Hofer-Zehnder semicapacity of cotangent bundles and symplectic submanifolds

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HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES AND SYMPLECTIC SUBMANIFOLDS LEONARDO MACARINI Abstract.We introduce the concept of Hofer-Zehnder G -semicapacity (or G -sensitive Hofer-Zehnder capacity)and prove that given a geometrically bounded symplectic manifold (M,ω)and an open subset N ?M endowed with a Hamilton-ian free circle action ?then N has bounded Hofer-Zehnder G ?-semicapacity,where G ??π1(N )is the subgroup generated by the homotopy class of the orbits of ?.In particular,N has bounded Hofer-Zehnder capacity.We give two types of applications of the main result.Firstly,we prove that the cotangent bundle of a compact manifold endowed with a free circle action has bounded Hofer-Zehnder capacity.In particular,the cotangent bundle T ?G of any compact Lie group G has bounded Hofer-Zehnder capacity.Secondly,we consider Hamiltonian circle actions given by symplectic submanifolds.For instance,we prove the following generalization of a recent result of Ginzburg-G¨u rel [7]:almost all levels of a function de?ned on a neighborhood of a closed symplectic submanifold S in a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian ?ow,provided that the function is constant on S .1.Introduction We will consider here the problem of the existence of periodic orbits on prescribed energy levels of Hamiltonian systems.An approach to this problem was the introduc-tion of certain symplectic invariants related to symplectic rigidity phenomena.More precisely,H.Hofer and E.Zehnder introduced in [9]the following de?nition:De?nition 1.1.Given a symplectic manifold (M,ω),de?ne the Hofer-Zehnder ca-pacity of M by c HZ (M,ω)=sup H ∈H a (M,ω)

max H,

where H a (M,ω)is the set of admissible Hamiltonians H de?ned on M ,that is,?H ∈H (M )?C ∞(M,R ),where H (M )is the set of pre-admissible Hamilto-nians de?ned on M ,that is,H satis?es the following properties:0≤H ≤ H :=max H ?min H ,there exist an open set V ?M such that H |V ≡0and a compact set K ?M \?M satisfying H |M \K

≡ H ;?every non-constant periodic orbit of X H has period greater than 1.

2LEONARDO MACARINI

We will also consider here a modi?ed Hofer-Zehnder capacity introduced by D. McDu?and J.Slimowitz[16]sensitive to the presence of overtwisted critical points. It is given by

c HZ(M,ω)=sup H∈ H a(M,ω)max H,

where H∈ H a(M,ω)if H is admissible and has no overtwisted critical points,that is,

?the linearized?ow at the critical points of H has no non-trivial periodic orbits of period less than1.

Remark.It is an open question whether the capacities c HZ and c HZ coincide.

It is easy to prove that if a symplectic manifold(M,ω)has bounded Hofer-Zehnder capacity,that is,if c HZ(U,ω)<∞for every open subset U?M with compact closure,then given any Hamiltonian H:M→R with compact energy levels,there exists a dense subsetΣ?H(M)such that for every e∈Σthe energy hypersurface H?1(e)has a periodic solution.Actually,if there exists a neighborhood of an energy hypersurface H?1(e0)with?nite Hofer-Zehnder capacity then there exists?>0and a dense subset G?(e0??,e0+?)such that H?1(e)has periodic orbits for every e∈G. This result was improved in[14],where it is showed that there are periodic orbits on H?1(e)for almost all e∈(e0??,e0+?)with respect to the Lebesgue measure.The same holds for c HZ(M,ω).

