AA CONTACT CAMEL THEOREM
SIMON ALLAIS
Abstract.
We provide a contact analogue of the symplectic camel theorem that holdsin R n × S , and indeed generalize the symplectic camel. Our proof is based on thegenerating function techniques introduced by Viterbo, extended to the contact case byBhupal and Sandon, and builds on Viterbo’s proof of the symplectic camel. Introduction
In 1985, Gromov made a tremendous progress in symplectic geometry with his theoryof J -holomorphic curves [13]. Among the spectacular achievements of this theory, therewas his famous non-squeezing theorem: if a round standard symplectic ball B nr of radius r can be symplectically embedded into the standard symplectic cylinder B R × R n − ofradius R , then r ≤ R (here and elsewhere in the paper, all balls will be open). Otherproofs were given later on by the means of other symplectic invariants: see Ekeland andHofer [7, 8], Floer, Hofer and Viterbo [11], Hofer and Zehnder [14] and Viterbo [21]. In1991, Eliashberg and Gromov discovered a more subtle symplectic rigidity result: the cameltheorem [9, Lemma 3.4.B]. In order to remind its statement, let us first fix some notation.We denote by q , p , . . . , q n , p n the coordinates on R n , so that its standard symplectic formis given by ω = d λ , where λ = p d q + · · · + p n d q n = p d q . We consider the hyperplane P := { q n = 0 } ⊂ R n , and the connected components P − := { q n < } and P + := { q n > } of its complement R n \ P . We will denote by B nr = B nr ( x ) the round Euclidean ball ofradius r in R n centered at some point x ∈ R n , and by P R := P \ B nr (0) the hyperplane P with a round hole of radius R > n >
2, if there exists a symplectic isotopy φ t of R n anda ball B nr ⊂ R n such that φ ( B nr ) ⊂ P − , φ ( B nr ) ⊂ P + , and φ t ( B nr ) ⊂ R n \ P R forall t ∈ [0 , r ≤ R . The purpose of this paper is to prove a contact version of thistheorem.We consider the space R n × S , where S := R / Z . We will denote the coordinates onthis space by q , p , . . . , q n , p n , z , and consider the 1-form λ defined above also as a 1-formon R n × S with a slight abuse of notation. We denote by α := λ − d z the standardcontact form on R n × S . The set of contactomorphisms of ( R n × S , α ) will be denotedby Cont( R n × S ) and the subset of compactly supported contactomorphisms isotopic tothe identity will be denoted by Cont ( R n × S ). As usual, by a compactly supported Date : August 16, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Gromov non-squeezing, symplectic camel, generating functions. a r X i v : . [ m a t h . S G ] A ug S. ALLAIS contact isotopy of ( R n × S , α ) we will mean a smooth family of contactomorphisms φ t ∈ Cont( R n × S ), t ∈ [0 , R n × S .In 2006, Eliashberg, Kim and Polterovich [10] proved an analogue and a counterpart ofGromov’s non-squeezing theorem in this contact setting; given any positive integer k ∈ N and two radii r, R > πr ≤ k ≤ πR , there exists a compactly supportedcontactomorphism φ ∈ Cont( R n × S ) such that φ ( B nR × S ) ⊂ B nr × S if and onlyif r = R ; however, if 2 n > R < / √ π , then it is always possible to find sucha φ . In 2011, Sandon [15] extended generating function techniques of Viterbo [21] anddeduced an alternative proof of the contact non-squeezing theorem. In 2015, Chiu [6] gavea stronger statement for the contact non-squeezing: given any radius R ≥ / √ π , there isno compactly supported contactomorphism isotopic to identity φ ∈ Cont ( R n × S ) suchthat φ (Closure( B nR × S )) ⊂ B nR × S . The same year, an alternative proof of this strongnon-squeezing theorem was given by Fraser [12] (the technical assumption “ φ is isotopicto identity” is no longer needed in her proof).Our main result is the following contact analogue of the symplectic camel theorem: Theorem 1.1.
In dimension n + 1 > , if πr < (cid:96) < πR for some positive integer (cid:96) and B nR × S ⊂ P − × S , there is no compactly supported contact isotopy φ t of ( R n × S , α ) such that φ = id , φ ( B nR × S ) ⊂ P + × S , and φ t ( B nR × S ) ⊂ ( R n \ P r ) × S for all t ∈ [0 , . Notice that the squeezing theorem of Eliashberg-Kim-Polterovich implies that Theo-rem 1.1 does not hold if one instead assumes that πR < ψ t of R n and a ball B nR ⊂ R n such that ψ ( B nR ) ⊂ P − , ψ ( B nR ) ⊂ P + ,and ψ t ( B nR ) ⊂ R n \ P r for all t ∈ [0 , r < R . Withoutloss of generality, we can assume that ψ = id (see [17, Prop. on page 14]) and thatthe isotopy ψ t is compactly supported. By conjugating ψ t with the dilatation x (cid:55)→ νx , weobtain a new compactly supported symplectic isotopy ψ (cid:48) t with ψ (cid:48) = id and a ball B nνR ⊂ P − such that ψ (cid:48) ( B nνR ) ⊂ P + , and ψ (cid:48) t ( B nνR ) ⊂ R n \ P νr for all t ∈ [0 , ν > π ( νr ) > (cid:96) > π ( νR ) for some (cid:96) ∈ Z , and the contact lift of ψ (cid:48) t to R n × S contradicts our Theorem 1.1.Our proof of the contact camel theorem is based on Viterbo’s proof [21, Sect. 5] ofthe symplectic version, which is given in terms of generating functions. Viterbo’s proof israther short and notoriously difficult to read. For this reason, in this paper we providea self-contained complete proof of Theorem 1.1, beside quoting a few lemmas from therecent work of Bustillo [3]. The generalization of the generating function techniques to thecontact setting is largely due to Bhupal [2] and Sandon [15]. In particular, the techniquesfrom [15] are crucial for our work. Organization of the paper.
In Section 2, we provide the background on generatingfunctions and the symplectic and contact invariants constructed by means of them. In
CONTACT CAMEL THEOREM 3
Section 3, we prove additional properties of symplectic and contact invariants that will bekey to the proof of Theorem 1.1. In Section 4, we prove Theorem 1.1.
Acknowledgments.
I thank Jaime Bustillo who gave me helpful advice and a betterunderstanding of reduction inequalities. I am especially grateful to my advisor MarcoMazzucchelli. He introduced me to generating function techniques and gave me a lot ofadvice and suggestions throughout the writing process.2.
Preliminaries
In this section, we remind to the reader some known results about generating functionsthat we will need.2.1.
Generating functions.
Let B be a closed connected manifold. We will usually write q ∈ B points of B , ( q, p ) ∈ T ∗ B for the cotangent coordinates, ( q, p, z ) ∈ J B for the 1-jetcoordinates and ξ ∈ R N for vectors of some fiber space. A generating function on B is asmooth function F : B × R N → R such that 0 is a regular value of the fiber derivative ∂F∂ξ .Then, Σ F := (cid:26) ( q, ξ ) ∈ B × R N | ∂F∂ξ ( q ; ξ ) = 0 (cid:27) , is a smooth submanifold called the level set of F .Generating functions give a way of describing Lagrangians and Legendrians of T ∗ B and J B respectively. Indeed, ι F : Σ F → T ∗ B, ι F ( q ; ξ ) = ( q, ∂ q F ( q ; ξ ))and (cid:98) ι F : Σ F → J B, (cid:98) ι F ( q ; ξ ) = ( q, ∂ q F ( q ; ξ ) , F ( q ; ξ ))are respectively Lagrangian and Legendrian immersions. We say that F generates theimmersed Lagrangian L := ι F (Σ F ) and the immersed Legendrian L := (cid:98) ι F (Σ F ). In thispaper, we will only consider embedded Lagrangians and Legendrians.We must restrict ourselves to a special category of generating functions: Definition 2.1.
A function F : B × R N → R is quadratic at infinity if there exists aquadratic form Q : R N → R such that the differential d F − d Q is bounded. Q is uniqueand called the quadratic form associated to F .In the following, by generating function we will always implicitly mean generating func-tion quadratic at infinity. In this setting, there is the following fundamental result: Theorem 2.2 ( [16, Sect. 1.2], [21, Lemma 1.6]) . If B is closed, then any Lagrangiansubmanifold of T ∗ B Hamiltonian isotopic to the 0-section has a generating function, whichis unique up to fiber-preserving diffeomorphism and stabilization.
S. ALLAIS
The existence in this theorem is due to Sikorav, whereas the uniqueness is due to Viterbo(the reader might also see [19] for the details of Viterbo’s proof). The contact analogousis the following (with an additional statement we will need later on):
Theorem 2.3 ( [4, Theorem 3], [5, Theorem 3.2], [18, Theorems 25, 26]) . If B is closed,then any Legendrian submanifold of J B contact isotopic to the 0-section has a generatingfunction, which is unique up to fiber-preserving diffeomorphism and stabilization. More-over, if L ⊂ J B has a generating function and φ t is a contact isotopy of J B , then thereexists a continuous family of generating functions F t : B × R N → R such that each F t generates the corresponding φ t ( L ) . Min-max critical values.
