 1, R N ) : x(s) M for all s I }.

5 Closed geodesics in stationary manifolds 255 It is well known that 1 is a submanifold of H 1 (S 1, R N ) and for any x 1 the tangent space to 1 at x is such that T x 1 { ξ H 1 (S 1, R N ) : ξ(s) T x(s) M for all s I }. Assume Z = 1 H 1 (S 1, R). Clearly Z is a Hilbert manifold such that at any z Z the tangent space is T z Z T x 1 H 1 (S 1, R); moreover, on such a manifold the action integral f : z = (x, t) Z f (z) R is defined by setting f (z) = 1 ż, ż 2 L ds = 1 ( 2 ẋ, ẋ +2 δ(x ), ẋ ṫ β(x )ṫ 2 ) ds. Standard arguments allow to prove that f is a C 1 functional and for any z = (x, t) Z and ζ = (ξ, τ) T z Z, there results f (z)[ζ ] = ẋ, ξ ds δ(x ), ξ ṫds+ δ (x )[ξ ], ẋ ṫds β(x ),ξ ṫ 2 ds δ(x ), ẋ τ ds β(x ) ṫ τ ds. (2.1) It is easy to see that if z is a closed geodesic in M then z Z is a critical point of f ; moreover, the vice versa can be proved. Proposition 2.1. If z = (x, t) is a critical point of f in Z then it is a closed geodesic in M. Proof. Let z = (x, t) Z be a critical point of f. Then by (2.1) it is f (z)[(, τ)] = δ(x ), ẋ τ ds for all τ C (I, R); hence, a constant c R exists such that δ(x ), ẋ β(x ) ṫ = c almost everywhere in I. β(x ) ṫ τ ds = (2.2) By integrating, there results ( δ(x), ẋ /β(x)) ds c = K (x ) = (1/β(x)) ds R ; (2.3) whence ṫ = δ(x), ẋ K (x) β(x) almost everywhere in I. (2.4) So, arguing as in [13], it can be proved that x C 2 (I, R N ), t C 2 (I, R) and z is a closed geodesic in M. Now our problem is to find critical points of f in Z. More precisely, by Remark 1.9 we have just to look for critical points of f in Z = 1 H 1, with H 1 ={t H 1 (S 1, R) : t() = t(1) = }.

6 256 A.M. Candela, A. Salvatore But, unlike the Riemannian case, the action functional f is unbounded from above and from below in Z, so to overcome this difficulty we introduce a new functional, bounded from below, whose critical points can be related to those ones of f. Proposition 2.2. Let : x 1 (x) H 1 be the C1 function such that s δ(x ), ẋ K (x ) (x )(s) = dσ for all s I. (2.5) β(x ) The following assumptions are equivalent: (i) z Z is a critical point of f ; (ii) z = (x, t) is such that x 1 is a critical point of the functional J(x ) = 1 δ(x), ẋ 2 ( ẋ, ẋ + K 2 ) (x) ds 2 (2.6) β(x) β(x) and t = (x). Moreover, if (i) or (ii) holds, it is f (x, (x )) = J(x ). (2.7) Proof. Since f is a C 1 functional on Z, let us consider the partial derivatives of f in z = (x, t) given by f x (z)[ξ ] = f (z)[(ξ, )] for all ξ T x 1, f t (z)[τ ] = f (z)[(, τ)] for all τ H 1. Clearly, the critical points of f in Z have to be in the set N = {z Z : f t (z) }. It is obvious that (2.2) and (2.4) imply z = (x, t ) N t = (x ) ; (2.8) whence, N is the graph of the C 1 map and a smooth submanifold of Z. So the functional J in (2.6) is just the restriction of f to N; hence, it is a C 1 functional such that there results J (x )[ξ ] = f x (x, (x ))[ξ ] for every x 1,ξ T x 1. (2.9) Let z = (x, t) Z. As for any (ξ, τ) T z Z it is f (z)[(ξ, τ)] = f x (z)[ξ ] + f t (z)[τ ], then (2.8) and (2.9) complete the proof. Lemma 2.3. For any x 1 there results J(x ) 1 2 Hence, J(x) and ẋ, ẋ ds. (2.1) J(x ) = x is constant.