Let us consider here a re?nement of the original Hofer-Zehnder capacity by con-sidering periodic orbits whose homotopy class is contained in a given subgroup G of π1(M).More precisely,we de?ne the Hofer-Zehnder semicapacity as follows:

De?nition1.2.Given a symplectic manifold(M,ω)and a subgroup G?π1(M) de?ne the Hofer-Zehnder G-semicapacity(or G-sensitive Hofer-Zehnder capacity)of M by

c G HZ(M,ω)=sup

H∈H G a(M,ω)

max H,

where H G a(M,ω)is the set of G-admissible Hamiltonians de?ned on M,that is,?H∈H(M),that is,H is pre-admissible(see the de?nition1.1);

?every nonconstant periodic orbit of X H whose homotopy class belongs to G has period greater than1.

We de?ne c G HZ(M,ω)analogously considering G-admissible Hamiltonians without overtwisted critical points.

Remark.We call it a semicapacity because given a symplectic embeddingψ: (N,τ)→(M,ω)such that dim N=dim M it cannot be expected,in general,that

c G HZ(N,τ)≤cψ?G

HZ (M,ω),whereψ?:π1(N)→π1(M)is the induced homomorphism

on the fundamental group.However,we can state the following weak monotonicity property(for a proof see[13]):given a symplectic embeddingψ:(N,τ)→(M,ω) such that dim N=dim M then

cψ?1?H

HZ

(N,τ)≤c H HZ(M,ω),

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES3 for every subgroup H?π1(M).In particular,ifψ?is injective we have that (M,ω).

c G HZ(N,τ)≤cψ?G

HZ

Note that obviously,

(M,ω)≤c G HZ(M,ω),

c HZ(M,ω)=cπ1(M)

HZ

for every subgroup G?π1(M).Moreover,it can be show that,like the Hofer-Zehnder capacity,if the Hofer-Zehnder G-semicapacity is bounded then there are periodic orbits with homotopy class in G on almost all energy levels for every proper Hamiltonian[14].

De?nition1.3.A symplectic manifold(M,ω)is called geometrically bounded if there exists an almost complex structure J on M and a Riemannian metric g such that ?J is uniformlyω-tame,that is,there exist positive constants c1and c2such that

ω(v,Jv)≥c1 v 2and|ω(v,w)|≤c2 v w

for all tangent vectors v and w to M;

?the sectional curvature of g is bounded from above and the injectivity radius is bounded away from zero.

Closed symplectic manifolds are clearly geometrically bounded;a product of two geometrically bounded symplectic manifolds is also such a manifold.It was showed in[5,11]that the cotangent bundle of a compact manifold endowed with any twisted symplectic form is geometrically bounded.A twisted symplectic structure on a cotan-gent bundle T?M is a symplectic form given byω0+π??,whereω0is the canonical symplectic form,π:T?M→M is the canonical projection and?is a closed2-form on M.

De?nition1.4.Given a symplectic manifold(M,ω)de?ne its index of rationality by

m(M,ω)=inf [u]ω;[u]∈π2(M)satis?es [u]ω>0 .

If the above set is empty,we de?ne m(M,ω)=∞.If m(M,ω)>0we call(M,ω)a rational symplectic manifold.

The main result here states that the existence of a free Hamiltonian circle action implies the boundedness of the Hofer-Zehnder semicapacity.It removes a technical hypothesis in a previous theorem in[13].

Main Theorem.Let(M,ω)be a geometrically bounded symplectic manifold and N?M an open subset that admits a free Hamiltonian circle action?.Then,given any open subset U i?→N with compact closure,

c i?1?G?HZ(U,ω)<∞,

where i?:π1(U)→π1(N)is the induced homomorphism on the fundamental group and G??π1(N)is the subgroup generated by the orbits of the circle action.Moreover,

(U,ω)<∞.

if M is rational,then c i?1?G?

HZ

4LEONARDO MACARINI

The main idea in the proof of the Main Theorem is to relate the Hofer-Zehnder

capacity of a symplectic manifold endowed with a free Hamiltonian action with the

Hofer-Zehnder capacity of its symplectic reduction.More precisely,let P be a sym-

plectic manifold with a free Hamiltonian G-action and bounded Hofer-Zehnder ca-pacity.When its reduced symplectic manifold M also has bounded Hofer-Zehnder

capacity?Of course,it does not hold in general,since every symplectic manifold M

can be obtained as a reduction of M×T?S1with respect to the obvious circle action on T?S1and this symplectic manifold has bounded Hofer-Zehnder capacity by the Main Theorem.On the other hand,the example of Zehnder[20]shows that there

exist compact symplectic manifolds with in?nite Hofer-Zehnder capacity.