In the following, F : B × R N → R is a smooth functionquadratic at infinity of associated quadratic form Q (generating functions are a specialcase). Let q be Morse index of Q (that is the dimension of its maximal negative subspace).We will denote by E the trivial vector bundle B × R N and, given λ ∈ R , E λ the sublevelset { F < λ } ⊂ E .In this paper, H ∗ is the singular cohomology with coefficients in R and 1 ∈ H ∗ ( B ) willalways denote the standard generator of H ( B ) ( B is connected). Let C > F is contained in {| F | < C } . A classical Morse theoryargument implies that (cid:0) E C , E − C (cid:1) is homotopy equivalent to B × ( { Q < C } , { Q < − C } )and the induced isomorphism given by K¨unneth formula: T : H p ( B ) (cid:39) −→ H p + q (cid:0) E C , E − C (cid:1) (2.1)does not depend on the choice of C . So we define H ∗ ( E ∞ , E −∞ ) := H ∗ (cid:0) E C , E − C (cid:1) . Wealso define H ∗ (cid:0) E λ , E −∞ (cid:1) := H ∗ (cid:0) E λ , E − C (cid:1) .Given any non-zero α ∈ H ∗ ( B ), we shall now define its min-max critical value by c ( α, F ) := inf λ ∈ R (cid:8) T α (cid:54)∈ ker (cid:0) H ∗ (cid:0) E ∞ , E −∞ (cid:1) → H ∗ (cid:0) E λ , E −∞ (cid:1)(cid:1)(cid:9) . One can show that this quantity is a critical value of F by classical Morse theory. Proposition 2.4 (Viterbo [21]) . Let F : B × R N → R and F : B × R N → R begenerating functions quadratic at infinity normalized so that F ( q , ξ (cid:48) ) = F ( q , ξ (cid:48)(cid:48) ) = 0 atsome pair of critical points ( q , ξ (cid:48) ) ∈ crit( F ) and ( q , ξ (cid:48)(cid:48) ) ∈ crit( F ) that project to thesame q . Then: (1) if F and F generate the same Lagrangian, then c ( α, F ) = c ( α, F ) for all non-zero α ∈ H ∗ ( B ) , (2) if we see the sum F + F as a generating function of the form F + F : B × R N + N → R , ( F + F )( q ; ξ , ξ ) = F ( q ; ξ ) + F ( q ; ξ ) , then c ( α (cid:94) β, F + F ) ≥ c ( α, F ) + c ( β, F ) , for all α, β ∈ H ∗ ( B ) whose cup product α (cid:94) β is non-zero. CONTACT CAMEL THEOREM 5 (3) if µ ∈ H dim( B ) ( B ) denotes the orientation class of B , then c ( µ, F ) = − c (1 , − F ) . Proof.
Point (1) follows from the uniqueness statement in theorem 2.2 See [21, Prop. 3.3]for point (2) and [21, cor. 2.8] for point (3). (cid:3)
When the base space is a product B = V × W , one has the following Proposition 2.5 ( [21, Prop. 5.1], [3, Prop. 2.1]) . Let F : V × W × R N → R bea generating function and let w ∈ W . Consider the restriction F w : V × R N → R , F w ( v ; ξ ) = F ( v, w ; ξ ) (quadratic at infinity on the base space V ), then (1) if µ is the orientation class of W , then for all non-zero α ∈ H ∗ ( V ) , c ( α ⊗ , F ) ≤ c ( α, F w ) ≤ c ( α ⊗ µ , F ) , (2) if F does not depend on the w -coordinate, for all non-zero α ∈ H ∗ ( V ) and non-zero β ∈ H ∗ ( W ) , c ( α ⊗ β, F ) = c ( α, F w ) . Generating Hamiltonian and contactomorphism.
Let Ham c ( T ∗ M ) be the setof time-1-flows of time dependent Hamiltonian vector field. Given ψ ∈ Ham c ( T ∗ M ), itsgraph gr ψ = id × ψ : T ∗ M (cid:44) → T ∗ M × T ∗ M is a Lagrangian embedding in T ∗ M × T ∗ M . Inorder to see gr ψ ( T ∗ M ) as the 0-section of some cotangent bundle, let us restrict ourselvesto the case M = R n × T k . First we consider the case k = 0 then we will quotient R n + k by Z k in our construction. Consider the linear symplectic map τ : T ∗ R n × T ∗ R n → T ∗ R n , τ ( q, p ; Q, P ) = (cid:18) q + Q , p + P , P − p, q − Q (cid:19) which could also be seen as ( z, Z ) (cid:55)→ (cid:0) z + Z , J ( z − Z ) (cid:1) where J is the canonical complexstructure of R n (cid:39) C n . The choice of the linear map is not important to deduce results ofSubsections 2.1 and 2.4 (in fact, [15], [20] and [21] give different choices). However, we donot know how to show the linear invariance of Subsection 3.2 without this specific choice.The Lagrangian embedding Γ ψ := τ ◦ gr ψ defines a Lagrangian Γ ψ ( T ∗ M ) ⊂ T ∗ R n isotopic to the zero section through the compactly supported Hamiltonian isotopy s (cid:55)→ τ ◦ gr ψ s ◦ τ − where ( ψ s ) is the Hamiltonian flow associated to ψ . As Γ ψ ( T ∗ M ) coincideswith the 0-section outside a compact set, one can extend it to a Lagrangian embedding onthe cotangent bundle of the compactified space L ψ ⊂ T ∗ S n .In order to properly define L ψ for ψ ∈ Ham c (cid:0) T ∗ ( R n × T k ) (cid:1) , let (cid:101) ψ ∈ Ham Z k ( T ∗ ( R n + k ))be the unique lift of ψ which is also lifting the flow ( ψ s ) with (cid:101) ψ = id. The application Γ (cid:101) ψ gives a well-defined Γ ψ : T ∗ ( R n × T k ) (cid:44) → T ∗ ( R n × T k × R k ). We can then compactify thebase space: R n × T k × R k ⊂ B where B equals either S n × T k × S k or S n × T k anddefine L ψ ⊂ T ∗ B . S. ALLAIS
In order to define F ψ : B × R N → R , take any generating function of L ψ normalizedsuch that the set of critical points outside ( R n × T k × R k ) × R N has critical value 0 (theset is connected since L ψ coincides with 0-section outside T ∗ ( R n × T k × R k )).We now extend the construction of L ψ to the case of contactomorphisms. Let Cont ( J M )be the set of contactomorphisms isotopic to identity through compactly supported contac-tomorphisms. Given any ψ ∈ Ham c ( T ∗ M ), its lift (cid:98) ψ : J M → J M, (cid:98) ψ ( x, z ) = ( ψ ( x ) , z + a ψ ( x ))belongs to Cont ( J M ), where a ψ : T ∗ M → R is the compactly supported function satis-fying ψ ∗ λ − λ = d a ψ . In [2], Bhupal gives a mean to define a generating function F φ associated to such contac-tomorphism φ for M = R n × T k in a way which is compatible with ψ (cid:55)→ (cid:98) ψ in the sensethat F (cid:98) ψ ( q, z ; ξ ) = F ψ ( q ; ξ ). Given any φ ∈ Cont ( J R n ) with φ ∗ ( d z − λ ) = e θ ( d z − λ ), (cid:99) gr φ : J R n → J R n × J R n × R , (cid:99) gr φ ( x ) = ( x, φ ( x ) , θ ( x ))is a Legendrian embedding if we endow J R n × J R n × R with the contact structureker( e θ ( d z − λ ) − ( d Z − Λ)), where ( q, p, z ; Q, P, Z ; θ ) denotes coordinates on J R n × J R n × R and Λ = (cid:80) i P i d Q i .For our choice of τ , we must take the following contact identification (cid:98) τ : J R n × J R n × R → J R n +1 , (cid:98) τ ( q, p, z ; Q, P, Z ; θ ) = (cid:18) q + Q , e θ p + P , z ; P − e θ p, q − Q, e θ −
1; 12 ( e θ p + P )( q − Q ) + Z − z (cid:19) so that Γ φ := (cid:98) τ ◦ (cid:99) gr φ is an embedding of a Legendrian compactly isotopic to the 0-sectionof J R n +1 . The construction of Γ φ descends well from R n + k to R n × T k taking the lift of φ ∈ Cont ( J ( R n × T k )) which is contact-isotopic to identity.In fact we will rather be interested by T ∗ M × S (cid:39) J M/ Z ∂∂z and φ ∈ Cont ( T ∗ M × S )which can be identified to the set of Z ∂∂z -equivariant contactomorphism of J M isotopic toidentity. The construction descends well to the last quotient and we obtain a well-definedLegendrian embedding Γ φ : T ∗ ( R n × T k ) × S (cid:44) → J (cid:0) R n × T k × R k × S (cid:1) .We can then compactify the base space R n × T k × R k × S ⊂ B × S , define L ψ ⊂ J ( B × S ) and take as F φ any generating function of L φ normalized such that the set ofcritical points outside ( R n × T k × R k × S ) × R N has critical value 0.2.4. Symplectic and contact invariants.
