Yanqiao DINGJianxun HUDepartment of Mathematics,Sun Yat-Sen University,Guangzhou 510275,China E-mail:dingyq6@mail2.sysu.edu.cn;stsjxhu@mail.sysu.edu.cn

THE WEINSTEIN CONJECTURE IN PRODUCT OF SYMPLECTIC MANIFOLDS∗

In this paper,using pseudo-holomorphic curve method,one proves the Weinstein conjecture in the product P1×P2of two strongly geometrically bounded symplectic manifolds under some conditions with P1.In particular,if N is a closed manifold or a noncompact manifold of finite topological type,our result implies that the Weinstein conjecture in ℂℙ2× T∗N holds.

Weinstein conjecture;J-holomorphic sphere;geometrically bounded

2010 MR Subject Classification53D35;58D10

1　Introduction

Let M be a symplectic manifold with symplectic form ω.A hypersurface S⊂M is said to be of contact type if there exists a vector field X defined on some neighborhood U of S such that(i)X is transversal to S and(ii)LXω=ω.

For any hypersurface S in symplectic manifold M,there exists a 1-dimensional characteristic line bundle LS⊂TS defined by

Let ξ be a section of the characteristic line bundle.The Weinstein conjecture claims that if S is a compact hypersurface of contact type,then S carries at least one closed orbit of ξ,see[24].

In 1987,Viterbo[22]proved the Weinstein conjecture in(ℝ2n,ωo)with the standard symplectic form ωo.Later,Hofer and Viterbo[1o]showed that the Weinstein conjecture is true for (T∗M,−dλ),where λ is the Liouville form on the cotangent bundle T∗M of a compact manifold M.Floer,Hofer and Viterbo[5]proved the stabilized Weinstein conjecture for(P×ℂl,ω⊕ωo) under the assumption[ω]=o on π2(P).In 1992,Hofer and Viterbo[11]introduced the pseudo-holomorphic curve method into the study of the Weinstein conjecture in the presence of holomorphic spheres.They proved the Weinstein conjecture in ℂℙn,S2×P,if P is a compact symplectic manifold with some conditions.Lu[17]extended the results of Hofer and Viterbo to the strongly geometrically bounded(SGB)symplectic manifolds.He showed the Weinstein conjecture holds in S2×T∗N,if N is a closed manifold or a noncompact manifold of finitetopological type.Liu and Tian completely proved the stabilized version Weinstein conjecture in[13].Lu[16]proved the Weinstein conjecture in the uniruled manifolds or the product of closed symplectic manifold and an uniruled manifold based on Liu-Tian’s result.

Since the product of regular almost complex structures is not regular in general(see[15]), the method of[11]can not be applied directly to any product manifolds.Making use of the regularity criterion in[19],we proved that there exists a regular almost complex structure, which is the product of regular almost complex structures,on the product of some 4-dimensional manifolds and symplectic manifolds.So this makes it possible to use the method of[11]to study the Weinstein conjecture for the product manifolds.In this paper,one proves the Weinstein conjecture in the product P1×P2of two SGB symplectic manifolds under some conditions with P1.In particular,if N is a compact manifold or a noncompact manifold of finite topological type,our result implies that the Weinstein conjecture in ℂℙ2×T∗N holds.

Next,we will introduce some notations to describe our result.Let(V,ω)be a symplectic manifold and F(V,ω)the space of all smooth almost complex structures which are compatible with ω on(V,ω).For J∈F(V,ω),define m(V,ω,J)in(o,+∞]by

where〈ω,[u]〉=RS2u∗ω which depends only on the free homotopy class[u]of u.Define m(V,ω)∈[o,+∞]by

where[S2,V]stands for the free homotopy class from S2into V.A free homotopy class α is said to be ω-minimal if m(V,ω)=〈ω,α〉and〈ω,α〉>o.Let α be an ω-minimal free homotopy class such that there exists a J∈F(V,ω)satisfies m(V,ω,J)=〈ω,α〉.Define H(α,J,Σo,Σ∞) to be the set of all u∈C∞(S2,V)such that

where Σo,Σ∞are two disjoint smooth submanifolds of V and closed as subsets.We also assume that one of Σoand Σ∞is compact.