Let us consider the easiest case,namely when G=S1.In what follows,we will

always denote the symplectic gradient of a Hamiltonian H by X H.Given a pre-admissible Hamiltonian H on M we can construct a pre-admissible Hamiltonian H on P whose reduced dynamics coincides with the dynamics of H.The problem is to give conditions to ensure that the reduced periodic orbit on M is not trivial,that is,that the periodic orbit of X H is not tangent to the orbits of the group action. An approach to solve this problem is to use the fact that besides the existence of a non-trivial periodic orbit,the Hofer-Zehnder capacity gives an upper bound T for the period of the orbit.Thus,we can try to construct a circle action with su?ciently great period in such a way that the Hofer-Zehnder capacity of the support of X H is su?ciently small(compared to the period of the action)and that the component of X H tangent to the circle action remains bounded.It would ensure that if a periodic orbit of X H is tangent to the orbits of the action then its period is necessarily greater than T.

When M has a free Hamiltonian circle action we construct such a circle action on

M×T?S1but,instead of M,its symplectic reduction is given by a?nite quotient M/Z n for n su?ciently great(Theorem2.1).We then get the boundedness of the Hofer-Zehnder semicapacity of M/Z n(resp. c G HZ)by Hofer-Viterbo theorem[8](resp. McDu?-Slimowitz theorem[16])generalized for geometrically bounded symplectic manifolds by G.Lu[11,12].It was essentially achieved in[13].However,we need a theorem to relate the Hofer-Zehnder capacity of M and its?nite symplectic quotient M/Z n.This is the content of Theorem2.2proved in Section4.

1.1.Applications to cotangent bundles.Now,let us consider some applications of the Main Theorem.Considering the lifted action to the cotangent bundle,we have the following corollary:

Corollary1.1.Let M be a manifold with a free circle action?:S1×M→M. Then T?M endowed with the canonical symplectic form has bounded Hofer-Zehnder G?-semicapacity,where G??π1(M)=π1(T?M)is the subgroup generated by the orbits of the action.

Now,let G be a compact Lie group and consider the action of the maximal torus in G.It gives the following immediate corollary:

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES5 Corollary1.2.Let G be a compact Lie group.Then T?G has bounded Hofer-Zehnder capacity.

1.2.Applications to symplectic submanifolds.The next corollary is a general-ization of a recent result of V.Ginzburg and B.G¨u rel[7]for weakly exact symplectic manifolds.Their proof is completely di?erent and is based on symplectic homology. Moreover,they consider only energy hypersurfaces that bound a compact domain containing the symplectic submanifold S while our result holds for any energy hyper-surface that does not intersect S.

Corollary1.3.Let(M,ω)be a geometrically bounded symplectic manifold and S?M a closed symplectic submanifold.There exists a neighborhood U of S such that if H is a proper Hamiltonian on U constant on S then H has periodic orbits with contractible projection on S on almost all energy levels.

Remark.The only condition on U is that U\S admits a free Hamiltonian circle action.Its existence follows by the symplectic neighborhood theorem[15].

As an immediate corollary we have the following result about the existence of periodic orbits for magnetic?ows(for an introduction to magnetic?ows see[6,13]): Corollary 1.4.Let(M,?)be a closed symplectic manifold and H:T?M→R be the standard kinetic energy Hamiltonian for a Riemannian metric on M.Then the Hamiltonian?ow of H with respect to the twisted symplectic formω0+π??has periodic orbits with contractible projection on M on almost all su?ciently low energy levels.