The symplectic invariants presented here aredue to Viterbo [21]. The generalization to the contact case is due to Sandon [15].throughout this subsection, B denotes a compactification of T ∗ ( R n × T k ). Given any ψ ∈ Ham c (cid:0) T ∗ ( R n × T k ) (cid:1) and any non-zero α ∈ H ∗ ( B ), consider c ( α, ψ ) := c ( α, F ψ ) . CONTACT CAMEL THEOREM 7
Proposition 2.6 (Viterbo, [21, Prop. 4.2, Cor. 4.3, Prop. 4.6]) . Let ( ψ t ) be a compactlysupported Hamiltonian isotopy of T ∗ ( R n × T k ) with ψ = id and ψ := ψ . Let H t : T ∗ ( R n × T k ) → R be the Hamiltonians generating ( ψ t ) . Given any non-zero α ∈ H ∗ ( B ) , (1) There is a one-to-one correspondence between critical points of F and fixed points x of ψ such that t (cid:55)→ ψ t ( x ) is a contractible loop when t ∈ [0 , given by ( x, ξ ) (cid:55)→ x .Moreover, if ( x α , ξ α ) ∈ crit( F ψ ) satisfies F ψ ( x α , ξ α ) = c ( α, F ψ ) , then c ( α, ψ ) = a ψ ( x α ) = (cid:90) ( (cid:104) p ( t ) , ˙ q ( t ) (cid:105) − H t ( ψ t ( x α ))) d t, where ( q ( t ) , p ( t )) := ψ t ( x α ) . The value a ψ ( x ) will be called the action of the fixedpoint x . (2) If H t ≤ , then c ( α, ψ ) ≥ . (3) If ( ϕ s ) is a symplectic isotopy of T ∗ ( R n × T k ) , then s (cid:55)→ c ( α, ϕ s ◦ ψ ◦ ( ϕ s ) − ) isconstant. (4) If µ is the orientation class of B , c (1 , ψ ) ≤ ≤ c ( µ, ψ ) with c (1 , ψ ) = c ( µ, ψ ) ⇔ ψ = id ,c ( µ, ψ ) = − c (1 , ψ − ) . These results were not stated with this generality in [21] but the proofs given by Viterboimmediately generalize to this setting.Given any open bounded subset U ⊂ T ∗ ( R n × T k ) and any non-zero α ∈ H ∗ ( B ), Viterbodefines the symplectic invariant c ( α, U ) := sup ψ ∈ Ham c ( U ) c ( α, ψ ) . This symplectic invariant extends to any unbounded open set U ⊂ T ∗ ( R n × T k ) by takingthe supremum of the c ( α, V ) among the open bounded subsets V ⊂ U . Proposition 2.7 (Bustillo, Viterbo) . For all open bounded sets
U, V ⊂ T ∗ ( R n × T k ) andany non-zero α ∈ H ∗ ( B ) , (1) if ( ϕ s ) is a symplectic isotopy of T ∗ ( R n × T k ) , then s (cid:55)→ c ( α, ϕ s ( U )) is constant, (2) U ⊂ V implies c ( α, U ) ≤ c ( α, V ) , (3) if µ and µ are the orientation classes of the compactification of T ∗ ( R n × T k ) and R k respectively, then for any neighborhood W of ∈ R k , c ( µ , U ) (cid:54) c ( µ ⊗ µ ⊗ , U × W × T k ) . (4) if B n +2 kr ⊂ T ∗ ( R n × T k ) is an embedded round ball of radius r and µ is theorientation class of B , then c ( µ, B n +2 kr ) = πr . Proof.
Point (1) is a consequence of Proposition 2.6 (3). Point (2) is a consequence ofthe definition as a supremum. Point (3) is proved in the proof of [3, Prop. 2.3]. Indeed,Bustillo makes use of (3) to deduce his Proposition 2.3 by taking the infimum of c ( µ ⊗ µ ⊗ S. ALLAIS , U × V × T k ) among neighborhoods U ⊃ X and W ⊃ { } (using Bustillo’s notations).We refer to [1, Sect. 3.8] for a complete proof of (4). (cid:3) Now, we give the contact extension of these invariants. Given any φ ∈ Cont (cid:0) T ∗ ( R n × T k ) × S (cid:1) and any non-zero α ∈ H ∗ ( B × S ), consider c ( α, φ ) := c ( α, F φ ) . The following Proposition is due to Sandon. Since our setting is slightly different, weprovide precise references for the reader’s convenience.
Proposition 2.8 (Sandon, [15]) . (1) Given any φ ∈ Cont ( T ∗ ( R n × T k ) × S ) , if µ isthe orientation class of B × S , then c ( µ, φ ) = 0 ⇔ c (1 , φ − ) = 0 . (2) Given any φ ∈ Cont ( T ∗ ( R n × T k ) × S ) and any non-zero α ∈ H ∗ ( B × S ) , if F φ is a generating function of φ , then (cid:6) c (cid:0) α, φ − (cid:1)(cid:7) = (cid:100) c ( α, − F φ ) (cid:101) . (3) Given any φ ∈ Cont ( T ∗ ( R n × T k ) × S ) , any non-zero α ∈ H ∗ ( B × S ) , if ( ψ s ) isa contact isotopy of T ∗ ( R n × T k ) × S , then s (cid:55)→ (cid:100) c ( α, ψ s ◦ φ ◦ ( ψ s ) − ) (cid:101) is constant. (4) For each ψ ∈ Ham c ( T ∗ ( R n × T k )) for each non-zero cohomology class α ∈ H ∗ ( B ) ,if d z denotes the orientation class of S , then c (cid:16) α ⊗ , (cid:98) ψ (cid:17) = c (cid:16) α ⊗ d z, (cid:98) ψ (cid:17) = c ( α, ψ ) . Proof.
Let F φ be the generating function of φ ∈ Cont ( T ∗ ( R n × T k ) × S ). According toduality formula in Proposition 2.4 (3), c ( µ, φ ) = − c (1 , − F φ ). Points (1) and (2) then followfrom [15, lemmas 3.9 and 3.10] taking L = 0-section and Ψ = (cid:98) τ ◦ (cid:91) gr φ − ◦ (cid:98) τ − : c (1 , F φ − ) = 0 ⇔ c (1 , − F φ ) = 0and (cid:100) c (1 , F φ − ) (cid:101) = (cid:100) c (1 , − F φ ) (cid:101) . Point (3) is a consequence of [15, lemma 3.15] applied to c t = c ( α, ψ t ◦ φ ◦ ( ψ t ) − ). Point (4)is given by the proof of [15, Prop. 3.18]. Indeed, let i a : ( E a , E −∞ ) (cid:44) → ( E ∞ , E −∞ ) and (cid:101) i a :( (cid:101) E a , (cid:101) E −∞ ) (cid:44) → ( (cid:101) E ∞ , (cid:101) E −∞ ) be the inclusion maps of sublevel sets of F ψ and F (cid:98) ψ respectively.Then (cid:101) E a = E a × S and, after identifying H ∗ ( (cid:101) E a , (cid:101) E −∞ ) with H ∗ ( E a , E −∞ ) ⊗ H ∗ ( S ), theinduced maps in cohomology (cid:101) i a ∗ is given by (cid:101) i a ∗ = i ∗ a ⊗ id . Thus (cid:101) i a ∗ ( α ⊗ β ) = ( i ∗ a α ) ⊗ β is non-zero if and only if i ∗ a α is non-zero, where β ∈ { d z, } . (cid:3) CONTACT CAMEL THEOREM 9
Let µ be the orientation class of B × S , given any open bounded subset U ⊂ T ∗ ( R n × T k ) × S and any non-zero α ∈ H ∗ ( B × S ), consider c ( α, U ) := sup φ ∈ Cont ( U ) (cid:100) c ( α, φ ) (cid:101) and γ ( U ) := inf (cid:8)(cid:6) c (cid:0) µ, φ (cid:1)(cid:7) + (cid:6) c (cid:0) µ, φ − (cid:1)(cid:7) | φ ∈ Cont (cid:0) T ∗ ( R n × T k ) × S (cid:1) such that φ ( U ) ∩ U = ∅ (cid:9) , These contact invariants extend to any unbounded open set U ⊂ T ∗ ( R n × T k ) × S bytaking the supremum among the open bounded subsets V ⊂ U . Proposition 2.9 (Sandon [15]) . For all open bounded sets
U, V ⊂ T ∗ ( R n × T k ) × S andany non-zero α ∈ H ∗ ( B × S ) , (1) if ( ϕ s ) is a contact isotopy of T ∗ ( R n × T k ) × S , then s (cid:55)→ c ( α, ϕ s ( U )) is constant, (2) U ⊂ V implies c ( α, U ) ≤ c ( α, V ) , (3) given any open subset W ⊂ T ∗ ( R n × T k ) , for each non-zero class β ∈ H ∗ ( B ) , if d z denotes the orientation class of S , then c ( β ⊗ d z, W × S ) = (cid:100) c ( β, W ) (cid:101) . Proof.