Under certain conditions,there are almost complex structureswhich are close to J enough with respect to the C1-topology,such thatis a smooth compact free S1-manifold.Such ais called a regular almost complex structure(see Definition 2.2)at the situation(α,Σo,Σ∞).The set of regular compatible almost complex structure is denoted by Freg(V,ω).For any regularclose to J enough,the compact smooth S1-manifolds H1and H2belong to the same free S1-cobordism class.whereis in a sufficiently small neighborhood of J.The definition of the d-index is not depend on the choice of

The following is our main result of this paper.

Theorem 1.1Let(P1,ω1),(P2,ω2)be two SGB symplectic manifolds with dimP1=4. α1∈[S2,P1]is an ω1-minimal free homotopy class which can be represented by an embedded J1-holomorphic sphere such that

where J1∈Freg(P1,ω1).are two disjoint nonempty compact submanifolds of P1.is a nonempty compact submanifold of P2.Suppose that there is a smooth Hamiltonian H:P1×P2→ℝ such that

where the open neighborhoods U(Σo)and U(Σ∞)are disjoint and such that

As in[11],it is easy to prove the Weinstein conjecture in P1×P2from Theorem 1.1.

Corollary 1.2Let(P1×P2,ω1⊕ω2),α1∈[S2,P1],Σ∞satisfy the hypothesis of Theorem 1.1.Then any stable compact smooth hypersurface S in (P1×P2,ω1⊕ω2)separating Σofrom Σ∞possesses at least one periodic Hamiltonian trajectory.

2　Product of Regular Almost Complex Structures

In this section,we will prove a result on the product of regular almost complex structures based on the notations and results of[19].Let(V,ω)be a symplectic manifold and J∈F(V,ω). For a smooth map u:S2→V,the space of smooth vector fields ξ(z)∈Tu(z)V along u will be denoted by Ωo(S2,u∗TV)and the space of smooth J-antilinear 1-forms on S2with values in u∗TV by Ωo,1(S2,u∗TV).Then the vertical differential ofDu:Ωo(S2,u∗TV)→Ωo,1(S2,u∗TV),have the following expression:

Remark 2.1Formula(2.1)is given by[19,Proposition 3.1.1]which is proved if V is compact.Actually,(2.1)is also true for noncompact manifold.

A J-homomorphic sphere u:S2→V is said to be multiply covered if there exists a J-holomorphic sphere u′:S2→V,and a holomorphic branched covering φ:S2→S2such that

The curve u is called simple if it is not multiply covered.

Definition 2.2An almost complex structure J on V is called regular at the situation (α,Σo,Σ∞),if for every u∈W2,2(S2,V)which satisfies condition(1.1)Duis onto.In particular for a regular J the set H(α,J,Σo,Σ∞)is a smooth S1-manifold.

Remark 2.3W2,2(S2,V)means the space of the continuous maps from S2to V which are in local coordinate charts represented by functions in W2,2(Ω),where Ω⊂ℝ2is an open set and W2,2(Ω)is the standard Sobolev space.By elliptic regularity theory every u∈W2,2(S2,V) which satisfies condition(1.1)is smooth.

There is a regularity criterion in[19]which is very important for us.

Lemma 2.4(see[19,Lemma 3.3.2])Let E→S2be a complex vector bundle of rank n and

be a real linear Cauchy-Riemann operator.Suppose that there exists a splitting E=L1⊕···⊕Lninto complex line bundles such that each subbundle L1⊕···⊕Lk,k=1,···,n,is invariant under D.Then D is surjective if and only if c1(Lk)≥−1 for every k.

Remark 2.5Ωo(S2,E)denotes the space of all smooth vector fields ξ(z)∈Ez.Ωo,1(S2,E) denotes the space of smooth J-antilinear 1-forms on S2with values in E.Let πk:E→Lkdenote the projection onto the kth summand.Then the subbundle L1⊕···⊕Lkis invariant under D means that if i>k,πi(Dξj)=o,∀ξj∈Ωo(S2,Lj),j=1,···,k.Here and throughout this section we identify the first Chern class c1(L)of L with the corresponding Chern number〈c1(L),[S2]〉.The operator Duis obviously a real linear Cauchy-Riemann operator.