7 Closed geodesics in stationary manifolds 257 Proof. By (2.3) and the Hölder inequality it follows that K 2 (x ) then (2.1) holds. 1 β(x) ds δ(x),ẋ 2 β(x) ds, As already remarked in Section 1, by Proposition 2.2 and Lemma 2.3 it follows that any non-trivial closed geodesic in M has to be spacelike. In order to prove the existence of critical points of J let us introduce some results of the Ljusternik Schnirelman theory (for more details, see, e.g., [16, 17]). Definition 2.4. Let X be a topological space. Given A X, the Ljusternik Schnirelman category of A in X, denoted by cat X (A), is the least number of closed and contractible subsets of X covering A. If it is not possible to cover A with a finite number of such sets it is cat X (A) =+. We assume cat(x) = cat X (X). In order to state the classical Ljusternik Schnirelman Multiplicity Theorem we need the following definition. Definition 2.5. Let be a Riemannian manifold and g : R a C 1 functional. We say that a sequence (x n ) n N is a Palais Smale sequence, briefly (PS) sequence, if sup g(x n ) < + and lim n N n + g (x n ) = (here g (x n ) goes to in the norm induced on the cotangent bundle by the Riemannian metric on ). Moreover, g satisfies the Palais Smale condition, briefly (PS), ifany(ps) sequence has a convergent subsequence. Theorem 2.6. Let be a smooth Riemannian manifold and g : R ac 1 functional which satisfies (PS). Let us assume that is complete or every sublevel of g in is complete. If k N, k >, let us define Ɣ k = { A : cat ( A) k}, c k = inf sup A Ɣ k x A g(x ). If Ɣ k and c k R, then c k is a critical value of g. (2.11) Remark 2.7. Let and g be as in Theorem 2.6. If g is bounded from below, then for any c R it is cat (g c ) < +, where g c ={x : g(x) c}. The following result, due to Fadell and Husseini (cf. [7]), allows to evaluate the Ljusternik Schnirelman category of the space of loops 1.

8 258 A.M. Candela, A. Salvatore Proposition 2.8. Let M be a connected finite-dimensional manifold which is not contractible in itself. Suppose that its fundamental group π 1 (M ) is not infinite with finitely many conjugacy classes. Then cat( 1 ) =+ and 1 has compact subsets of arbitrarily high category. 3. Penalization arguments Let (M,, L ), M = M R, be a stationary manifold with strictly space-convex boundary. Since M is not complete and, eventually, not bounded, the functional J does not satisfy the (PS) condition. Indeed we can consider a sequence (x n ) n N of constant curves in 1 such that x n x M or d(x n, x ) + as n + for a certain x M (when M is not bounded, too): (x n ) n N is a (PS) sequence without subsequences converging in 1. Thus we will introduce a family of penalized functionals ( f ε ) ε> in such a way that every J ε, associated to f ε, satisfies the (PS) condition. Here and in the following, we assume that M is not bounded and there exists U C 2 (M, R + ) such that (1.8) holds (otherwise the proof of Theorem 1.7 is simpler). Fixed ε>, let ψ ε : R + R + be a C 2 cut-function defined as follows: if s 1 ε, ψ ε (s) = + µ n ( s 1 ) n if s > 1 (3.1) n! ε ε, n=3 where µ = max{1,λ}, λ given in (1.8). It is easy to prove that ψ ε (s) µψ ε (s) for any s R +, (3.2) ψ ε (s) ψ ε (s) for any ε ε, s R +. (3.3) For any x M, define ( ) 1 φ ε (x ) = ψ ε, φ 2 (x) U ε (x ) = ψ ε (U(x )), (3.4) where φ is as in Remark 1.6. Let us penalize the action functional f in the following way f ε (z) = f (z) + φ ε (x ) ds + U ε (x ) ds, z = (x, t ) Z. By standard arguments f ε is of class C 1 ; moreover, arguing as in Proposition 2.1, any critical point z ε of f ε is C 2 and solves the following boundary problem D s ż ε = 2ψ ε ż ε () =ż ε (1), ( 1 2 (z ε ) ) L (z ε ) 3 (z ε ) ψ ε ( U (z ε )) L U (z ε ), (3.5)

