estimates of the number of periodic trajectories on a class of lorentz manifolds

Nonlinear Analysis 34 (1998) 927–939

Estimates of the number of periodic trajectorieson a class of Lorentz manifolds

Juan MolinaI.C.T.P., Strada Costiera 11, 34100 Trieste, Italy; and Dipartimento di Matematica, Universit�a di Pisa,

Via Filippo Buonarroti 2, 56127 Pisa, Italy

Received 18 April 1996; accepted 18 December 1996

Keywords: Lorentzian manifold; T -periodic geodesics; Critical points; Morse theory

1. Introduction

We set M =RN ×R and we equip it with a Lorentz structure, i.e. a symmetricnondegenerate (0,2) tensor �eld g on M of constant index 1.In other words, g smoothly assigns a scalar product g(z)[·; ·] to each point z=(x; t)

∈M (de�ned on the tangent space TzM ≡RN ×R), such that every matrix representa-tion of g(z) has exactly one negative eigenvalue.Let z(s)= (x(s); t(s)); s∈ I = [0; 1] be a smooth curve on M =RN ×R; z is called

a T -periodic trajectory if it is a geodesic for the metric g in (1.1.2) below † , and itsatis�es the condition:

x(0)= x(1); x(0)= x(1)

t(0)= 0; t(1)=T

t(0)= t(1):

(1.0.1)

It is well known that if z(s) is a geodesic on M; E(z(s)) =Ez = g(z(s))[z(s); z(s)] isa constant with respect to s (see 1.1.2 below); the curve z is said to be time-like,light-like or space-like if Ez¡0 or, respectively, Ez =0; Ez¿0 (see [1]).Suitable Lorentz manifolds are used in Relativity theory in order to describe the

physical space–time, when M =R3×R.

†We recall that z(s) is a geodesic if Dsz(s)= 0; where z(s) is the tangent vector to z at z(s); and Dsz isthe covariant derivative of z in the direction of z; with respect the Lorentz metric g.

0362-546X/98/$19.00 ? 1998 Published by Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(97)00576 -2

928 J. Molina / Nonlinear Analysis 34 (1998) 927–939

A time-like periodic trajectory is physically interpreted as the periodic orbit of aparticle of positive mass. A null geodesic represents the world line of a light ray.Space-like geodesics are not trajectories of a particle, but they are important in orderto study global geometric properties of Lorentz manifolds.Note that there are no time-like or light-like (not trivial) closed curves. In fact, if

z(s)= (x(s); t(s)) is a closed curve, from the Rolle Theorem there exists s0 ∈ [0; 1] suchthat t(s0)= 0; then

Ez(s0) = 〈�(x)x(s0); x(s0)〉 − �(x(s0))t(s0)2 = 〈�(x)x(s0); x(s0)〉:It is easy to see that if xj ∈RN is a critical point of �; z(s)= (xj; sT ) is a trivial

T -periodic trajectory.The research for nontrivial T -periodic trajectories has been investigated in the last

years by many authors such as Benci, Fortunato, Giannoni, Greco, Masiello, Pisani(see [2, 3] and the references quoted therein).In this work we will consider the following Lorentzian structure, for z=(x; t)∈

RN ×R and (�; �)∈RN ×R; letg(z)[(�; �); (�; �)]= g(x)[(�; �); (�; �)]= 〈�(x)�; �〉 − �(x)�2 (1.0.2)

where 〈�(x)�; �〉= ∑Ni; j=1 �i; j(x)�i�j and �(x)¿0 for all x.

We assume that:

�∈C2(RN ;R);

�i; j ∈C2(RN ;R); i; j; ∈{1; : : : ; n};

lim|x|→+∞

@�i; j(x)@xk

=0;

there exist �¿0 :∑i; j

�i; j(x)�i�j ≥ �∑i

�2i ;

lim|x|→+∞

�(x)= sup{�(x) : x∈RN}=d; d¿0

lim|x|→+∞

grad �(x)= 0:

(1.0.3)

In this paper we apply the Morse theory to the functional de�ned on �1(S1;RN )(see Section 2) by

J (x)=12

∫ 1

0〈�(x)x; x〉ds− T 2

2

(∫ 1

0

ds�(x)

)−1: (1.0.4)

We improve results obtained in [2] by using the Morse Theory as in [4, 5] and [6].In order to establish our result we need the following Lemma.