Using Lemma 2.4,we can give a sufficient condition which guarantees a product regular almost complex structure is still regular.First,we will introduce some notations.The number of all self-intersections of a curve u will be denoted by

We denote by c1(A)=〈c1(TM),A〉for A∈H2(M;ℤ),where c1(TM)is the first Chern class of TM,by Ao·A1the intersection number of two classes Aoand A1,and by χ(Σ)the Euler characteristic of a closed Riemann surface Σ.

Lemma 2.6(adjunction inequality[19,Theorem 2.6.4])Let(M,J)be an almost complex 4-manifold and A∈H2(M;ℤ)be a homology class that is represented by a simple J-holomorphic curve u:Σ→M.Then

with equality if and only if u is an immersion with only transverse self-intersections(i.e.,if zo/=z1and u(zo)=u(z1)=:x,then TxM=Imdu(zo)⊕Imdu(z1)).

Remark 2.7TxM=Imdu(zo)⊕Imdu(z1)means that TxM equals the direct sum of the subspaces du(TzoΣ)and du(Tz1Σ).

We have the following proposition.

Proposition 2.8Let(P1,ω1)be a symplectic 4-manifold and(P2,ω2)a symplectic manifold.Assume α1∈[S2,P1]is an ω1-minimal free homotopy class which can be represented by an embedded J1-holomorphic sphere u such thatare two disjoint nonempty compact submanifolds ois a nonempty compact submanifold ofwhereis regular at the situationin P1and J2then the product almost complex structure J=J1×J2is regular at the situationin P1×P2.

ProofFirst it is easy to see every J1-holomorphic sphere u which represents α1is simple. In fact,if u is multiply covered there exists a J1-holomorphic sphere u′:S2→P1,and a holomorphic branched covering φ:S2→S2such that

giving a contradiction to our assumption that α1is ω1-minimal.Assume u represents the homology class A∈H2(P1;ℤ),i.e.,u∗([S2])=A.Then all the J1-holomorphic spheres represent α1will represent A.

Since A∈H2(P1;ℤ)is represented by an embedded J1-holomorphic sphere u which is also simple,by the adjunction inequality we can get

For every simple J1-holomorphic sphere v:S2→P1which represents A,we have

The equality of the adjunction inequality holds for v.Thus every simple J1-holomorphic sphere v which represents A is an embedded curve.We can get every J1-holomorphic sphere represents α1is an embedded curve.

where ω=ω1⊕ω2.The J-holomorphic α sphere has the formwhere

u∈W2,2(S2,P1)and satisfies

We have the splitting

It follows from the definition of Du(2.1)that

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for every vector field ζ∈Vect(S2).For the embedded curve u,the complex subbundle

is invariant under Du.Now let L1⊂u∗TP1be the orthogonal complement of Lowith respect to any Hermitian inner product of u∗TP1.Then by Lemma 2.4

because J1is regular at the situationin P1.In the product manifold(P1× P2,ω1⊕ω2),ω1(·,J1·)+ω2(·,J2·)defines a product metric on P1×P2.Let∇be the Levi-Civita connection on P1×P2and∇ithe Levi-Civita connection on Pi,i=1,2,respectively. By the relation between∇and∇i,i=1,2,we know in the product manifold P1×P2,

Thus the subbundles Lo,Lo⊕L1,are invariant under D˜utoo.In the trivial bundle S2×TpoP2, each subbundle L2⊕···⊕L1+j,j=1,···,n,is obviously invariant under D˜u.c1(Lj)≥−1, j=2,···,n+1.By Lemma 2.4 again,we know D˜uis surjective.

Remark 2.9From the arguments of Lemma 3.3.3,Corollary 3.3.4 and Corollary 3.3.5 in [19],we can get the above proposition easily.