9 Closed geodesics in stationary manifolds 259 where is as in (1.6) and U(z) = U(x) if z = (x, t). Remark 3.1. Since the penalization terms do not depend on the variable t, there results f ε (z)[(,τ)] = f (z)[(,τ)] for all τ H 1. Then, Proposition 2.2 holds for the functionals f ε and J ε by using the same map, where J ε (x ) = J(x ) + φ ε (x ) ds + Moreover, since ψ ε is positive, (2.1) implies J ε (x ) 1 2 U ε (x ) ds, x 1. ẋ, ẋ ds for all x 1. (3.6) In order to relate the critical points of J ε to those ones of J we need the following technical lemma. Lemma 3.2. Let U C 2 (M, R + ), λ>, R > and x M be such that (1.8) holds and {x M : d(x, x ) R} is complete. Then there exist some positive constants c 1, c 2, c 3 such that for any x M there results U(x ), U(x ) 1/2 λ d(x, x ) c 1, U(x ) 1 2 λ d2 (x, x ) c 2 d(x, x ) c 3. Proof. See [4, Lemma 2.2]. Remark 3.3. The function U satisfying (1.8) is introduced in order to give a control on M at infinity; hence, it is not restrictive to assume that U is bounded on bounded sets and it is possible to choose Clearly, there results U sup { U(x ) : x M, d(x, x ) R + 1 }. U(x ) > U d(x, x ) R + 1. Proposition 3.4. Let (M,, L ) be a stationary manifold with strictly space-convex boundary such that the hypotheses (1.7) (1.11) hold. Taken L, M >, there exists ε = ε (L, M) > such that for every ε ],ε [ if x ε 1 satisfies J ε (x ε ) =, L J ε (x ε ) M, (3.7) then J(x ε ) = J ε (x ε ), J (x ε ) =. Proof. It is enough to prove that there exist ε 1, ε 2 > such that any x ε 1 which satisfies (3.7) is such that sup U(x ε (s)) < 1 s I ε if ε ε 1, (3.8) inf s I φ2 (x ε (s)) >ε if ε ε 2. (3.9)

10 26 A.M. Candela, A. Salvatore First, let us prove (3.8). Arguing by contradiction, assume that there exist ε n (x n ) n N 1 such that and J ε n (x n ) =, L J εn (x n ) M for all n N (3.1) and sup U(x n (s)) 1. s I ε n As U is bounded on bounded sets (see Remark 3.3) it follows that sup { d(x n (s), x ) : s I, n N } =+. (3.11) On the other hand, (3.6) and (3.1) imply ( ) ẋ n, ẋ n ds is bounded ; (3.12) thus (3.11) and (3.12) give n N inf d(x n (s), x ) + as n +. (3.13) s I Clearly, by (1.9) and (3.13) there exists n 1 N such that for n n 1 it is inf s I φ(x n (s)) ρ> εn ; whence, (3.1) and (3.4) imply φ εn (x n (s)) = φ ε n (x n (s)) = for all s I. (3.14) Let us consider n n 1. By (3.1) and Remark 3.1, defined t n = (x n ), z n = (x n, t n ) is a C 2 critical point of f εn, where by (3.14) it is f εn (z) = f (z) + U ε n (x) ds for all z = (x, t) Z. In particular, for any ξ T xn 1 it results = f ε n (z n )[(ξ, )] = + δ(x n ), ξ ṫ n ds 1 2 ẋ n, ξ ds + δ (x n )[ξ ], ẋ n ṫ n ds β(x n ),ξ ṫ 2 n ds + U εn (x n ),ξ ds ; whence, (2.5) and simple calculations prove that x n is a C 2 solution of the following equation D s ẋ n + δ(x n ),Dsẋn β(x n ) δ(x n ) = δ (x n )[ẋ n ], ẋ n β(x n ) + ( δ(x n ), ẋ n K(x n ) ) β(x n ),ẋ n β 2 (x n ) ( + ṫ n δ (x n ) δ (x n ) ) [ẋ n ] 1 2 ṫ 2 n β(x n ) + ψ ε n (U(x n )) U(x n ), δ(x n ) δ(x n ) (3.15) where δ (x n ) is the adjoint of δ (x n ). Taken x M, let us define the linear operator A(x ) : v T x M A(x )[v] = δ(x),v β(x) δ(x ) T x M.

