SYMPLECTIC TOPOLOGY AS THE GEOMETRY OF GENERATING FUNCTIONS.

by Claude Viterbo

Abstract. In this paper we introduce a very simple approach to symplectic topology, using the notion of generating functions quadratic at infinity of Lagrange submanifolds in a cotangent bundle. We first prove that such a generating function is essentially unique. Then, a finite dimensional variational study of the generating function yields invariants of the Lagrange submanifold. We prove a certain number of properties of this invariants. If our Lagrange submanifold is the graph of a compact supported symplectic map ψ of ^{2n} we get two numbers c_{+}(ψ) and c_{−}(ψ). We prove many properties of these invariants, C^{0} continuity, continuity with respect to a generating Hamiltonian

etc... . These invariants are used to get estimates on the number of periodic orbits and to define symplectic invariants for sets. The symplectic invariants thus defined satisfy the axioms of capacities. They also behave very nicely with respect to symplectic reduction, thus allowing us to give simple proofs of the camel problem, and a conjecture by Y. Eliashberg.

Ceremade, U.A. 749 du C.N.R.S., Université de Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris Cedex 16, FRANCE.

Introduction

In the last years important progress in symplectic topology have been achieved using either of the following approaches:

The Gromov-Floer approach studies pseudo holomorphic spheres in symplectic manifolds to infer results on the symplectic topology of the manifold. In particular, results on the geometry of Lagrange submanifolds, existence of fixed points of symplectomorphisms isotopic to the identity, have thus been obtained for which we refer to [G 1] [Fl 1], [Fl 2]. This approach is essentially the only one available when working in arbitrary symplectic manifolds. In a certain sense we might say that this approach considers symplectic topology as the study of pseudo-holomorphic curves in almost complex manifolds.

The other approach, which is the one we are interested in here, could be called the Conley-Zehnder approach. It is mainly concerned with periodic orbits of Hamiltonian systems. This can be a very efficient way of dealing with symplectic topology questions, at least in ^{2n}, as has been sufficiently demonstrated for instance by Ekeland and Hofer in [E-H 2], [E-H 3], [V 3] and the author [V 2]. The method is based on studying the topology of the action functional, this is what we now explain. Let H(t, x) be a time dependent Hamiltonian on ^{2n}, we search 1-periodic solutions of ẋ = J∇H(t, x) = X_{H}(t, x) where dH(t, x)ξ = ω(X_{H}(t, x), ξ) = ⟨J∇H(t, x), ξ⟩ (if we identify ^{2n} to ^{n}, then J is the matrix of multiplication by i). Such periodic solutions can be obtained as critical points of the action functional:

A_{H}(x) = ∫_{S^{1}} ⟨Jx, ẋ⟩ − H(t, x)dt.

The usual way to get these critical points is by taking minmax on some classes of sets invariant by the flow. While this would be ackward to describe in this introduction, we point out that it becomes the usual minmax theory if we replace A_{H} by a finite dimensional reduction A_{H}^{N}. The capacities of H are then the critical value associated to a minmax on certain cohomology classes (cf.[V 3]). It is thus fundamental to compute the topology of the sublevel sets {x | A_{H}^{N}(x) ≤ λ }. Thus we can say that this approach considers symplectic topology as the topology of the action functional.

Our approach is based on the remark that the action functional is nothing else than a special generating function (for the definition of this we refer to section 1). This was already exploited in [V 1] to show that all the indices defined for periodic solutions of Hamiltonian systems are, up to a constant, all equal, and that this indices have a very geometric definition. The natural generalization of such a result is to show that the sublevel sets of a generating function are, after a suitable suspension, diffeomorphic (provided we restrict ourselves to generating functions with quadratic phase cf. section 1). Thus we can get rid of action functionals, and work with any generating function of the time one flow of the Hamiltonian, which is a much more flexible tool. The ideas developed here originated from a question that Yasha Eliashberg asked to Helmut Hofer and myself. Hofer’s answer is contained in [H]. The present paper is an expanded version of our own answer which is corollary 4.8. In the course of the proof which was based on the usual study of the ”action functional topology” it appeared that chosing between the ”broken geodesic” reduction of Chaperon-Laudenbach-Sikorav, and the Lyapounov-Schmidt reduction of Conley-Zehnder, meant having to chose between two equally desirable properties, C^{0} continuity of the capacities (prop. 4.6), and the ”triangle inequality” (prop. 4.8). The advantage of our approach is that we get both properties, moreover most of the proofs are then obvious. The paper is organized as follows.

In section 1 we define generating functions, and prove that they are essentially unique.

Section 2 associates to every Lagrange submanifold L of T^{∗}B and cohomology class u in H^{∗}(B) a real number c(u, L). In this section and the next one we prove basic properties of this numbers.

In section 4 we associate to every compact supported symplectomorphism ψ of 2n isotopic to the identity (in the set of compact supported symplectomorphisms) a Lagrange submanifold Γ_{ψ} of T^{∗}S^{2n} which is the compactification of its graph. The construction of section 3 then yields two numbers that we shall denote by c_{+}(ψ) and c_{−}(ψ). This allows us to define positive symplectomorphisms (by the condition c_{−}(ψ) = 0) and to prove several remarkable properties of these maps.