Point (1) is a direct consequence of Proposition 2.8 (3). Point (2) is a consequenceof the definition as a supremum. Point (3) follows from the proof of [15, Prop. 3.20].Indeed, inequality c ( β ⊗ d z, W × S ) ≥ (cid:100) c ( β, W ) (cid:101) is due to Proposition 2.8 (4) whereas theother one is due to the fact that, for all φ ∈ Cont ( W × S ), one can find ψ ∈ Ham c ( W )such that φ ≤ (cid:98) ψ in Sandon’s notations (see her proof for more details). (cid:3) Some properties of symplectic and contact invariants
Estimation of γ (cid:0) T ∗ C × B n − R × S (cid:1) . Here, we will prove the following
Lemma 3.1.
Let
R > be such that πR (cid:54)∈ Z , b > , n > and C := R /d Z . Then γ (cid:0) T ∗ C × B n − R × S (cid:1) ≤ (cid:6) πR (cid:7) . Remark 3.2.
This Lemma fails for n = 1. The use of Lemma 3.1 will be the step wherewe will need the assumption that 2 n + 1 > Lemma 3.3.
Let x ∈ ∂B n − R , x ∈ B n − R and r := | x − x | . We set θ ( r ) ∈ [0 , π ] to besuch that cos (cid:18) θ ( r )2 (cid:19) = r R Then any rotation ρ : R n − → R n − of angle θ ( r ) centered at x sends x outside B n − R , i.e. ρ ( x ) (cid:54)∈ B n − R . bx x aa (cid:48) α∂D∂B n − r ( x ) c ρ ( x ) Figure 1.
Configuration in the plane P πR + ε R R + δh + ε h ε ε δ R − δ Figure 2.
Approximating h + ε by a smooth compactly supported h ε . Proof.
Let x ∈ ∂B n − R , x ∈ B n − R and r := | x − x | . Take any rotation ρ : R n − → R n − of angle θ ( r ), where θ ( r ) is defined as above. Let P ⊂ R n − be the affine plane spannedby x , x and ρ ( x ). The round disk B n − R ∩ P has a radius smaller than R and lies in anopen round disk D of radius R with x ∈ ∂D centered at c ∈ P . Therefore, it is enoughto show that ρ ( x ) (cid:54)∈ D . Let a, a (cid:48) be the two points of ∂D ∩ ∂B n − r ( x ), b be the secondpoint of ∂D ∩ ( x c ) and α be the unoriented angle (cid:91) ax a (cid:48) ∈ [0 , π ] (see Figure 1). As [ x b ] isa diameter of ∂D , the triangle abx is right at a , thus α = (cid:100) ax b satisfiescos (cid:16) α (cid:17) = ax bx = r R .
Hence, α = θ ( r ) and ρ ( x ) (cid:54)∈ D . (cid:3) Proof of Lemma 3.1.
We exhibit a family of ψ ε ∈ Ham c ( R n − ) satisfying • ψ ε (cid:0) B n − R (cid:1) ∩ B n − R = ∅ , • c (1 , ψ ε ) = 0, • ∀ ε > c ( µ, ψ ε ) ≤ πR + ε , CONTACT CAMEL THEOREM 11 where µ is the orientation class of the compactified space S n − . Consider the radialHamiltonian H ( x ) = − h ( r ) where h : [0 , + ∞ ) → R is defined by: h ( u ) = 12 (cid:90) R θ ( √ v ) d v − (cid:90) min( u, R )0 θ ( √ v ) d v If ψ designates the time-1-flow associated to H , for all r ∈ [0 , R ] and any x ∈ B n − R suchthat | x | = r , ψ ( x ) is the image of x by some 0-centered rotation of angle − h (cid:48) ( r ) = θ ( r ).Thus ψ ( x ) (cid:54)∈ B n − R and ψ ( B n − R ) ∩ B n − R = ∅ . Nevertheless, ψ is not well defined as H is not smooth in the neighborhood of x = 0 and | x | = 2 R . For every small ε >
0, we thenconstruct a family of smooth h ε : [0 , + ∞ ) → R approximating h in the following way (seeFigure 2): there exists δ = δ ( ε ) ∈ (0 , R ) such that • h ε is compactly supported on [0 , R + δ ], • h (cid:48) ( u ) ≤ h (cid:48) ε ( u ) ≤ π for all u ∈ [0 , + ∞ ), • h ε ( u ) = πR + ε − π u for all u ∈ (cid:2) , δ (cid:3) , • h ε ( u ) = h ( u ) + ε for all u ∈ [ δ, R − δ ].Hamiltonians H ε ( x ) := − h ε ( | x | ) are smooth functions so their time-1-flow ψ ε are welldefined. As H ε ≤ c (1 , ψ ε ) = 0. The only fixed point with non-zero action is 0 so c ( µ, ψ ) ≤ − H ε (0) = − H (0) + ε (0 has action − H ε (0)) and − H (0) = h (0) = 12 (cid:90) R θ ( √ v ) d v + ε. Changing the variable x = √ v R and writing θ ( s ) = 2 arccos (cid:0) s R (cid:1) ,12 (cid:90) R θ ( √ v ) d v = 8 R (cid:90) x arccos( x ) d x Then, an integration by parts and writing x = sin α give (cid:90) x arccos( x ) d x = 12 (cid:90) x √ − x d x = 12 (cid:90) π sin α d α = π , thus c ( µ, ψ ε ) ≤ πR + ε as expected.Now, from the family ψ ε we deduce a second, ϕ ε ∈ Ham c ( T ∗ C × R n − ), satisfying:(1) ϕ ε (cid:0) C × (cid:0) − ε , ε (cid:1) × B n − R (cid:1) ∩ (cid:0) C × (cid:0) − ε , ε (cid:1) × B n − R (cid:1) = ∅ ,(2) c (1 , ϕ ε ) = 0,(3) ∀ ε > c ( µ, ϕ ε ) ≤ πR + ε ,where µ is the orientation class of the compactified space C × S × S n − . Let U ε := C × (cid:0) − ε , ε (cid:1) × B n − R and χ : R → [0 ,
1] be a smooth compactly supported functionwith χ | [ − ε , ε ] ≡
1. We then introduce the compactly supported negative Hamiltonians K ε : C × R × R n − → R defined by: K ε ( q , p , x ) := χ ( p ) H ε ( x ) , ∀ ( q , p , x ) ∈ C × R × R n − , so that ϕ ε ( q , p , x ) = ( q , p , ψ ε ( x )) for p ∈ (cid:0) − ε , ε (cid:1) , thus ϕ ε ( U ε ) ∩ U ε = ∅ as wanted.Moreover, since K ε is negative, c (1 , ϕ ε ) = 0. The function χ can be chosen so that it is even and decreasing inside supp χ ∩ ( ε , + ∞ ) with an arbitrarily small derivative. For | p | > ε and ( q , p , x ) ∈ supp ϕ ε , the q -coordinate of ϕ ε ( q , p , x ) is thus slightly differentfrom q . Thus, the only fixed points with a non-zero action are the ( q , p , | p | ≤ ε .The action is still given by − K ε (0) = h (0) + ε = πR + ε , thus c ( µ, ϕ ε ) ≤ πR + ε .Take the contact lift of the previous family: (cid:99) ϕ ε ∈ Cont ( T ∗ C × R n − × S ). Property(1) of ϕ ε implies (cid:99) ϕ ε (cid:18) C × (cid:18) − ε , ε (cid:19) × B n − R × S (cid:19) ∩ (cid:18) C × (cid:18) − ε , ε (cid:19) × B n − R × S (cid:19) = ∅ . (3.1)On the one hand, Proposition 2.8 (4) and property (2) of ϕ ε gives c (cid:0) , (cid:99) ϕ ε (cid:1) = c (1 , ϕ ε ) = 0 . Thus, if d z denotes the orientation class of S , Proposition 2.8 (1) gives c (cid:16) µ ⊗ d z, (cid:0)(cid:99) ϕ ε (cid:1) − (cid:17) = 0 . (3.2)On the other hand, Proposition 2.8 (4) and property (3) of ϕ ε gives, c (cid:0) µ ⊗ d z, (cid:99) ϕ ε (cid:1) = c ( µ, ϕ ε ) ≤ πR + ε. (3.3)Equations (3.2) and (3.3) then imply (cid:108) c (cid:16) µ ⊗ d z, (cid:99) ϕ ε (cid:17)(cid:109) + (cid:108) c (cid:16) µ ⊗ d z, (cid:0)(cid:99) ϕ ε (cid:1) − (cid:17)(cid:109) ≤ (cid:6) πR + ε (cid:7) . Thus, since (cid:99) ϕ ε verifies (3.1), γ (cid:0) U ε × S (cid:1) ≤ (cid:6) πR + ε (cid:7) . Since πR (cid:54)∈ Z , x (cid:55)→ (cid:100) x (cid:101) is continuous at πR and any open bounded set V ⊂ T ∗ C × B n − R × S is included in U ε × S for a small ε , we conclude that γ (cid:0) T ∗ C × B n − R × S (cid:1) ≤ (cid:6) πR (cid:7) . (cid:3) Linear symplectic invariance.