3　Holomorphic Spheres

Let us recall the definition of geometrically bounded manifold(cf.[2,8,17]).

Definition 3.1Let(M,ω)be a symplectic manifold without boundary.we will call it geometrically bounded if there exists an almost complex structure J and a complete Riemannian metric g on M such that the following properties are satisfied:

2.the sectional curvature Kg≤C(a positive constant)and the injectivity radius i(M,g)> o.

Definition 3.2(see[17,Definition 2.4])In Definition 3.1 if we require J∈F(M,ω),then the symplectic manifold(M,ω)is called strongly geometrically bounded(SGB).

It is well known that the closed symplectic manifolds are SGB and a product of two SGB symplectic manifolds is SGB.It is easy to prove the symplectic manifolds which at infinity are isomorphic to the symplectization of a closed contact manifold are SGB(cf.[4]).The standard cotangent bundles as well as the twisted cotangent bundles over closed manifolds are SGB(cf. [4,17]).

Let(P1,ω1,J1,g1),(P2,ω2,J2,g2)be two SGB symplectic manifolds such that dimP1=4. V=P1×P2,ω=ω1⊕ω2,J=J1×J2,g=g1⊕g2.Then(V,ω,J,g)is a SGB symplectic manifold.Assumebe a free homotopy class which is defined in Proposition 2.8 such that

From the definition of m(V,ω,J),we can get that a J-holomorphic sphere which represents α is simple.

Consider the Banach manifold B consisting of all maps u∈W2,2(S2,V)such that with D={z||z|≤1}

where Σo,Σ∞are two disjoint smooth submanifolds without boundary of V and closed as subsets in V.We also assume that one of Σoand Σ∞is compact.Denote bythe vector bundle whose fiber over(z,v)∈S2×V consists of all linear maps φ:TzS2→TvV such that J(v)φ=−φ◦i.Given u:S2→V we denote byS2→S2×V the”graphfor the pull back bundle.Let ε be the Banach bundle ε→B whose fiber εu=W1,2at u∈W2,2(S2,V)consists of all W1,2sections ofThe nonlinear Cauchy Riemann operatordu+J◦du◦i,can be considered as a smooth section of ε→B,and its zero set is H(α,J,Σo,Σ∞).By elliptic regularity theory every u∈B with=o is smooth.Hofer and Viterbo proved some propositions[11, Propositions 2.3,2.4 and 2.7]for the compact manifold V which guaranteed the d-index is well defined and made the existence of closed orbit possible.Note that the geometrically bounded symplectic manifolds are the tame almost complex manifolds in[21].Sikorav[21]showed the Gromov’s compactness theorem is true in the tame almost complex manifolds.Lu proved a prior compactness property[17,Proposition 2.5]for the SGB symplectic manifold.Utilizing the prior compactness and assumption(1.2),Lu[17]showed[11,Propositions 2.3,2.4]also hold true for the case of SGB symplectic manifold if the neighborhood UJand Freg(V,ω)∩UJof J in these Propositions were replaced by U(J,δ,fro)and Freg(V,ω)∩U(J,δ,fro).The definition of U(J,δ,fro)was given in[17].In the following,U(J,δ,fro)is abbreviated to U.So the d-index d(α,J,Σo,Σ∞)is well defined in the SGB symplectic manifold.

Thus we have the SGB version of[11,Propositions 2.1,2.3 and 2.4]in the following.

Proposition 3.3Let(V,ω)be a SGB symplectic manifold,J∈F(V,ω),m(V,ω,J)=〈ω,α〉,Let Σo,Σ∞be described above,then there exists an open neighborhood U of J such that

(2)Freg(V,ω)∩U is dense in U.

is a compact S1-manifold with boundary

Let H:V→ℝ be a smooth map and gJ(·,·)=ω(·,J·)the Riemannian metric.We denote by∇H the gradient of H with respect to the metric gJ.For suitable neighborhoods U(Σo),U(Σ∞)of Σo,Σ∞respectively,suppose H|U(Σo)≡ho,H|U(Σ∞)≡h∞,ho

where φ is the unique complex antilinear map TzS2→TvV satisfying the following:

1.If z=o or∞,φ is the zero map.