The next application is based on the following result of P.Biran[1]which enable us to represent a K¨a hler manifold as a disjoint union of two basic components whose symplectic nature is very standard:

Theorem1.1(P.Biran[1]).Let(M2n,?)be a closed K¨a hler manifold with[?]∈H2(M,Z)andΣ?M a complex hypersurface whose homology class[Σ]∈H2n?2(M) is the Poincar′e dual to k[?]for some k∈N.Then,there exists an isotropic CW-complex??(M,?)whose complement-the open dense subset(M\?,?)-is symplectomorphic to a standard symplectic disc bundle(E0,1

ωcan is uniquely characterized by the requirements that its restriction to the zero k

6LEONARDO MACARINI

sectionΣequals?|Σ,the?bers ofπ:E0→Σare symplectic and have area1/k andωcan is invariant under the obvious circle action along the?bers.It is called standard because the symplectic type of(E0,ωcan)depends only on the symplectic type of(Σ,?|Σ)and the topological type of the normal bundle NΣ[1].

Let us recall that the pair(M,Σ)is called subcritical[2,3]if M\Σis a subcritical Stein manifold,that is,if there exists a plurisubharmonic Morse function?on M\Σsuch that index p(?)

Since(E0\Σ,ωcan)has an obvious Hamiltonian free circle action,we have the following immediate corollary of the Main Theorem:

Corollary1.5.According to the notation of the Theorem1.1,we have that

c G HZ(E0\Σ,?)<∞,

where G?π1(E0\Σ)is the subgroup generated by the orbits of the obvious S1-action on E0\Σ.In particular,the periodic orbits are contractible in M.Moreover,if(M,Σ) is subcritical,then

(M\Σ,?)<∞,

c i?G

HZ

where E0\Σi?→M\Σ.

Remarks.

?The result for subcritical manifolds follows by the fact that if dim?<1

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES7? H >2π( H1|U +√

n),

then H′ >2π( H1|U +√

n)<∞.

When M is rational,the hypothesis on the critical points is not necessary and hence

(U,ω)≤2πn( H1|U +√

c i?1?G?

HZ

8LEONARDO MACARINI

3.Proof of Theorem2.1

If H has no overtwisted critical points,let n0=n0( H1|U )be given by

√n)}.

n0=inf{n∈N;

If H has overtwisted critical points and M is rational,let n0=n0( H1|U ,m(M,ω))

√n)and

be the in?mum of n∈N such that

m(M,ω)≥(2π/n)( H1|U +√

√

n)H1(x)+

n,the induced Z n-action is given byρ√n.On the other hand,√n and,since H

andΦcommute,we have thatρ1/√

1

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES9 given by

J(x,θ,μ)=(1/√nΦ(θ,μ).

√nΦ(θ,μ)generate a circle action But,since the Hamiltonians(1/

√

with the same period1/

10LEONARDO MACARINI

But note that

ωW (π?2dθ(ξ)X H 1(?θ(z )(z )),π?2dθ(η)X H 1(?θ(z )(z )))=0

and

ωW (dH 1(z )ξY (?θ(z )(z )),dH 1(z )ηY (?θ(z )(z )))=0,

since the vectors are colinear.On the other hand,

ωW (π?2dθ(ξ)X H 1(?θ(z )(z )),dH 1(z )ηY (?θ(z )(z )))=0

and

ωW (dH 1(z )ξY (?θ(z )(z )),π?2dθ(η)X H 1(?θ(z )(z )))=0,

because the vectors are orthogonal with respect to the product decomposition P =M ×T ?S 1and by the de?nition of ωW .