A symplectomorphism ϕ : T ∗ ( R n × T k ) → T ∗ ( R n × T k ) will be called linear when it can be lifted to a linear map (cid:101) ϕ : R n + k ) → R n + k ) .Throughout this subsection, we fix a linear symplectomorphism ϕ : T ∗ ( R n × T k ) → T ∗ ( R n × T k ) of the form ϕ ( q , q ) = ( ϕ ( q ) , ϕ ( q )) , ∀ ( q , q ) ∈ R n + k × T k , for some linear maps ϕ : R n + k → R n + k and ϕ : T k → T k . Let B be either S n × T k or S n + k , such that B × T k is a compactification of T ∗ ( R n × T k ). We denote by µ ∈ H ∗ ( B )the orientation class of B . Proposition 3.4.
For any open subset U ⊂ T ∗ ( R n × T k ) and for any non-zero α ∈ H ∗ ( T k ) ,we have c ( µ ⊗ α, ϕ ( U )) = c ( µ ⊗ ϕ ∗ α, U ) . CONTACT CAMEL THEOREM 13
Proposition 3.4 is a consequence of the following statement:
Lemma 3.5.
For any ψ ∈ Ham c ( T ∗ ( R n × T k )) and for any non-zero α ∈ H ∗ ( T k ) , c ( µ ⊗ α, ψ ) = c ( µ ⊗ ϕ ∗ α, ϕ − ◦ ψ ◦ ϕ ) . In order to prove Lemma 3.5, we will need suitable generating functions for ψ and ϕ − ◦ ψ ◦ ϕ : Lemma 3.6.
Let F : B × T k × R N → R be a generating function of ψ ∈ Ham c ( T ∗ ( R n × T k )) . There exists a diffeomorphism ϕ (cid:48) : B → B such that, if Φ : B × T k × R N → B × T k × R N denotes the diffeomorphism Φ( q , q ; ξ ) = ( ϕ (cid:48) ( q ) , ϕ ( q ); ξ ) , then F := F ◦ Φ is a generating function of ϕ − ◦ ψ ◦ ϕ ∈ Ham c ( T ∗ ( R n × T k )) . Proof of Lemma 3.6.
Let ψ ∈ Ham c ( T ∗ ( R n × T k )) and R > ψ ⊂ B n + kR × T k . Since ϕ : R n + k → R n + k is linear and invertible, one can find a diffeomorphism ϕ (cid:48) : R n + k → R n + k such that ϕ (cid:48) ( x ) = ϕ ( x ) for all x ∈ B n + kR ∪ ϕ − ( B n + kR ) and ϕ (cid:48) ( x ) =( x , . . . , x n + k − , ± x n + k ) outside some compact set, thus ϕ (cid:48) can naturally be extendedinto a diffeomorphism ϕ (cid:48) : B → B . Let F : B × T k × R N → R be a generatingfunction of ψ , we then define the diffeomorphism Φ : B × T k × R N → B × T k × R N by Φ( q , q ; ξ ) := ( ϕ (cid:48) ( q ) , ϕ ( q ); ξ ). The function F := F ◦ Φ is a generating functionsince ∂F ∂ξ = ∂F ∂ξ ◦ Φ. Let ( q ; ξ ) ∈ Σ F and q := ( ϕ (cid:48) × ϕ )( q ). First, let us assume that q ∈ ϕ − ( B n + kR × T k ) (so q = ϕ ( q )), since ( q ; ξ ) ∈ Σ F , there exists x ∈ T ∗ ( R n × T k )such that( q ; ∂ q F ( q ; ξ ) · v ) = (cid:18) x + ψ ( x )2 ; (cid:104) J ( ψ ( x ) − x ) , v (cid:105) (cid:19) , ∀ v ∈ R n + k ) . Let x = ϕ − ( x ). On the one hand, by linearity of ϕ − , q = x + ϕ − ◦ ψ ◦ ϕ ( x )2 , on the other hand, ∂ q F ( q ; ξ ) · v = (cid:104) J ( ψ ( x ) − x ) , ϕ ( v ) (cid:105) for all v ∈ R n + k ) and ϕ − is alinear symplectomorphism, thus ∂ q F ( q ; ξ ) · v = (cid:10) J ϕ − ( ψ ( x ) − x ) , v (cid:11) = (cid:10) J ( ϕ − ◦ ψ ◦ ϕ ( x ) − x ) , v (cid:11) , ∀ v ∈ R n + k ) . Now, let us assume that q (cid:54)∈ ϕ − ( B n + kR × T k ). If q is at infinity, then ∂ q F ( q ; ξ ) = 0 sinced F = 0 at any point at infinity. If q ∈ R n + k × T k , let x ∈ T ∗ ( R n × T k ) be associatedto q as above. Since x + ψ ( x )2 (cid:54)∈ B n + kR × T k , necessarily, ψ ( x ) = x (cid:54)∈ B n + kR × T k so ∂ q F ( q ; ξ ) = 0 and ( q ; ξ ) is a critical value of F . Hence ( q ; ξ ) is a critical value of F and( q ; ∂ q F ( q ; ξ )) = ( q ; 0) = (cid:18) x + ϕ − ◦ ψ ◦ ϕ ( x )2 ; J ( ψ ( x ) − x ) (cid:19) , where x = ( ϕ (cid:48) × ϕ ) − ( x ) (cid:54)∈ supp( ϕ − ◦ ψ ◦ ϕ ). Conversely, if x ∈ T ∗ ( R n × T k ), the associated ( q ; ξ ) ∈ Σ F is given by (( ϕ (cid:48) × ϕ ) − ( q ); ξ )where ( q ; ξ ) ∈ Σ F is associated to x = ( ϕ (cid:48) × ϕ )( x ) ∈ T ∗ ( R n × T k ). (cid:3) Proof of Lemma 3.5.
Let F : B × T k × R N → R be a generating function of ψ ∈ Ham c ( T ∗ ( R n × T k )). Let ϕ (cid:48) : B → B and Φ : B × T k × R N → B × T k × R N bethe diffeomorphisms defined by Lemma 3.6 such that F := F ◦ Φ is a generating functionof ϕ − ◦ ψ ◦ ϕ . Let us denote by E and E the domains of the generating functions F and F respectively. For all λ ∈ R , Φ gives a diffeomorphism of sublevel sets Φ : E λ → E λ . Inparticular, it induces an homology isomorphism Φ ∗ : H ∗ ( E λ , E −∞ ) → H ∗ ( E λ , E −∞ ). Wethus have the following commutative diagram: H l ( B × T k ) ( ϕ (cid:48) × ϕ ) ∗ (cid:15) (cid:15) T (cid:47) (cid:47) H l + q ( E ∞ , E −∞ ) Φ ∗ (cid:15) (cid:15) i ∗ ,λ (cid:47) (cid:47) H l + q ( E λ , E −∞ ) Φ ∗ (cid:15) (cid:15) H l ( B × T k ) T (cid:47) (cid:47) H l + q ( E ∞ , E −∞ ) i ∗ ,λ (cid:47) (cid:47) H l + q ( E λ , E −∞ )where the T j ’s denote the isomorphisms induced by the K¨unneth formula (2.1) and the i ∗ j,λ ’s are the morphisms induced by the inclusions i j,λ : ( E λj , E −∞ j ) (cid:44) → ( E ∞ j , E −∞ j ). Thecommutativity of the right square is clear. As for the left square, it commutes because π ◦ Φ = ( ϕ (cid:48) × ϕ ) ◦ π , where π : B × T k × R N → B × T k is the canonical projection. Let α be a non-zero class of H l ( T k ) and µ be the orientation class of H ∗ ( B ). Since the verticalarrows are isomorphisms, i ∗ ,λ T ( µ ⊗ α ) is non-zero if and only if i ∗ ,λ T ( ϕ (cid:48) × ϕ ) ∗ ( µ ⊗ α ) isnon-zero. Since ϕ (cid:48) is a diffeomorphism, ( ϕ (cid:48) ) ∗ µ = ± µ , thus ( ϕ (cid:48) × ϕ ) ∗ ( µ ⊗ α ) = ± µ ⊗ ϕ ∗ α and i ∗ ,λ T ( µ ⊗ α ) is non-zero if and only if ± i ∗ ,λ T ( µ ⊗ ϕ ∗ α ) is non-zero. Therefore, c ( µ ⊗ α, F ) = c ( µ ⊗ ϕ ∗ α, F ) . (cid:3) Reduction lemma.
In this subsection, we work on the space T ∗ ( R m × T l × T k ) × S and the points in this space will be denoted by ( q, p, z ), where q = ( q , q ) ∈ ( R m × T l ) × T k and p = ( p , p ) ∈ R m + l × R k . Let B be a compactification of T ∗ ( R m × T l ). Given any openset U ⊂ T ∗ ( R m × T l × T k ) × S and any point w ∈ T k , the reduction U w ⊂ T ∗ ( R m × T l ) × S at q = w is defined by U w := π ( U ∩ { q = w } ) , where π : T ∗ ( R m × T l ) × { w } × R k × S → T ∗ ( R m × T l ) × S is the canonical projection. Lemma 3.7.