11 Closed geodesics in stationary manifolds 261 Clearly, it is δ(x ) 2 A(x ) for any x M β(x ), (3.16) with h(x) = sup { h(x)[v] : v T x M, v =1} for any linear and continuous operator h(x) : T x M T x M ; hence, by (1.7), (1.1) and (3.16) it is lim A(x ) = d(x,x ) + which implies the existence of R 1 R such that, taken x M verifying d(x, x ) R 1, the operator I + A(x) is invertible. Let B(x) = (I + A(x)) 1 be its inverse. In particular, by (3.13) there exists n 2 N, n 2 n 1, such that for n n 2 and for all s I there results d(x n (s), x ) R 1 R (3.17) and B(x n (s)) is well defined. Thus (3.15) becomes D s ẋ n = δ (x n )[ẋ n ], ẋ n B(x β(x n ) n )[δ(x n )] + ( δ(x n ), ẋ n K(x n ) ) β(x n ),ẋ n B(x β 2 (x n ) n )[δ(x n )] + ṫ n B(x n ) [ (δ (x n ) δ (x n ))[ẋ n ] ] 1 2 ṫ 2 n B(x n )[ β(x n )] + ψ ε n (U(x n )) B(x n )[ U(x n )]. Let us define u n (s) = U(x n (s)). By (3.5) it is u n () = u n (1), then by (1.8) and (3.17) the previous equation implies ( = ü n (s) ds = H U R (x n )[ẋ n, ẋ n ] + U(x n ), D s ẋn ) ds We claim that λ = λ lim n ẋ n, ẋ n ds + ẋ n, ẋ n ds 1 ṫ 2 2 n + ṫ 2 n U(x n ), D s ẋ n ds δ (x n )[ẋ n ], ẋ n β(x n ) ( δ(xn ), ẋ n K(x n ) ) β(x n ), ẋ n β 2 (x n ) ṫ n U(xn ), B(x n ) [ (δ (x n ) δ (x n ))[ẋ n ] ] ds U(xn ), B(x n )[ β(x n )] ds ψ ε n (U(x n )) U(x n ), B(x n )[ U(x n )] ds. U(xn ), B(x n )[δ(x n )] ds U(xn ), B(x n )[δ(x n )] ds ds =. (3.18)