2.If z/=o and/=∞,φ maps the tangent vector z∈TzS2=Here we took the identity chart S2⊃ℂ≃ℂ to distinguish in TzS2for z∈ℂ the tangent vector z.

If u∈B then the associated graph mapmaps z∈S2into W⊂S2×V.

Consequently we can define h(u)∈ε by

Now we define a parameter depending family of smooth section of ε→B by

Clearly,fλis S1−equivalent for every λ and fλis a Fredholm section in the sense that at every zero u of fλthe linearisation Dfλ:TuB→εuis Fredholm.Consider the set

By elliptic regularity theory,C⊂[o,+∞)×C∞(S2,V).let Cλ={u|(λ,u)∈C}.Then Cois a compact smooth manifold with a free smooth S1-action,and Co=H(α,J,Σo,Σ∞).Lu[17] showed that if the manifold V is SGB,[11,Proposition 2.7]was also true.

Proposition 3.4(see[17,Proposition 3.1])Let α∈[S2,V],Σo,Σ∞,J and H be as above,and let C be compact.Then

i.e.,H(α,J,Σo,Σ∞)is the boundary of a smooth compact manifold M equipped with a free S1-action,so that the action on∂M coincides with the action on H.

As in[11]and[17],we have the following SGB version of[11,Theorem 3.4],which was first observed by Lu in[17].

Proposition 3.5Let(V,ω)be a SGB symplectic manifold.Σo,Σ∞are described above. J∈F(V,ω)such that m(V,ω,J)≥〈ω,α〉,where α∈[S2,V].Let ε→B be the Hilbert space bundle defined above.Let H:V→ℝ be a smooth map such that

Let C be defined above.Then

(1)if(λ,u)∈C,then λ∈[o,λ∞],λ∞=(h∞−ho)−1〈ω,α〉;

(2)for every multi index β there is a constant Cβ>o such that for every(λ,u)∈C, v=u◦φ,here φ:S1×ℝ→ℂ,φ(t,s)=e2π(s+it).

(3)there exists ε>o such that for every(λ,u)∈C we have:if v(s)(S1)/⊂U(Σo)then

If v(s)(S1)/⊂U(Σ∞)then

ProofSince the proof is similar to[11,Theorem 3.4],we only outline a sketch here.Lu [17]first proved thatwas contained in a compact subset of V,see[17,Theorem 2.9].This result guarantees the argument in the proof of[11,Theorem 3.4]can be generalized to SGB case.Furthermore,we can prove this Proposition.

4　Proof of Main Theorem

In order to get the relation between the d-index of P1×P2with the d-index of Pi,i∈{1,2}, we need a regular almost complex structure J=J1×J2,where Ji∈Freg(Pi,ωi),i∈{1,2}. However,the product of regular almost complex structures is not regular in general.Thus Proposition 2.8 is necessary for our case.Now we can prove Theorem 1.1.

ProofLet α∈[S2,P1×P2]be of the form,where pois a fixed point and α1∈[S2,P1]is defined in the hypothesis of Theorem 1.1.On P1×P2we take the product almost complex structure J=J1×J2,where J1∈Freg(P1,ω1),J2∈Freg(P2,ω2). Then

By Proposition 2.8,J is regular at the situation(α,Σo,Σ∞).From d=[∅]and m(P1×P2,ω1⊕ω2,J)≤m(P2,ω2),we have d(α,J,Σo,Σ∞)/=[∅].

In the following,we use the idea of[11]to prove Theorem 1.1.From Proposition 3.4,we can get C is noncompact.We can assume{(λk,uk)}⊂C such that

For every(λ,u)∈C,We define v=u◦φ,where φ:S1×ℝ→ℂ,φ(t,s)=e2π(s+it).Definewhere ω=ω⊕ω.From12Proposition 3.5,we have

where so(v)=sup{s|v((−∞,s]×S1)⊂U(Σ)}.The last inequality is proved by[11,Lemma 3.1],which is also true here.