Let µ be the orientation class of B × S k × S and be the generator of H ( T k ) . For any open bounded set U ⊂ T ∗ ( R m × T l × T k ) × S and any w ∈ T k , c ( µ ⊗ , U ) ≤ γ ( U w ) . It is an extension to the contact case of Viterbo-Bustillo’s reduction lemma [3, Prop.2.4] and [21, Prop. 5.2]. We will follow Bustillo’s proof as close as contact structure allowsus to do.
CONTACT CAMEL THEOREM 15
Let µ be the orientation class of B × S k × S and 1 be the generator of H ( T k ) andfix an open bounded set U ⊂ T ∗ ( R m × T l × T k ) × S and a point w ∈ T k . Remark thatone can write µ = µ ⊗ µ , where µ and µ are the orientation classes of B × S and S k respectively. By definition of the contact invariants, it is enough to show that, given any ψ ∈ Cont ( U ) and any ϕ ∈ Cont ( T ∗ ( R m × T l ) × S ) such that ϕ ( U w ) ∩ U w = ∅ , (cid:100) c ( µ ⊗ , ψ ) (cid:101) ≤ (cid:6) c (cid:0) µ , ϕ (cid:1)(cid:7) + (cid:6) c (cid:0) µ , ϕ − (cid:1)(cid:7) . Thus, we fix a contact isotopy ψ t defined on T ∗ ( R m × T l × T k ) × S and compactlysupported in U such that ψ = id and ψ =: ψ ∈ Cont ( U ) and we fix a contactomorphism ϕ ∈ Cont ( T ∗ ( R m × T l ) × S ) such that ϕ ( U w ) ∩ U w = ∅ . Let F t : ( B × S k × T k × S ) × R N → R be a continuous family of generating functions for the Legendrians L t := L ψ t ⊂ J ( B × S k × T k × S ) given by Theorem 2.3, F := F and K : ( B × S ) × R N (cid:48) → R be a generating function of ϕ . By the uniqueness statement of Theorem 2.3, one maysuppose that F ( x ; ξ ) = Q ( ξ ) where Q : R N → R is a non-degenerated quadratic formwithout loss of generality. Recall that F tw : ( B × S k × S ) × R N → R denotes the function F tw ( q , p, z ; ξ ) := F t ( q , w, p, z ; ξ ) and let (cid:101) K : ( B × S k × S ) × R N (cid:48) → R be the generatingfunction defined by (cid:101) K ( x, y, z ; η ) := K ( x, z ; η ). In order to prove Lemma 3.7, we will usethe following Lemma 3.8.
Given t ∈ [0 , , let c t := c ( µ, F tw − (cid:101) K ) which is a continuous R -valuedfunction. Then we have the following alternative: • either ∀ t ∈ [0 , , c t (cid:54)∈ Z • or ∃ (cid:96) ∈ Z such that ∀ t ∈ [0 , , c t = (cid:96) .In particular, (cid:108) c (cid:16) µ, − (cid:101) K (cid:17)(cid:109) = (cid:108) c (cid:16) µ, F w − (cid:101) K (cid:17)(cid:109) . Proof of Lemma 3.8.
The reduced function F tw generates L tw ⊂ J ( B × S k × S ). L t is theimage of the immersion (plus points in the 0-section at infinity):Γ ψ t ( q, p, z ) = (cid:32) q + Q t , e θ t p + P t , z ; P t − e θ t p, q − Q t , e θ t −
1; 12 (cid:16) e θ t p + P t (cid:17) (cid:0) q − Q t (cid:1) + Z t − z (cid:33) , writing ψ t ( q, p, z ) = ( Q t , P t , Z t ). Therefore, L tw is the set of points (plus points in the0-section at infinity): (cid:32) q + Q t , e θ t p + P t , z ; P t − e θ t p , q − Q t , e θ t −
1; 12 (cid:16) e θ t p + P t (cid:17) (cid:0) q − Q t (cid:1) + Z t − z (cid:33) for points ( q, p, z ) that verify q + Q t = w . In the remaining paragraphs, we will use notations p = e θ p + P and q = q + Q .Suppose there exists (cid:96) ∈ Z and t ∈ [0 ,
1] such that c t = (cid:96) . Then it is enough to provethat t (cid:55)→ c t is locally constant. In order to do so, we will follow Bustillo’s proof [3, lemma (cid:0) q t , p t , z t ; ξ t , η t (cid:1) be the critical point of( F t w − (cid:101) K )( q , p , z ; ξ, η ) = F t ( q , w, p , z ; ξ ) − K ( q , p , z ; η )associated to the min-max value c t = (cid:96) . By continuity of the min-max critical point,we may suppose that K is a Morse function in some neighborhood of (cid:0) q t , p t , z t ; η t (cid:1) byperturbing K without changing its value at this point. Writing x t := (cid:0) q t , p t , z t (cid:1) , such acritical point verifies ∂F t w ∂x = ∂ (cid:101) K∂x and ∂F t w ∂ξ = ∂ (cid:101) K∂η = 0 . These equations define two points ( x t , ∂ x F t w , F t w ) = ( x t , ∂ x (cid:101) K, F t w ) ∈ L t w and ( x t , ∂ x (cid:101) K, (cid:101) K ) ∈ L ϕ × T ∗ S k which only differ in the last coordinate by a (cid:96) ∂∂a factor: (cid:0) x t , ∂ x F t w , F t w (cid:1) = (cid:16) x t , ∂ x (cid:101) K, (cid:101) K + (cid:96) (cid:17) = (cid:16) x t , ∂ x (cid:101) K, (cid:101) K (cid:17) + (cid:96) ∂∂a . We will denote by ( q t , p t , z t ) ∈ T ∗ ( R m × T k + l ) × S the point whose image is Γ ψ t ( q t , p t , z t ) =(( q t , w, p t , z t ; ξ t ) , ∂ x F t , F t ) and we will denote ( Q t , P t , Z t ) = ψ t ( q t , p t , z t ). Since( x t , ∂ x (cid:101) K, (cid:101) K ) ∈ L ϕ × T ∗ S k , ∂ p (cid:101) K = 0 so q t = Q t (= w ).Remark that ϕ ( U w ) ∩ U w = ∅ together with ( x t , ∂ x F t w , F t w ) ∈ (cid:0) L ϕ + (cid:96) ∂∂a (cid:1) × T ∗ S k impliesthat either (cid:0) q t , p t , z t (cid:1) (cid:54)∈ U w or (cid:0) Q t , P t , Z t (cid:1) (cid:54)∈ U w . In order to see it, we go back tothe definition of generating function on T ∗ ( R m × T l ) × S given in Subsection 2.3. Let π : J R m + l → T ∗ ( R m × T l ) × S be the quotient projection and consider the Z l × Z -equivariantlift of ϕ : (cid:101) ϕ ∈ Cont( J R m + l ) and (cid:102) U w := π − ( U w ) ⊂ J R m + l . Since ϕ ( U w ) ∩ U w = ∅ , wehave that (cid:101) ϕ ( (cid:102) U w ) ∩ (cid:102) U w = ∅ so L (cid:101) ϕ ∩ (cid:98) τ ( (cid:102) U w × (cid:102) U w × R ) = ∅ . But (cid:102) U w + ∂∂z = (cid:102) U w and (cid:98) τ ( x, X + ∂∂z , θ ) = (cid:98) τ ( x, X, θ ) + ∂∂a for all ( x, X, a ) ∈ J R m + l × J R m + l × R , intersection L (cid:101) ϕ ∩ (cid:98) τ ( (cid:102) U w × (cid:102) U w × R ) = ∅ is thus equivalent to (cid:18) L (cid:101) ϕ + (cid:96) ∂∂a (cid:19) ∩ (cid:98) τ (cid:16) (cid:102) U w × (cid:102) U w × R (cid:17) = ∅ (definition of (cid:98) τ : J R m + l × J R m + l × R → J R m + l )+1 is given in Subsection 2.3). Hence,given any point ( u, v, a ) ∈ L (cid:101) ϕ + (cid:96) ∂∂a , the corresponding ( x, X, θ ) = (cid:98) τ − ( u, v, a ) verifiesthat either x (cid:54)∈ (cid:102) U w or X (cid:54)∈ (cid:102) U w . This property descends to quotient: ( x t , ∂ x F t w , F t w ) ∈ (cid:0) L ϕ + (cid:96) ∂∂a (cid:1) × T ∗ S k implies that either ( q t , p t , z t ) (cid:54)∈ U w or ( Q t , P t , Z t ) (cid:54)∈ U w .Since q t = Q t = w , it follows that either ( q t , p t , z t ) (cid:54)∈ U or ( Q t , P t , Z t ) (cid:54)∈ U . Since ψ t has its support in U , they both imply that (cid:0) Q t , P t , Z t (cid:1) = ψ t (cid:0) q t , p t , z t (cid:1) = (cid:0) q t , p t , z t (cid:1) (cid:54)∈ U. Hence, ( q t , p t , z t ) is outside the support of ψ t , thus, the associated point ( q t , w, p t , z t ; ξ t ) ∈ Σ F t is critical of value F t ( q t , w, p t , z t ; ξ t ) = 0. Thus, we have seen that m t := (cid:0) q t , p t , z t ; ξ t , η t (cid:1) verifies ∂F t w ∂x = ∂ (cid:101) K∂x = 0 , ∂F t w ∂ξ = ∂ (cid:101) K∂η = 0 and F t w = 0 , CONTACT CAMEL THEOREM 17 so it is a critical point of − (cid:101) K , as wished, with the same critical value − (cid:101) K = F t w − (cid:101) K .Let t (cid:55)→ m t be the continuous path of critical value of t (cid:55)→ F tw − (cid:101) K obtained by min-max.It remains to show that c t = ( F tw − (cid:101) K )( m t ) is equal to (cid:96) in some neighborhood of t . Since (cid:101) K does not depend on ξ , ∂F t ∂ξ (cid:0) q t , w, p t , z t ; ξ t (cid:1) = 0so the point ( q t , w, p t , z t ; ξ t ) remains inside the level set Σ F t . If H : [0 , × ( T ∗ ( R m × T l × T k ) × S ) → R denotes the compactly supported Hamiltonian map associated to ( ψ t ), ι F t (cid:0) q t , w, p t , z t ; ξ t (cid:1) ∈ U c ⊂ (supp H ) c , and (supp H ) c is an open set so, for all t in a small neighborhood of t , ι F t ( q t , w, p t , z t ; ξ t ) ∈ (supp H ) c thus F t ( q t , w, p t , z t ; ξ t ) = 0 and ( q t , p t , z t ; ξ t ) remains a critical point of F tw .Thus, in a small neighborhood of t , since m t is a critical point of F tw − (cid:101) K and F tw (witha slight abuse of notation), t (cid:55)→ ( q t , p t , z t ; η t ) is a continuous path of critical value for K .But K is a Morse function in some neighborhood of (cid:0) q t , p t , z t ; η t (cid:1) , thus this continuouspath is constant and K ( q t , p t , z t ; η t ) ≡ − (cid:96) .Finally, we have seen that, in some neighborhood of t , F t ( q t , w, p t , z t ; ξ t ) ≡ K ( q t , p t , z t ; η t ) ≡ − (cid:96) , thus c t = F t (cid:0) q t , w, p t , z t ; ξ t (cid:1) − K (cid:0) q t , p t , z t ; η t (cid:1) ≡ (cid:96). In particular, since t (cid:55)→ (cid:100) c t (cid:101) is constant, one has (cid:108) c (cid:16) µ, F w − (cid:101) K (cid:17)(cid:109) = (cid:108) c (cid:16) µ, F w − (cid:101) K (cid:17)(cid:109) . But F w ( x ; ξ ) = Q ( ξ ) where Q is a non-degenerated quadratic form, so F w − (cid:101) K is a stabi-lization of the generating function − (cid:101) K , thus Proposition 2.4 (1) implies c ( µ, F w − (cid:101) K ) = c ( µ, − (cid:101) K ). (cid:3) Proof of Lemma 3.7.
By Proposition 2.5 (1), c ( µ ⊗ , ψ ) := c ( µ ⊗ , F ) ≤ c ( µ, F w ) . The triangular inequality of Proposition 2.4 (2) applied to µ = µ (cid:94) c (cid:16) µ, F w (cid:17) ≤ c (cid:16) µ, F w − (cid:101) K (cid:17) − c (cid:16) , − (cid:101) K (cid:17) . By Proposition 2.4 (3) and Proposition 2.5 (2), we have − c (1 , − (cid:101) K ) = c ( µ, (cid:101) K ) = c ( µ , K ).Hence c (cid:16) µ ⊗ , ψ (cid:17) ≤ c (cid:16) µ, F w − (cid:101) K (cid:17) + c (cid:16) µ , K (cid:17) , and thus, (cid:108) c (cid:16) µ ⊗ , ψ (cid:17)(cid:109) ≤ (cid:108) c (cid:16) µ, F w − (cid:101) K (cid:17)(cid:109) + (cid:108) c (cid:16) µ , K (cid:17)(cid:109) . According Lemma 3.8, (cid:100) c ( µ, F w − (cid:101) K ) (cid:101) = (cid:100) c ( µ, − (cid:101) K ) (cid:101) so (cid:108) c (cid:16) µ ⊗ , ψ (cid:17)(cid:109) ≤ (cid:108) c (cid:16) µ, − (cid:101) K (cid:17)(cid:109) + (cid:108) c (cid:16) µ , K (cid:17)(cid:109) . Since K generates ϕ , (cid:100) c ( µ , − K ) (cid:101) = (cid:100) c ( µ , ϕ − ) (cid:101) , according to Proposition 2.8 (2). ThusProposition 2.5 (2) gives (cid:100) c ( µ, − (cid:101) K ) (cid:101) = (cid:100) c ( µ , ϕ − ) (cid:101) . Finally, by definition, c ( µ , K ) = c ( µ , ϕ ). (cid:3) Contact camel theorem
In this section, we will prove Theorem 1.1. We work on the space R n × S in dimension2 n + 1 >
3, we denote by q , p , . . . , q n , p n , z coordinates on R n × S so that the Liouvilleform is given by λ = p d q := p d q + · · · + p n d q n and the standard contact form of R n × S is α = p d q − d z . Let τ t ( x ) = x + t ∂∂q n be the contact Hamiltonian flow of R n × S associatedto the contact Hamiltonian ( t, x ) (cid:55)→ p n . Lemma 4.1.
Let R and r be two positive numbers and B nR × S ⊂ P − × S . If there existsa contact isotopy ( φ t ) of ( R n × S , α ) supported in [ − c/ , c/ n × S for some c > suchthat φ = id , φ ( B nR × S ) ⊂ P + × S and φ t ( B nR × S ) ⊂ ( R n \ P r ) × S for all t ∈ [0 , ,then there exists a smooth family of contact isotopy s (cid:55)→ ( ψ st ) with ψ st ∈ Cont( R n × S ) and ψ s = id associated to a smooth family of contact Hamiltonians s (cid:55)→ ( H st ) supportedin [ − c/ , c/ n − × R × S , such that, for all s ∈ [0 , , all t ∈ R and all x ∈ R n × S , ψ sc ( x ) = x + c ∂∂q n , (4.1) ψ st (cid:18) x + c ∂∂q n (cid:19) = ψ st ( x ) + c ∂∂q n , ∀ t ∈ R (4.2) ψ st + c = ψ sc ◦ ψ st . (4.3) Moreover, for all t ∈ R , ψ t = τ t whereas ψ t := ψ t satisfies ψ t (cid:0) B nR × S (cid:1) ⊂ (cid:32) R n \ (cid:91) k ∈ Z (cid:18) P r + kc ∂∂q n (cid:19)(cid:33) × S , ∀ t ∈ R . (4.4) Proof.
Assume there exists such a ( φ t ). Let K t : R × ( R n × S ) → R be the compactlysupported contact Hamiltonian associated to ( φ t ). By hypothesis, K t is supported in[ − c/ , c/ n , thus one can define its c ∂∂q n -periodic extension K (cid:48) t : R × ( R n × S ) → R and the associated contact isotopy ( φ (cid:48) t ). The contactomorphism φ (cid:48) t : R n × S → R n × S satisfies φ (cid:48) t (cid:16) x + c ∂∂q n (cid:17) = φ (cid:48) t ( x ) + c ∂∂q n .For all s ∈ [0 , ψ st ) with ψ st ∈ Cont( R n × S ) and ψ s = id defined as follow (look also at Figure 3). Given x ∈ R n , trajectory γ ( t ) = ψ st ( x )first follows t (cid:55)→ φ (cid:48) t ( x ) from t = 0 to t = s . Then γ follows t (cid:55)→ τ t ( φ (cid:48) s ( x )) from t = 0 to t = 1 /
4. Then t (cid:55)→ (cid:101) φ (cid:48) t ( τ / ◦ φ (cid:48) s ( x )) from t = 0 to t = s , where ( (cid:101) φ t ) = ( τ c/ ◦ φ − t ◦ τ − c/ ) isthe contact Hamiltonian flow associated to the translated contact Hamiltonian application (cid:101) K (cid:48) t = − K (cid:48) s − t ◦ τ − c/ . Finally, γ follows t (cid:55)→ τ t ( (cid:101) φ (cid:48) s ◦ τ / ◦ φ (cid:48) s ( x )) from t = 0 to t = 3 /
4. Wenormalize time such that s (cid:55)→ ψ st gives an isotopy of smooth contact Hamiltonian flows of c ∂∂t -periodic contact Hamitonians H st . CONTACT CAMEL THEOREM 19 B r × S P R × S P R × S + ∂∂q n τ c/ τ c/ φ (cid:48) (cid:101) φ (cid:48) Figure 3.