Lemma 1.1. Let K�= {x1; : : : ; xr} be the set of critical points of �. If m(xi; J ) rep-resents the Morse index (see De�nition 3:4) of xi respect to the functional J on�1(S1;RN ); then r is an odd number.

J. Molina / Nonlinear Analysis 34 (1998) 927–939 929

Moreover we have thatr∑

i= 1

(−1)m(xi ; J ) = 1:

Proof. It will be done in Section 4.Hence, we can de�ne

X = {x∈K� :m(x; J ) is even}Y = {x∈K� :m(x; J ) is odd}:

(1.1.1)

If we put k1 = r+12 and k2 = r−1

2 ; the sets

X = {y1; : : : ; yk2 ; yk1}Y = {u1; : : : ; uk2}

(1.1.2)

are ordered in such a way that

m(y1; J ) ≤ · · · ≤ m(yk1 ; J )and

m(u1; J ) ≤ · · · ≤ m(uk2 ; J ):(1.1.3)

Finally, we set

mj =m(yj; J ); j=1; : : : ; k1

�j =m(uj; J ); j=1; : : : ; k2:(1.1.4)

Now we can announce our results:

Theorem 1.2. Let � and � be as in (1:1:3).Assume that x1; : : : ; xr are the critical points of � where r is an odd number; and

�(xj)¡d for j=1; : : : ; r.Moreover assume that the matrices �(xj) are invertible; for all j=1; : : : ; r; and

suppose that T 6=2n�√2=√�kj where n∈N and �kj are the eigenvalues of the matrices

(�(xj))− 12 �′′(xj)(�(xj))−

12 for k =1; : : : ; N:

Then if n(T ) represents the number of nontrivial T -periodic trajectories countedwith multiplicity; we have that

n(T ) ≥k1∑j=2

12|mj − �j−1|+ 12m1 (1.2.1)

where mj and �j are as in (1:1:7) and (1:1:8).


Remark 1.3. If � has only one critical point x1; then in (1.1.9) we get

n(T ) ≥ m(x1)2

:

On the other hand from Remark 2.3 below it follows that

limT→+∞

m(x1; T )=+∞:Then, from Theorem 1.2 there exist in�nitely many nontrivial T -periodic trajectories

(time-like). In particular we get the result obtained by [2] (Theorem 1.6).The paper is organized as follows: In Ssection 2 we give the variational description

of the problem; in Section 3 we brie y review some basic facts on Morse theory, andwe state the Morse type relations and we will prove the fundamental Lemma 3.15 andTheorem 3.16. Finally in Section 4 we prove Theorem 1.2 and Lemma 1.1.

2. The functional framework

We de�ne the Hilbert spaces:

�1(S1;RN )= {x∈AC(I;RN ) : x∈L2(I;RN ) and x(0)= x(1)}(where S1 = R

Z and I = [0; 1])

H (0; T )= {t ∈AC(I;R) : t ∈L2(I;R); t(0)= 0; t(1)=T}and we set

�=�1(S1;R)×H (0; T ):For z=(x; t)∈� we de�ne the action integral f by

f(z)=12

∫ 1

0{〈�(x)x; x〉 − �(x)t2}:

It is easy to see that f is a smooth functional and that its critical points are theT -periodic trajectories.Moreover if we de�ne the functional

J : �1→R by

J (x)=12

∫ 1

0〈�(x)x; x〉ds− T 2

2

(∫ 1

0

ds�(x)

)−1:

Benci, Fortunato and Giannoni (see [2]) have proved the following

Theorem 2.1. z=(x(s); t(s)) s∈ I = [0; 1] be a smooth curve on M =RN ×R. Thefollowing statements are equivalent:(i) z is a critical point of f on �


(ii) x is a critical point of J on �1 and

t(s)=T(∫ 1

0

ds�(x)

)−1 1�(x)

:

Proof. See Theorem 2.2 of [2].