If(λ,u)∈C,then

We define two sequences of numbers by

Note that vkdenotes the map induced by ukon the cylinder.Clearly

In fact,since(4.1)holds,the nonlinearity u→h(u)is well behaved and one can use Bubble off analysis to obtain the solution u of(4.2).

where ω=ω1⊕ω2.If sk→o or+∞,we have

This contradiction shows that sk∈for all k for some suitable a>o independent of k.

Hence,from the definition ofand the fact thatu it follows that{uk}is convergent itself.However,this contradicts our assumption on{(λk,uk)}.Therefore we know that

Hence,we can find a sequence{sk},

such that with xk:=vk(sk,·),

Eventually taking a subsequence we may assume

It is obvious that x∈C∞(S1,P1×P2).We first assume λS2→P1be the map induced from uk:S2→P1×P2by the projection onto the first factor.Then Now let:Z→P1be the map induced fromin the cylinder.Since∇H vanishes on Σoand Σ∞is holomorphic in the neighbourhood of all z such thatis close to

Since〈ω1,α1〉>o,we have a contradiction.So we must have

and(4.3)still holds.

In the following,we will show that x is nonconstant.Arguing indirectly let us assume x≡const∈P1×P2.Denote bythe P1−component of vk.If we have x=mo∈Σoandu niformly.Since h|U(Σo)=o,this contradicts the definition of.Similarly,

is also impossible.Therefore,we have for some τ>o,

This shows that x has to be nonconstant.Eventually we have H(x(t))∈(ho,h∞).This proves the theorem.

5　Applications

We will give some applications of Theorem 1.1 in this section.Note that given the standard complex structure i on ℂℙ2any two different points determine up to M¨obius transformationa unique holomorphic sphere u.There is an embedding u:holomorphic.={y},where x,y are different points in u(S2).Then with α1=[u],where u(S2)is the holomorphic curve running through x and y,we have

We note here that i is a regular complex structure.Now let P1=ℂℙ2,P2be a SGB symplectic

Corollary 5.1Let Σo,Σ∞,P2be as above,then any stable compact smooth hypersurface S in ℂℙ2×P2separating Σofrom Σ∞possesses at least one periodic Hamiltonian trajectory.

It is well known that the standard cotangent bundles(T∗N,ω)over closed manifolds N is SGB with[ω]|π2(T∗N)=o(cf.[4,17]).

Liouville manifold(ˆM,ˆλ)is a SGB symplectic manifold withLet us recall the definition of Liouville manifold now.A 1-form α on a manifold Σ is called a contact form for ξ:=kerα,if dα is nondegenerate on ξ.In this case ξ is called a contact structure. A compact exact symplectic manifold with boundary(M,λ)is called a Liouville domain,if (Σ:=∂M,α:=λ|∂M)is a contact submanifold.We know every Liouville domain carries a Liouville vector field X defined by ιXω=λ,and the contact condition implies that X points outward at the boundary.We can paste the positive end of a symplectization(Σ×[o,∞),d(etα)) along the boundary Σ.Then we obtain a complete Liouville manifold,which is denoted by (ˆM,ˆλ).

As in[1,23],we introduce the following notation.

Definition 5.2A noncompact manifold M is said to be of finite topological type,if there is a compact domain Ω⊂M such that M˚Ω is diffeomorphic to∂Ω×[1,∞).

Actually,if M is a subset of a closed manifold or if M is of finite topological type the cotangent bundles(T∗M,ω)with standard symplectic structure are geometrically bounded. This is first pointed out by Audin,Lalonde and Polterovich[2,PP.286].Lu[17]also claimed the cotangent bundle of a finite topological type manifold with twisted symplectic structure is SGB and omit the proof.One can find other properties of cotongent bundle in[6].In the following,we will give a different proof of this for the completeness of our results.Our proof uses the idea of[4,Proposition 2.2].

Proposition 5.3Let M be a manifold of finite topological type,then the cotangent bundle(T∗M,ω)with standard symplectic structure is SGB.