We have then that,

(ψ?ωW )z (ξ,η)=ωW ((d?)θ(z )(z )ξ,(d?)θ(z )(z )η)+ωW ((d?)θ(z )(z )ξ,π?2dθ(η)X H 1(?θ(z )(z )))

?ωW ((d?)θ(z )(z )ξ,dH 1(z )ηY (?θ(z )(z )))+ωW (π?2dθ(ξ)X H 1(?θ(z )(z )),(d?)θ(z )(z )η)

?ωW (dH 1(z )ξY (?θ(z )(z )),(d?)θ(z )(z )η)=

=(ωW )z (ξ,η)+π?2dθ(η)(i X H 1(?θ(z )(z ))ω)(d?θ(z )(z )ξ)?dH 1(z )η(i Y (?θ(z )(z ))ω)(d?θ(z )(z )ξ)

+π?2

dθ(ξ)(i X H 1(?θ(z )(z ))ω)(d?θ(z )(z )η)?dH 1(z )ξ(i Y (?θ(z )(z ))ω)(d?θ(z )(z )η),where the last equality follows from the fact that ??t ωW =ωW .Now,note that

π?2dθ(η)(i X H 1(?θ(z )(z ))ω)(d?θ(z )(z )ξ)=π?2dθ(η)dH 1(z )ξ,

because i X H 1ωW =dH 1and ??dH 1=dH 1.On the other hand,we have that

dH 1(z )η(i Y (?θ(z )(z ))ω)(d?θ(z )(z )ξ)=dH 1(z )ηπ?2dθ(ξ),

because i Y ωW =π?2dθand ??π?2dθ=π?2dθ(note that π?2dθhere is the pullback of the angle form dθon S 1to P ).Consequently,

(ψ?ωW )z (ξ,η)=(ωW )z (ξ,η)+

+π?2dθ(η)dH 1(z )ξ?dH 1(z )ηπ?2dθ(ξ)+π?2dθ(ξ)dH 1(z )η?dH 1(z )ξπ?2dθ(η)

=(ωW )z (ξ,η),

as desired.

Now,we claim that the quotient projection πμ: J

?1(μ)→ J ?1(μ)/S 1is given by π1?ψ?1| J ?1(μ),where π1:P →N is the projection onto the ?rst factor.In e?ect,note that,by the previous lemma,ψsends the orbits of X J to the orbits of X Φ.On the other hand,π1is the quotient projection at Φ?1(μ)=ψ?1( J

?1(μ))with respect to the trivial circle bundle given by the orbits of X Φ.

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES 11

To show that σμis equal to τ,note that

π?μτ=(ψ?1)?i ?Φ?1(μ)π?1τ

=(ψ?1)?i ?Φ?1(μ)ωW

=i ?ψ(Φ?1(μ))ωW

=i J ?1(μ)ωW ,

where the third equality follows from the fact that ψ?ωW =ωW . 3.2.Construction of a ρ-invariant pre-admissible Hamiltonian H on N ×T ?S 1from H .Consider the smooth map π:P →N/Z n de?ned by

π|J ?1(μ)=πμ,where πμis the quotient projection.Now,let ˉH :=H ?τ?1be the induced Hamiltonian on N/Z n .We will construct from ˉH a ρ-invariant Hamiltonian de?ned on P .Fix a su?ciently small constant δ>0and de?ne this new Hamiltonian by H (z )=(ˉH +α(J (z ))(m (ˉH )?ˉH ))( π(z )),where α:R →R is a C ∞function such that 0≤α≤1,α(μ)=1?μ/∈(δ,1?δ),α(μ)=0?μ∈(12+δ)and |α′(μ)|≤2+2δfor every μ∈[0,1].Thus, H |J ?1(μ)is the lift of ˉH

μ:=ˉH +α(μ)(m (ˉH )?ˉH )=(1?α(μ))ˉH +α(μ)m (ˉH )by the quotient projection πμ.1/21

01Figure 2.Graph of the function α.

The ?rst obvious property of H is that m ( H )=m (ˉH )=m (H ).In e?ect,m ( H )=sup μ∈R m (ˉH +α(μ)(m (ˉH )?ˉH ))=m (ˉH ),since 0≤α≤1and α(μ)=1?μ/

∈(δ,1?δ).It is easy to see that H ∈H ( U,ω)where U is given by (see ?gure 3) U = 0≤μ≤1

π?1μ(τn (U )).