Remark 2.2. Suppose that xj ∈RN is a critical point of � (therefore xj is a criticalpoint of J on �1(S1;RN )), then for any y; v∈�1(S1;RN ):

J ′′(xj)(y; v)=∫ 1

0〈�(xj)y; v〉ds− T 2

2

∫ 1

0�′′(xj)(y; v)ds:

Thus, from J ′′(xj)(y; v)= 0 for any v∈�1(S1;RN ) we have that�y +

T 2

2(�(xj))−1�′′(xj)y=0

y(0)=y(1)

y(0)= y(1):

(2.2.1)

Then if we put y=(�(xj))− 12 q we can see that the resolution of the system (2.3.1) is

equivalent to the resolution of the following system�q+

T 2

2(�(xj))−

12 �′′(xj)(�(xj))−

12 q=0

q(0)= q(1)

q(0)= q(1):

(2.2.2)

3. Theoretical elements from the Morse theory

In this section we present some facts of Morse theory in the form which will besuitable for the applications of this paper, and we will prove Theorem 3.16 which isfundamental for our result. For a complete exposition about this topic we refer to [6]and [7].Let E be a real Hilbert space, an open subset of E and let f∈C2(;R): We

denote by Kf the set of critical point of f; that is,

Kf = {x∈ :f′(x)= 0}:For a; b∈R; a¡b; we denote byfb= {x∈ : f(x) ≤ b}fba = {x∈ : a ≤ f(x) ≤ b}:


De�nition 3.1. We say that f satis�es the Palais–Smale (PS condition in (a; b)⊂R;(b possibly +∞) if any sequence {xn}n in fba such that:

f(xn) is bounded in (a; b) and :f′(xn)→ 0:

has a converging subsequence.

De�nition 3.2. Let H∗(·; ·;F) be a homology theory, with F a �eld, and let (X; A) bea topological couple. We de�ne the Poincar�e polynomial as

Pt(X; A)=∞∑q=0

{dimHq(X; A; F)}tq: (3.2.1)

Notice that Pt(X; A) can be a formal series.In the following de�nition we will consider for x∈ and t ∈R the Cauchy problemd�dt=V (�)

�(0; x)= x:

where V is given by

V (x)=−∇f(x)

‖∇f(x)‖+ 1 : (3.2.2)

Now, for any closet set X ⊂ we de�ne the exit set �(X ); by�(X )= {x ∈ @X : for all �0 there exists �∈ (0; �0) : �(�; x) =∈X }

and we set

�= {X ⊂ :�(X ) is closed}:A set X in � is called a Conley Block.

De�nition 3.3. For X in � we de�ne the index of X , It(X ); as follows:

It(X )=Pt(X;�(X )):

De�nition 3.4. For x∈Kf; we de�ne the Morse index, m(x); of x respect to f; bym(x; f)=max{dimV :V ∈ S}

where S is the set of linear subspaces V of E such that f′′(x)(u; u)¡0 for anyu∈V − {0} .We say that the critical point x of f is not degenerate, if

Ker f′′(x)= {0}:


De�nition 3.5. We say that f is a Morse function, f∈M (); if the set Kf has onlynondegenerate critical points and moreover f satis�es the (PS) condition on f(). Inthis case the Morse polynomial for K ⊂Kf is

m�(K;f)=∑x∈K

�m(x;f):

Let S be the family of formal series in one variable with coe�cients in N∪{+∞};with the usual sum and product.

De�nition 3.6. If P ∈ S we setcj(P)= aj if and only if P(�)=

∑j

aj�j: (3.6.1)

It is possible to introduce the following total order in S (see [6]): Let P;Q∈S; wesay that P ≤ Q if there exists n∈N such that cj(P)= cj(Q) for 0 ≤ j ≤ n − 1 andcn(P)¡cn(Q).