ProofSince M is of finite topological type,we may assume there is a compact domainΩ⊂M such that M˚Ω is diffeomorphic to∂Ω×[1,∞).Assume the diffeomorphism is h:∂Ω×[1,∞)→M˚Ω.Denote Λ=M˚Ω,Λs=h(∂Ω×[s,∞)),s≥1,∂Λs=h(∂Ω×{s}).

First we will define the Riemannian metric on T∗M.Let φtbe the flow on T∗M formed by fiberwise dilations by the factor et.Choose a fiberwise convex hypersurface Σ⊂T∗M|Ω, enclosing the compact domain Ω.Note that Σ has contact type for ω.Let U be the closure of the unbounded part of the complement toOn the closure of the bounded part of the complement to Σ in T∗M|Ω,we can choose a compatible almost complex structure J.Let g be the Riemannian metric determined by ω and J,i.e.,g(·,·)=ω(·,J·)(wealso require that the radical vector is g-orthogonal to Σ).Now we can extend these structures to U so that

i.e.,g,just as ω,is homogeneous of degree one with respect to the dilations,and

Then the metric g,the almost complex structure J and the standard symplecture ω are compatible on U.Hence are compatible on T∗M|Ω.To define the Riemannian metric on T∗M|Λ, letbe the flow of the vector field∂son∂Ω×[1,∞),i.e.,

Now extend those structures to T∗M|Λso that

We know the standard symplectic structure ω also satisfies(ψs♯)∗ω=ω.Then ω,J,and g are compatible on T∗M|Λ.Thus we get a compatible triple(ω,J,g)on T∗M.

The metric g is obviously complete.Indeed,define

It is clear the integral curves φt(x),for t>o and x∈Σe,are minimizing geodesics of g.The distance from x todt,goes to∞as t→∞.Let|s1−s2|be positive and small.Assume xdist(x,y)→∞.Let γ(s)be a curve with γ(o)=x,γ(1)=y.From(5.4),we have the lengthwhereis determined by the compact parts of T∗M|∂Λs1and T∗M|∂Λs2.ThusFrom equation(5.3),we know a curve γ(t)from T∗M|∂Λs1to T∗M|∂Λs2has the same length with the curve ψs♯(γ(t))from T∗M|∂Λs+s1to T∗M|∂Λs+s2.ψs♯is a symplectomorphism.Thus we have Therefore,every bounded subset of T∗M is contained in a compact subset and is relatively compact.By Hopf-Rinow Theorem,this is equivalent to completeness.

From[7,Lemma 1]and the definition of metric(5.1),it follows that the sectional curvature of g goes to zero as x→∞in U.Thus the sectional curvature of g is bounded from above on T∗M|Ω.From(5.3)we know the sectional curvature of g on T∗M|Λis determined by the sectional curvature of g on T∗M|Ωwhich is bounded from above.We get that the sectional curvature of g is bounded from above on T∗M.

The calculus of the geodesics is given in the following lemma.

Lemma 5.4With the metric g and notations defined in Proposition 5.3,a curve γs(x) through,is a geodesic if and only ifis a geodesic throughfor any o

ProofWe only give the proof of the first assertion here,since the second can be proved similarly.Let π:T∗M→M be the projection of the cotangent bundle.Assume x∈U12such that π(x)∈Ω.The metric is defined by(φt)∗g=etg.Thus we have

Choose a local coordinate chart(V,φ)of M such that π(x)∈V and(π−1(V),h′)is a local trivialization of T∗M,i.e.,

is a local coordinate chart of x in T∗M.Let(x1,···,xn,y1,···,yn)be the local coordinates and denote

The Christoffel symbols corresponding to the Riemannian metric g is given by

The push forward of the vector fields can be given by

Now suppose γs(x)is a curve through x.In local coordinates γs(x)is given by

Equation of geodesics in the local coordinates

We know γs(x)is a geodesic if and only ifis a geodesic.

From Corollary 5.1 and Proposition 5.3,it is easy to get Corollary 1.3.

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∗June 23,2015;revised March 21,2016.The second author was partially supported by NSFC (11371381,11521101).†

Yangqiao DING.