Construction of ψ . Identity (4.1) comes from the fact that ψ sc = τ c/ ◦ (cid:101) φ s ◦ τ c/ ◦ φ s and, by definition of( (cid:101) φ t ), (cid:101) φ s = τ c/ ◦ φ − s ◦ τ − c/ . Identity (4.2) comes from the fact that contactomorphism ψ st is a composition of c ∂∂q n -equivariant contactomorphisms. Identity (4.3) is implied by c ∂∂t -periodicity of Hamiltonian H st . Inclusion (4.4) comes from the hypothesis on the contactisotopy ( φ t ). (cid:3) Let r, R > (cid:96) satisfying πr < (cid:96) < πR and let B nR × S ⊂ P − × S . Suppose by contradiction that there exists a contact isotopy( φ t ) of ( R n × S , α ) supported in [ − c/ , c/ n × S for some c > φ = id, φ ( B nR × S ) ⊂ P + × S and φ t ( B nR × S ) ⊂ ( R n \ P r ) × S for all t ∈ [0 , r ∈ ( (cid:96) − , (cid:96) ). Consider the family ofcontact isotopy s (cid:55)→ ( ψ st ) given by Lemma 4.1 and denote by ( H st ) the associated familyof Hamiltonian supported in [ − c/ , c/ n × R × S . We define λ st : R n × S → R by( ψ st ) ∗ α = λ st α . Let us consider:Ψ s : R × R × R n +1 → R × R × R n +1 , Ψ s ( t, h, x ) = ( t, λ st ( x ) h + H st ◦ ψ st ( x ) , ψ st ( x )) . According to (4.1), (4.2) and (4.3), for all ( t, h, x ) ∈ R × R × R n +1 , ∀ k, l ∈ Z , Ψ s (cid:18) t + lc, h, x + kc ∂∂q n (cid:19) = Ψ s ( t, h, x ) + lc ∂∂t + ( k + l ) c ∂∂q n . Thus Ψ s descends to a map Ψ s : T ∗ C × T ∗ ( R n − × C ) × S → T ∗ C × T ∗ ( R n − × C ) × S where C := R /c Z . Lemma 4.2.
The family s (cid:55)→ Ψ s is a contact isotopy of the contact manifold ( T ∗ C × T ∗ ( R n − × C ) × S , ker( d z − p d q + h d t )) . Proof.
We write ψ st ( q, p, z ) = ( Q t ( q, p, z ) , P t ( q, p, z ) , Z t ( q, p, z )) , since d Z t − P t d Q t = ( ψ st ) ∗ ( d z − p d q ) = λ st ( d z − p d q ), we have(Ψ s ) ∗ ( d z − p d q + h d t ) = d Z t − P t d Q t + ˙ Z t d t − P t ˙ Q t d t + ( λ st h + H st ◦ ψ st ) d t = λ st ( d z − p d q + h d t ) + (cid:16) ˙ Z t − P t ˙ Q t + H st ◦ ψ st (cid:17) d t. But, since H st is the contact Hamiltonian of the isotopy ψ st supported in [ − c/ , c/ n × R × S , P t ˙ Q t − ˙ Z t = H st ◦ ψ st . Finally,(Ψ s ) ∗ ( d z − p d q + h d t ) = λ st ( d z − p d q + h d t ) . (cid:3) For technical reasons, we replace T ∗ R n × S by its quotient T ∗ ( R n − × C ) × S andconsider our B nR × S inside this quotient (since c can be taken large while R is fixed, thisidentification is well defined). Let us consider the linear symplectic map: L : T ∗ C × T ∗ ( R n − × C ) → T ∗ C × T ∗ ( R n − × C ) ,L ( t, h, x, q n , p n ) = ( q n − t, − h, x, q n , p n − h )and denote by (cid:98) L = L × id the associated contactomorphism of T ∗ C × T ∗ ( R n − × C ) × S .Let us consider U := (cid:98) L (cid:0) Ψ (cid:0) T ∗ C × B nR × S (cid:1)(cid:1) . We compactify the space T ∗ C × T ∗ ( R n − × C ) as C × S × S n − × C × S (cid:39) S n − × T × C . Let µ and d z be the orientation class of S n − × T and S respectively and d q n and d t be the canonical base of H ( C ). Lemma 4.3.
One has the following capacity inequality: c ( µ ⊗ d t ⊗ d z, U ) ≥ (cid:6) πR (cid:7) . Proof.
Let α := µ ⊗ d t . Since s (cid:55)→ (cid:98) L ◦ Ψ s is a contact isotopy, Proposition 2.9 (1) implies c ( α ⊗ d z, U ) = c (cid:16) α ⊗ d z, (cid:98) L (cid:0) Ψ (cid:0) T ∗ C × B nR × S (cid:1)(cid:1)(cid:17) . But (cid:98) L (Ψ ( T ∗ C × B nR × S )) = L (Φ ( T ∗ C × B nR )) × S where Φ : T ∗ C × T ∗ ( R n − × C ) → T ∗ C × T ∗ ( R n − × C ) is the linear symplectic map:Φ ( t, h, x, q n , p n ) = ( t, h + p n , x, q n + t, p n ) . Thus, using Proposition 2.9 (3), c ( α ⊗ d z, U ) = (cid:6) c (cid:0) α, L (cid:0) Φ (cid:0) T ∗ C × B nR (cid:1)(cid:1)(cid:1)(cid:7) . In order to conclude, let us show that c (cid:0) µ ⊗ d t, L (cid:0) Φ (cid:0) T ∗ C × B nR (cid:1)(cid:1)(cid:1) ≥ πR . By the linear symplectic invariance stated in Proposition 3.4, c (cid:0) µ ⊗ d t, L ◦ Φ (cid:0) T ∗ C × B nR (cid:1)(cid:1) = c (cid:0) µ ⊗ A ∗ d t, T ∗ C × B nR (cid:1) , where A : C → C is the linear map A ( t, q n ) = ( q n , q n + t ). We have A ∗ d t = d q n , therefore c (cid:0) µ ⊗ d t, L ◦ Φ (cid:0) T ∗ C × B nR (cid:1)(cid:1) = c (cid:0) µ ⊗ d q n , T ∗ C × B nR (cid:1) . CONTACT CAMEL THEOREM 21
The cohomology class µ ⊗ d q n can be seen as the tensor product of the orientation class µ of the compactification S n − × S × C of T ∗ ( R n − × C ) by the orientation class d h of the compactification S of the h -coordinate and the generator 1 of H ( C ) (for the t -coordinate). Indeed, d q n ∈ H ( C ) can be identify to d q n ⊗ ∈ H ( C ) ⊗ H ( C ) (writingalso d q n for the orientation class of C by a slight abuse of notation). Hence, if µ S n − andd p n are the orientation classes of S n − and of the compactification S of the p n -coordinaterespectively, then µ ⊗ d q n = ( µ S n − ⊗ d h ⊗ d p n ) ⊗ ( d q n ⊗
1) = ( µ S n − ⊗ d q n ⊗ d p n ) ⊗ d h ⊗ µ ⊗ d h ⊗ . Now, according to Proposition 2.7 (3), c (cid:0) ⊗ d h ⊗ µ , C × R × B nR (cid:1) ≥ c (cid:0) µ , B nR (cid:1) . Finally, Proposition 2.7 (4) implies c (cid:0) µ ⊗ d t, L (cid:0) Φ (cid:0) T ∗ C × B nR (cid:1)(cid:1)(cid:1) ≥ c (cid:0) µ , B nR (cid:1) = πR . (cid:3) Proof of Theorem 1.1.
Let us apply Lemma 3.7 with m = n − l = k = 1 and theorientation class µ ⊗ d t ⊗ d z to the exhaustive sequence of open bounded subsets definedby U ( k ) := (cid:98) L (Ψ ( C × ( − k, k ) × B nR )); taking the supremum among k >
0, we find: c ( µ ⊗ d t ⊗ d z, U ) ≤ γ ( U ) , (4.5)where U ⊂ T ∗ C × T ∗ R n − × S is the reduction of U at q n = 0. Now Ψ ( T ∗ C × B nR × S ) ⊂ T ∗ C × (cid:83) t ∈ [0 ,c ] ψ t ( B nR × S ) so U ⊂ (cid:98) L T ∗ C × (cid:91) t ∈ [0 ,c ] ψ t ( B nR × S ) . (4.6)Let V := (cid:83) t ∈ [0 ,c ] ψ t ( B nR × S ) ∩ { q n = 0 } and π : T ∗ R n − × { } × R × S → T ∗ R n − × S be the canonical projection. Since (cid:98) L does not change the q n -coordinate and coordinates of T ∗ R n − , inclusion (4.6) implies U ⊂ T ∗ C × π ( V ) . But (4.4) implies V ⊂ ( B nr (0) ∩ { q n = 0 } ) × S and π ( B nr (0) ∩ { q n = 0 } ) = B n − r (0),thus U ⊂ T ∗ C × B n − r (0) × S . (4.7)Since πr (cid:54)∈ Z , by Lemma 3.1, γ (cid:0) T ∗ C × B n − r (0) × S (cid:1) ≤ (cid:6) πr (cid:7) , thus, Lemma 4.3, inclusion (4.7) and inequality (4.5) gives (cid:6) πR (cid:7) ≤ γ ( U ) ≤ γ (cid:0) T ∗ C × B n − r (0) × S (cid:1) ≤ (cid:6) πr (cid:7) , a contradiction with πR > (cid:96) > πr . (cid:3) References
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Simon Allais, ´Ecole Normale Sup´erieure de Lyon, UMPA46 all´ee d’Italie, 69364 Lyon Cedex 07, France
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