De�nition 3.7. Let {Pj}j be a sequence in S. We say that

P= limn→∞Pn if and only if for all j∈Ncj(Pn)→ cj(P) when n→∞: (3.7.1)

Identifying the formal series∑

j aj�j with the sequence {aj}; one obtains that the

topology given by (3.7.1) is equivalent to the product �∞i=0 Xi where Xi=N∪{+∞}

for all i. Hence S is compact. Denoting cl(A) the closure of A⊂S; we have thefollowing.

Proposition 3.8. For any set A⊂S

R= inf A=:min cl(A)

R=supA=:max cl(A)

exist and are unique.


De�nition 3.9. For any �¿0 and A⊂E; we putN�(A)= {x∈E :d(x; A)¡�}M�f()= {g∈M () : g(x)=f(x) for x =∈N�(Kf())}(note that for �1¡�2; M

�1f ()⊂M�2

f ()):

Finally we put

F()= {f∈C2() :M�f() 6= ∅ for each �¿0}:


If A is closed we say that f∈F(A) if there exists an open neighbourhood N�(A) ofA such that f∈F(N�(A)).

Lemma 3.10. Suppose that f satis�es the (PS) condition in f(). Moreover assumethat for any x∈Kf(); the linear operator associated with Hf(x); the hessian of fat x; is a Fredholm operator (of index 0). Then f∈F().


Let K ⊂Kf. We say that K is an isolated critical set if there exists an open set!⊂ such that K =Kf ∩!. The open set ! will be called an isolating set for K .

De�nition 3.11. Let ! be an isolating set for K ⊂Kf. We de�ne, i�(K;f); the gener-alized Morse index of K; as

i�(K;f)= sup�¿0

infg∈M�

f (!)m�(Kg(!); g):

If x∈K is an isolated critical point, we say that the integer i1({x}; f) is the multi-plicity of x.

Remark 3.12. (See Theorem 5.8 of [6].)(i) Note that an isolated set K ⊂Kf contains at least i1(K;f) critical points if counted

with multiplicities.(ii) It is easy to see that i�(K;f) does not depend on the isolating set for K .(iii) If f is a Morse function in ; then K()=Kf ∩ is an isolated set, and

i�(K;f)=∑x∈K()

�m(x;f) =m�(K(); f):

(iv) Let f∈F(); K1 and K2 be two isolated compact sets in Kf with K1 ∩K2 = ∅;then it is easy to see that

i�(K1 ∪K2; f)= i�(K1; f)+ i�(K2; f):

Theorem 3.13. Let f be bounded from below; and suppose that f∈F() withcl()∈�. Then there exists a formal series Q(�) with integer nonnegative coe�-cients such that

i�(Kf())= I�(cl())+ (1+ �)Q(�):


Remark 3.14. Suppose that f∈M (int(fba )) with a; b �nite numbers in R. ThenTheorem 3.13 becomes the well-known classical Morse relation, that is,∑

x∈Kf(fba )�m(x;f) =P�(fb; fa)+ (1+ �)Q(�)


where P�(fb; fa) is the Poincar�e Polynomial and Q(�) is a polynomial with nonnegativeinteger coe�cients (see [6]).Now we will consider the compact Lie group S1 = R

[0;1] . We recall that f is S1

invariant if f(gx)=f(x) for all x∈ and for all g∈ S1. In this case the ow � in3.2.2 becomes also invariant, in the sense that (see [8])

�(t; gx)= g�(t; x) for all t ∈R; g∈ S1; x∈E:

Lemma 3.15. Let f be an S1 invariant functional of class C2(); A⊂. Moreoversuppose that:– A∈�.– A is S1 invariant; that is; for all g∈ S1 and for all x∈A; gx∈A.– K(A)∩Fix(S1)=�.– f is bounded from below.– f∈F(A).Then there exists a formal series PA(�) with integer coe�cients such that

I�(A)= (1+ �)PA(�): (3.15.1)

Moreover suppose that there exists a formal series P(�)=∑bs�s with integer coef-

�cients such thati�(K(A))= (1+ �)P(�)

and

bs¿0; bs−1≥ 0: (3.15.2)

Then there exist at least bs critical orbits in K(A) (counted with multiplicity) eachof them with generalized Morse index equal to s.