In e?ect,it is clear that 0≤ H ≤m ( H ).Moreover, H | V ≡0where V is the subset of P given by 12+δπ?1μ(τn (V ))and τn (V )?N/Z n is the open set such that

12LEONARDO MACARINI

ˉH|

τn(V)≡0becauseα(μ)=0?μ∈(12+δ).Note that the interior of V is not empty.

On the other hand, H≡m( H)outside the compact set K= δ≤μ≤1?δπ?1μ(τn(K))? U\? U,whereτn(K)?τn(U)\?τn(U)is the compact set such thatˉH|τn(U)\τn(K)≡m(ˉH).

Finally,note that if H has no overtwisted critical points then H also has no over-twisted critical points becauseˉHμhas no such critical points and the derivative of X H equals the identity in the direction tangent to the orbits ofρ.

3.3.Existence of a non-trivial periodic orbit of XˉH given by the reduction of a periodic orbit of X H.Let us now prove that there exists a non-trivial periodic orbitˉγof XˉH with period less than(2π/n)( H1|U +√

.

m(H)

Moreover,the homotopy class[γ]ofγbelongs to the subgroupπ1(S1)?π1(P). Proof.Since

√

3a+2r sinθ,

U))?R2is a connected two-dimensional compact submanifold with boundary,such that there exists an open

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES13 disk of radius R with L distinct points y j∈B2(R)(0≤L<∞)and an orientation preserving di?eomorphism

ψ:φ(τ2(U))→B2(R)\{y1,...,y L}

such that φ(τ2(U))ω0 = B2(R)\{y1,...,y L

}ω0 =πR2.

From a theorem of Dacorogna and Moser(see Lemma2.2in[19])this

ψcan

be required to be symplectic.Thus,consider the HamiltonianˉH:M×B2(R)→R given by

ˉH(x,y)= m(H)if y=y j for some j=1,...,L

H(x,φ?1ψ?1y)otherwise

which is obviously C∞,since H|U\K=m(H),where K?U is a compact subset such that K?U\?U.

By Proposition1.6of[16](extended for geometrically bounded symplectic mani-folds in[11]),XˉH has a contractible periodic orbitˉγwith period

T<

πR2

m(ˉH)

= τ2(U)ωS10

n

+

1

n =(2π/n)( H1|U +√

n?(1/n)H1|U:

Reduced dynamics of X H:

Note thatˉHμ=(1?α(μ))ˉH+α(μ)m(H)has the Hamiltonian vector?eld with respect toωn given by XˉH

μ

=(1?α(μ))XˉH.Thus,the reduced dynamics of X H at J?1(μ)is a reparametrization of the dynamics of XˉH.Since0≤α(μ)≤1,the nonconstant periodic orbits of XˉH

μ

have period greater than or equal to those of XˉH. Consequently,by Section3.1,it is su?cient to show that the periodic orbit of H given by Theorem3.1is not tangent to an orbit of?(such that it is projected onto a singularity of XˉH).

Non-triviality of the projected orbit:

14LEONARDO

MACARINI

n )

1(0)

Figure 3.The subset U

.Suppose that there exists a periodic orbit γof X H tangent to an orbit of ρ.We

will show that the period of γis strictly greater than T max ( U,

H )(see Theorem 3.1).Firstly,note that

T max ( U,

H )= τ2( U )ωS 10 n ) X J 2 X J .