12 262 A.M. Candela, A. Salvatore In fact, (1.7), (1.1) and (3.12) imply δ(x n ), ẋ n β(x n ) ds = o(1), δ(x n ), ẋ n 2 β(x n ) ds = o(1), then K (x n ) = o(1) by definition (2.3); hence, (3.18) follows by (2.5) (here o(1) is any infinitesimal sequence). Then the definition of J εn and (3.1) give δ(x ẋ n, ẋ n ds = 2J εn (x n ) n ), ẋ n 2 ds β(x n ) + K (x n ) β(x n ) ds 2 U εn (x n ) ds 2L 2 U εn (x n ) ds + o(1). Moreover, by using also the hypotheses (1.11) and arguing as in [5, Appendix], it can be proved that δ (x n )[ẋ n ], ẋ n U(xn ), B(x β(x n ) n )[δ(x n )] ds = o(1), ( δ(xn ), ẋ n K(x n ) ) β(x n ), ẋ n U(xn ), B(x β 2 (x n ) n )[δ(x n )] ds = o(1), ṫ 2 n ṫ n U(xn ), B(x n ) [ (δ (x n ) δ (x n ))[ẋ n ] ] ds = o(1), U(xn ), B(x n )[ β(x n )] ds = o(1), U(xn ), B(x n )[ U(x n )] 2 for n large enough. By (3.2) and the previous formulas there results λ ẋ n, ẋ n ds + 2 ψ ε n (U(x n )) ds + o(1) ( 2λL + 2 ψ ε n (U(x n )) λψ εn (U(x n )) ) ds + o(1) 2λL + o(1) which gives a contradiction. Whence, (3.8) holds and for n large enough it is U εn (x n (s)) = for all s I. (3.19) Now suppose that (3.9) does not hold, then there exist ε n and (x n ) n N in 1 such that (3.1) is satisfied and inf φ(x n (s)) ε n. (3.2) s I By the first part of this proof, if n is large enough (3.19) holds; hence, z n = (x n, (x n )) is a C 2

13 Closed geodesics in stationary manifolds 263 critical point of f εn satisfying the following equation ( ) 1 D s ż n = 2ψ L (z n ) ε n 2 (z n ) 3 (z n ). Arguing as in [1, Lemma 4.7] it is possible to prove that (x n ) n N converges in H 1 (S 1, R N ) to a curve x 1 (M M ) and z = (x, (x)) solves the equation D s ż(s) = γ (s) L (z(s)), where γ L 2 (I, R). Moreover, as in the proof of [1, Theorem 5.1] (see also [11, Lemma 3.8]), the condition (1.3) implies that z is a closed geodesic in M M. Let us remark that by [1, Remark 4.5] it is lim n + ( ) 1 ψ εn ds =, φ 2 (x n ) then J εn (x n ) J(x) as n + ; hence, by (2.7) and (3.1), z is a spacelike geodesic. Since (3.2) implies that z touches the boundary M, Definition 1.2 and Remark 1.3 give a contradiction. 4. Proof of the main theorem Let M = M R be a stationary manifold with strictly space-convex boundary such that the hypotheses of Theorem 1.7 hold. Lemma 4.1. Let φ be as in Remark 1.6 and assume that (1.9) holds. If (x n ) n N is a sequence in 1 such that ( ) ẋ n, ẋ n ds is bounded n N and there exists (s n ) n N I such that then lim n + φ(x n (s n )) =, 1 lim ds =+. n + φ 2 (x n ) Proof. Cf. [2, Lemma 3.2]. Lemma 4.2. Let ε> be fixed. For any c R the sublevel J c ε = {x 1 : J ε (x ) c} is a complete metric space; moreover, J ε satisfies the (PS) condition.