Proof. In [7] (Theorem 8.5) it has been shown that for all c∈R there exists a poly-nomial Pc(�) with integer coe�cients such that

I�(A∩fc)= (1+ �)Pc(�):Taking the limit for c→+∞ we get (3.15.1).From De�nition 3.9 there exist sequences {�j} of positive numbers with �j→ o and

{gj}∈M�jf (!A) such that

i�(K(A))= limj→∞

m�(Kgj (!(A)); gj)= limj→∞

∞∑r=0

ajr�r: (3.15.3)

From Theorem 3.13 we have that there exists a formal series Q(�)=∑

r qr�r with

nonnegative integer coe�cients such that

i�(K(A))= I�(A)+ (1+ �)Q(�): (3.15.4)


So if we suppose that there are no critical orbits with generalized Morse index equalto s we have that for all j su�ciently large

ajs =0: (3.15.5)

Now from (3.15.3) and (3.15.4) we obtain that qs= qs−1 = 0.On the other hand from the hypothesis we have that

(1+ �)∑

bs�s=(1+ �)[PA(�)+Q(�)]

with PA(�)=∑

r pr�r; and∑

bs�s= [PA(�)+Q(�)] (3.15.6)

therefore bs=ps. Thus we obtain that ps−1 =−ps; from this in (3.15.6) we obtainthat bs−1 =−ps=−bs¡0. This is a contradiction. Therefore there exists at least onecritical orbit with generalized Morse index equal to s.Now suppose that bs¿as; as= limj→+∞ a

js (note that as≥ 1).

Putting bs= bs− as in the above argument, we have thati�(K(A))− as�s = · · ·+ as−1�s−1 + as+1�s+1 + · · ·

= (1+ �)(· · ·+ bs−1�s−1 + (bs− as)�s+ · · ·): (3.15.7)

From the fact that (bs− as)¿0 we can conclude, as before, that the coe�cient of�s in the left hand side of (3.15.7) is di�erent from zero. This is absurd, and soas≥ bs.

Theorem 3.16. Let f be an S1 invariant functional of class C2 on ; and cl()∈�.Moreover suppose that– f is bounded from below.– f satis�es the (PS) condition.– Kf ∩Fix(S1) has only nondegenerate critical points (nondegenerate critical orbits)and Kf ∩Fix(S1)⊂fba with a and b a �nite real numbers.

– Every critical point x0 ∈Kf −Fix(S1) is such that the linear operator associatedwith Hf(x0); the hessian of f at x; is a Fredholm operator (of index 0).Then there exist two formal series P and Q with integer nonnegative coe�cients

such that:∑x∈Kf(X )∩Fix(S1)

�m(x) + (1+ �)P(�)= I�(X )+ (1+ �)Q(�): (3.16.1)

Moreover if P(�)=∑bs�s with bs 6=0 then there exist at least bs critical orbits

(counted with multiplicity) each of them with generalized Morse index equal to s.

Proof. In the equivariant case we have that the Theorem 3.13 is also true (see [7]or [10]), that is, there exists a formal series Q(�) with integer nonnegative coe�cientssuch that

i�(Kf())= I�(cl())+ (1+ �)Q(�): (3.16.2)


Moreover by (iii) of Remark 3.12 and Lemma 3.15 we have that

i�(Kf) = i�(Kf()∩Fix(S1))+ i�(Kf()−Fix(S1))=

∑x∈Kf∩Fix(S1)

�m(x;f) + (1+ �)P(�)

where P is a formal series with integer coe�cients. Now, putting P(�)=P1(�)+P2(�)with P1 a formal series with negative integer coe�cients and P2 ∈S; in (3.16.2) weobtain that

r∑i=1

�m(xi) + (1+ �)P2(�)= I�(X )+ (1+ �)(Q−P1)(�)

and we can use the Lemma 3.15.