Consequently,since the period of the orbits of ρis equal to

√n

X J 2 .But,by the de?nition of n ,

T =√ X H ,X J

n ) X J 2 ,

for δ>0su?ciently small.Thus,it is su?cient to prove that

X H ,X J

n )

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES15 In e?ect,note that

X H,X J

W?J 2

= ? H,?J

?J 2 ,

where in the second equation we used the fact that W de?nes an isometry.On the other hand,we have that for everyξ∈T z P,

d H(z)ξ=dˉH( π(z))d π(z)ξ+m(H)α′(J(z))dJ(z)ξ?ˉH( π(z))α′(J(z))dJ(z)ξ

?α(J(z))dˉH( π(z))d π(z)ξ

=(m(H)?ˉH( π(z)))α′(J(z))dJ(z)ξ+(1?α(J(z)))dˉH( π(z))d π(z)ξ. But note that,since γis tangent to a?ber,it is projected onto a singularity of XˉH,

that is,dˉH( π(z))=0for every z∈ γ.Hence,

X H,X J

?J 2

=(m(H)?ˉH( π(z)))|α′(J(z))|

≤(2+2δ)m(H),

as desired.

Consequently,by Section3.1,we conclude by symplectic reduction thatˉH has a non-trivial periodic orbitˉγof period less than or equal to

(2π/n)( H1|U +√

m(H)

.

But,sinceτ:N→N/Z n is a symplectic covering with n sheets,we have that the liftγofˉγis a periodic orbit of X H with period less than or equal to

2π( H1|U +√

m(H)

.

Finally,note that the periodic orbit γof X H given by Theorem3.1satis?es[ γ]∈π1(S1).Hence,the corresponding reduced periodic orbitˉγsatis?es[ˉγ]∈(τn)?G?since(πμ)?π1(S1)=(τn)?G?.Thus,since the periodic orbitγof X H is given by the lift ofˉγbyτn,we have that[γ]∈G?.

4.Proof of Theorem2.2

It is su?cient to prove Theorem2.2for n=2,since proceeding inductively we can apply the theorem below for N/Z n and conclude the result for any n=2m.

16LEONARDO MACARINI

Theorem4.1.Let M be a symplectic manifold endowed with a Z2free symplectic action,that is,a symplectomorphismψ:M→M such thatψ2=Id andψ(x)=x for every x∈M.Then,given an admissible Hamiltonian H on M(resp.H∈ H a(M,ω)),there exists an admissible Hamiltonian H′(resp.H′∈ H a(M,ω))such that

?H′is Z2-invariant;

?supp X H′=supp X H+H?ψ;

? H′ ≥(1/2) H .

Moreover,ifψis isotopic to the identity and H is G-admissible(resp.H∈ H G a(M,ω)) then H′is also G-admissible(resp.H′∈ H G a(M,ω)),for any subgroup G?π1(M). Proof.De?ne H=(1/2)(H+H?ψ).The?rst step in the proof is the following key proposition:

Proposition4.1.Given a periodic orbitγ:[0,T]→M of X H then X H and X H?ψare colinear alongγ.

Proof.Initially,note that X H and X H?ψcommute,since

{H,H?ψ}=ω(X H,X H?ψ)

=ω(X H,ψ?X H)

=ω(ψ?X H,ψ?ψ?X H)

=ω(ψ?X H,X H)=0,

where in the second and third equalities we used the fact thatψis symplectic and the last equality follows by the antisymmetry ofω.

Lemma4.1.If X H and X H?ψare not colinear alongγ,thenγis not isolated.In fact,there exists a1-parameter deformation ofγby periodic orbits of X H with the same period and energy,that is,there exists a family of geometrically distinct periodic orbitsγs:[0,T]→M for s∈(??,?)for some?>0such thatγ0=γand H(γs(t))= H(γ0(t))for every s∈(??,?).

Proof.Suppose that there exists t∈[0,T]such that X H(γ(t))and X H?ψ(γ(t))are linearly independent.Then,X H(γ(t))and X H?ψ(γ(t))are not colinear to X H(γ(t)). Since[X H,X H]=0we have thatφs?γ:[0,T]→M is a periodic orbit for every s∈R, whereφs is the?ow of X H.These periodic orbits are geometrically distinct for every s su?ciently small because X H(γ(t))and X H(γ(t))are linearly independent. Consider now the following linear application:

Φ:C∞(M)→C∞(M/Z2)

given byΦ(H)(x)=(1/2)(H+H?ψ)(π?1(x)),whereπ:M→M/Z2is the quotient projection.Note that we can identify C∞(M/Z2)with the subspace S?C∞(M)of ψ-invariant smooth functions and,with respect to this identi?cation,Φis just the projection.