14 264 A.M. Candela, A. Salvatore Proof. Let ε>and c R be fixed. Clearly, the sets { } { ẋ, ẋ ds : x J c 1 ε, φ ε (x )ds : x J c ε } and { } U ε (x )ds : x J c ε are bounded, then by Lemmas 4.1 and 3.2 it can be easily deduced that there exist r, µ> such that J c ε 1 ( B r,µ ), B r,µ = { x M : d(x, x ) r, φ(x ) µ }. Since B r,µ is a compact subset of M, then 1 (B r,µ ) is complete which implies that the closed subset Jε c is complete, too. Let us prove, now, that J ε verifies the (PS) condition. Let (x n ) n N 1 be a (PS) sequence, i.e., ( J ε (x n )) n N is bounded, (4.1) lim J n + ε (x n ) =. (4.2) By (4.1) and the previous remark there exist r, µ>such that x n 1 ( B r,µ ) for any n N and whence, ( ) ẋ n, ẋ n ds n N is bounded ; (x n ) n N is bounded in H 1 (S 1, R N ) (4.3) and there exists x 1 (B r,µ ) such that x n x weakly in H 1 (I, R N ) and uniformly in I up to subsequences. By [3, Lemma 2.1] it follows that there exist two bounded sequences (ξ n ) n N and (ν n ) n N in H 1 (I, R N ) such that x n x = ξ n + ν n, ξ n T xn 1 for any n N, ξ n weakly in H 1 ( I, R N ) and ν n strongly in H 1 ( I, R N ). (4.4) Moreover, taken t n = (x n ), by (2.5) and (4.3) the sequence (t n ) n N is bounded in H 1 (I, R), then, up to subsequences, t n t weakly in H 1 (I, R). By (4.2), (2.8) and (2.9) imply o(1) = J ε (x n )[ξ n ] = f ε (z n )[(ξ n, τ n )], (4.5) where z n = (x n, t n ) and τ n = t n t. By the definition of f ε, (4.5) becomes o(1) = ẋ n, ξ n ds + + δ (x n )[ξ n ], ẋ n ṫ n ds + δ(x n ), ξ n ṫ n ds δ(x n ), ẋ n τ n ds 1 β 2 (x n )[ξ n ] ṫ 2 n ds + β(x n ) ṫ n τ n ds φ ε (x n )[ξ n ] ds + U ε (x n )[ξ n ] ds.

15 Closed geodesics in stationary manifolds 265 Clearly, by (4.3) and (4.4) it follows that φ ε (x n )[ξ n ] ds = o(1), U ε (x n )[ξ n ] ds = o(1). Then, arguing as in [13, Lemma 3.2], it is possible to prove that ξ n inh 1 (I, R N ); whence, x n x strongly. Lemma 4.3. For any c R it results Proof. Setting cat 1( J c ) < +. F(x ) = ẋ, ẋ ds, by (2.1) it is J c F 2c ; moreover, since cat 1(F k )<+ for all k R (see [6, Lemma 4.1]), the monotonicity property of the Ljusternik Schnirelmann category gives the conclusion. Proof of Theorem 1.7. Let L > be fixed. Lemma 4.3 implies that there exists k N such that B J L for all B Ɣ k, where Ɣ k is defined as in (2.11) and J L ={x 1 : J(x) >L}. Since J L J ε,l for all ε>, it is B J ε,l for all B Ɣ k ; hence, L c ε,k, where c ε,k = inf sup J ε (x ). (4.6) B Ɣ k x B By Proposition 2.8 there exists a compact set K 1 such that cat 1(K ) k. By (3.3) it is J ε (x) J 1 (x) for all x 1 if ε 1, then by (4.8) for all ε 1 it follows L c ε,k M, where M = max J 1 (x ). (4.7) x K Since for all ε 1 Lemma 4.2 and Theorem 2.6 imply the existence of at least one critical point x ε of J ε such that J ε (x ε ) = c ε,k satisfies (4.9), Proposition 3.4 implies that, if ε is small enough, x ε is a critical point of J. Whence, z ε = (x ε, (x ε )) is a closed geodesic in M such that f (z ε ) = J(z ε ) L. Remark 4.4. Clearly, for any L > ifε is small enough there exists a closed geodesic z ε such that f (z ε ) L. In particular, there exists at least a closed spacelike geodesic. Unluckily, we have not a multiplicity result since the found geodesics may not be geometrically distinct. References [1] F. Antonacci and R. Sampalmieri, Closed geodesics on compact Lorentzian manifolds of splytting type, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) [2] V. Benci, Normal modes of a Lagrangian system constrained in a potential well, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) [3] V. Benci and D. Fortunato, On the existence of infinitely many geodesics on space time manifolds, Adv. Math. 15 (1994) 1 25.

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