4. Proof of the Theorem 1.2 and Lemma 1.1

From the hypothesis on T and the Remark 2.3 it follows that the xj; j=1; : : : ; r arenondegenerate critical (�xed) points of J on �1(S1;RN ).On the other hand in [2] it has been proved (assumptions 1.3 are relevant here) that

J satis�es the (PS) condition on

X = {x∈�1(S1;RN ) : J (x)≥−T 2d}:Moreover from (2.2.1), by a straightforward computation we can see that for everyx∈KJ −Fix(S1) the linear operator associated with the hessian of J at x is a Fredholmoperator (of index 0). Then from Theorem 3.16 we have that

r∑i=1

tm(xi) + (1+ t)P(t)= It(X )+ (1+ t)Q(t)

where P(t) and Q(t) are formal series with nonnegative integer coe�cients. On theother hand it is easy to see that X is contractable in itself, therefore

It(X )= 1:

Thus we get

r∑i=1

tm(xi)− 1= (1+ t)[Q(t)−P(t)] (4.1)

Now if we set Q(t)=∑

s qsts and P(t)=

∑s pst

s where ps and qs are nonnegativeintegers, and if n(T ) represents the number of nontrivial T-periodic trajectories countedwith multiplicity, from Theorem 3.16 we have that

n(T )≥ (q−p)−

where (q−p)− represents the sum of the negative coe�cients of Q(t)−P(t).


If we take t=(−1) in (4.1), we getr∑i=1

(−1)m(xi) = 1 (4.2)

and hence � should have an odd number of critical points (and this proves theLemma 1.1).Now, if we consider the sets X and Y as in (1.1.2), (1.1.3) and (1.1.4), we de�ne

the sets A and B as follows:

A= {i∈{2; : : : k1} :mi¿�i−1}:B= {i∈{2; : : : k1} :mi¡�i−1}

From (4.1) we get∑i∈A(tmi + t�i−1 ) +

∑i∈B(t�i−1 + tmi)+ tm1 − 1= (1+ t)[Q(t)−P(t)]: (4.3)

If we set

C =11+ t

∑i∈A(tmi + t�i−1 )

D=11+ t

∑i∈B(t�i−1 + tmi)

E=tm1 − 11+ t

we get

C =∑i∈A

mi−�i−1∑j=1

(−1) j+1tmi−j

D=∑i∈B

�i−1−mi∑j=1

(−1) j+1t�i−1−j (4.4)

E=m1∑i=1

(−1)i+1tm1−i

and

C +D+E=Q(t)−P(t):Now, from (1.1.3), (1.1.4) and (4.4) it is easy to see that C; D and E cannot containmonomials of the same power and coe�cients with opposite sign. Thus

n(T )≥C−+D−+E−


where C−; D−; E− represent, respectively, the negative coe�cients of C; D; and E.That is,

n(T )≥k1∑j=2

|mj − �j−1|2

+12m1:

References

[1] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York,1983.

[2] V. Benci, D. Fortunato, F. Giannoni, On the existence of multiple geodesics in static space-times, Ann.Inst. Henri Poincar�e 8(1) (1991) 79–102.

[3] C. Greco, In�nitely many spacelike periodic trajectories on a class of Lorentz manifolds, Rend. Sem.Mat. Padova 91 (1994) 251–263.

[4] V. Benci, D. Fortunato, Periodic solutions of asymptotically linear dynamical systems, NonlinearDi�erential Equations and Applications 1(3) (1994) 267–280.

[5] V. Benci, D. Fortunato, Estimate of the number of periodic solutions via the twist number, to appear.[6] V. Benci, F. Giannoni, Morse theory for C1 functionals and Conley blocks, Topological Methods in

NonLinear Analysis 4 (1994) 365–398.[7] V. Benci, A new approach to the Morse–Conley theory and some applications, Annali di Matematica

Pura ed Appl., (IV) 158 (1991) 231–305.[8] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989.[9] A. Marino, G. Prodi, Metodi Perturbativi nella teoria di Morse. Bolletino U.M.I. (4) 11 Suppl. fasc. 3

(1975) 1–32.[10] K. Chang, In�nite Dimensional Morse Theory and Multiple Solutions Problems, Birkhauser, Basel,

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estimates of the number of periodic trajectories on a class of lorentz manifolds

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