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES17 Now,suppose,by contradiction,that X H and X H?ψare not colinear alongγ.By the previous lemma,we have thatπ?γis not an isolated periodic orbit of XΦ(H).In fact,there exists a1-parameter deformation ofπ?γgiven by periodic orbits of the same period.

Lemma4.2(Lemma19of[17]).Given?>0,there exists F∈C∞(M/Z2)such that F?Φ(H) C∞

Now,let F be the corresponding function in S?C∞(M)given by the identi?cation of S with C∞(M/Z2).Write F= H+G such that G C∞

Φ(F)= F.

Since F?H C∞

Proof.Suppose,by contradiction,that H has an overtwisted critical point p.By Corollary3.5of[16],there exists a sequence of Hamiltonians H n such that H? H n C∞n→∞?→0and X H n has a non-trivial periodic orbit γn of period less than1 converging to p.By the discussion in the proof of the previous proposition,there exists a sequence H n such that H?H n C∞n→∞?→0and either X H n or X H n?ψhas a non-trivial periodic orbitγn of period<1given by a reparametrization of γn.Thus, p is also a critical point for both H and H?ψ.

Now,note that,since X H and X H?ψcommute,DX H(p)and DX H?ψ(p)also com-mute.Hence,if(1/2)(DX H+DX H?ψ)(p)has an eigenvalue±iλforλ>2π,the same holds either for DX H(p)or DX H?ψ(p),a contradiction. Now,letδ>0be a su?ciently small constant and f:[min H,max H]→R a smooth function such that|f′|≤1+2δ,f(x)=0for every x∈[min H,min H+δ] and f(x)= H for every x∈[max H?δ,max H]:

Consider the Hamiltonian given by

H′=f? H.

Note that,sinceψis proper and by the de?nition of f,H′is pre-admissible.Moreover, we have that H′ = H ≥(1/2) H ,because min H≤(1/2)max H and max H= max H,becauseψis proper.

Now,suppose that X H′has a periodic orbit of period T′<1and homotopy class in G.Then,since X H′(x)=f′( H(x))X H(x)and|f′|≤1+2δforδarbitrarily small, we have that X H also has a periodic orbitγof period T<1such that[γ]∈G. By the previous proposition,X H and X H?ψare colinear to X H alongγ.Conse-quently,since X H=(1/2)(X H+X H?ψ),we conclude thatγis also a periodic orbit

18LEONARDO MACARINI

HOFER-ZEHNDER SEMICAPACITY OF COTANGENT BUNDLES19

[12]G.Lu,Errata to the“The Weinstein conjecture on some symplectic mani-

folds containing the holomorphic spheres”,Kyushu J.Math54(2000).

[13]L.Macarini,Hofer-Zehnder capacity and Hamiltonian circle actions,

preprint2002math.SG/0205030,to appear in Comm.in Contemporary

Mathematics.

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existence of closed trajectories near a hypersurface,math.SG/0305148.

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ematical Monographs.The Clarendon Press,Oxford University Press,New

York,1998.

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Hamiltonian paths,Geom.Topol.5(2001),799–830.

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(1970),562–603.

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ifolds,Paci?c J.Math.193(2000),no.2,419–461.

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Series,Series C,In:Periodic Solutions of Hamiltonian Systems and Related

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Instituto de Matem′a tica Pura e Aplicada-IMPA,Estrada Dona Castorina,110-Jardim Bot?a nico,22460-320Rio de Janeiro RJ,Brasil

E-mail address:leonardo@impa.br

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