We also prove that a compact supported symplectomorphism has infinitely many periodic points in the interior of its support.

Section 5 is devoted to studying the behaviour of our invariants by symplectic reduction. The treatment is in no way complete and is only pursued as needed to give a simple proof of the ”camel problem”, section 6 deals with a property of ”simple hypersurfaces”.

It is a pleasure to thank Yasha Eliashberg and Dusa McDuff for many useful conversations. I also warmly thank , François Laudenbach for chasing an uncountable number of mistakes, and suggesting many improvements, and Marc Chaperon for simplifying the proof of prop. 4.6. Most of this paper was written while I was visiting the Forschungsinstitut für Mathematik of the E.T.H. in Zürich. I would like to thank Prof. Jürgen Moser for his kind invitation. I am also very grateful to Edi Zehnder with whom I had many useful discussions and who patiently listened to a preliminary version of some of the results of this paper.

1. Some properties of generating functions with quadratic phase.

We first recall some definitions about generating functions.

Let E →^{π} B be a vector bundle over the manifold B, S : E → a C^{2} function. We shall denote by \frac{∂S}{∂ξ} (e) the fiber derivative of S at e ∈ E. We assume the map \frac{∂S}{∂ξ} : E → E^{∗} to be transverse to 0, we may then define the manifold

Σ_{S} = = \{e ∈ E / \frac{∂S}{∂ξ} (e) = 0\} ,

and the map

\begin{aligned}i_{S} : & Σ_{S} → T^{∗}B \\ & e → (π(e) , ∂S(e)) \end{aligned}

Note that because \frac{∂S}{∂ξ} (e) = 0 for e in Σ_{S}, we can identify dS(e) to an element of T_{π(e)}^{∗}B. To simplify the notations we shall often take “abstract local coordinates” (b, ξ) on E, we then write i_{S}(b, ξ) = (b, \frac{∂S}{∂b} (b, ξ)).

It is clear that i_{S} is a Lagrange immersion. More generally, given a Lagrange immersion φ : L → T^{∗}B, we shall say that S is a generating function for φ if there is a diffeomorphism h : L → Σ_{S} such that i_{S} ∘ h = φ.

Without further assumptions, the question of existence and uniqueness of generating functions for a given Lagrange immersion is completely solved in terms of algebraic topology invariants of the immersion (cf. [Gi], [La]), however we shall consider the following class of generating functions.

Definition 1.1.

A generating function S is said to be quadratic at infinity if S(b, ξ) = q_{∞}(b, ξ) for |ξ| large enough, where q_{∞} is a nondegenerate quadratic form on each fiber.

We shall abbreviate “generating function quadratic at infinity” into “g.f.q.i.”. It seems that algebraic topological methods cannot decide whether a Lagrange immersion has a g.f.q.i. or not.

We refer to [Si 1], [Si 2] for examples and counterexamples, and quote the main result from there

Proposition 1.2.

The property of having a g.f.q.i. is invariant by Hamiltonian isotopy.

Note: By Hamiltonian isotopy we mean the time one flow of a time dependent Hamiltonian vector field.

The main new result of this section will be a uniqueness result for g.f.q.i.. Before we state it, we need to define equivalence of g.f.q.i. (cf. [Hö], [We]).

Definition 1.3.

Let S_{1} : E_{1} → , S_{2} : E_{2} → be two g.f.q.i.. We shall say that S_{1} and S_{2} are equivalent if there is a fiber preserving diffeomorphism Φ : E_{1} → E_{2}, such that S_{2} ∘ Φ = S_{1} + Cst. .

We will usually write Φ(b, ξ) = (b, φ(v, ξ)) so that S_{2}(b, φ(b, ξ)) = S_{1}(b, ξ) + C. Clearly S_{1} and S_{2} generate the same Lagrange immersion.

Definition 1.4.

Let S_{1} : E_{1} → be a g.f.q.i. and q_{2} : E_{2} → a nondegenerate quadratic form on the fibers. Then S_{2} : E_{1} ⊕ E_{2} → given by

S_{2}(b, ξ_{1}, ξ_{2}) = S_{1}(b, ξ_{1}) + q_{2}(b, ξ_{2})

will be called a stabilization of S_{1}.

Once again, S_{1} and S_{2} generate the same Lagrange immersion. Our main result in this section can be stated as

Proposition 1.5.

Let S_{1}, S_{2} be two g.f.q.i. generating the same Lagrange embedding i of B into T^{∗}B. Assume that i(B) = φ_{1}(O_{B}), where φ_{t} is a Hamiltonian isotopy of T^{∗}L. Then after stabilization, S_{1} is equivalent to S_{2}.

The proof will take up the end of this section. It is based on the following lemma

Lemma 1.6.

If S_{1}, S_{2} are g.f.q.i. for the zero section of T^{∗}B, then after stabilization S_{1} and S_{2} are equivalent.

Proof:

We may of course assume S_{1} to be a quadratic form in the fiber, that is S_{1}(b, ξ^{+}, ξ^{−}) = |ξ^{+}|^{2} − |ξ^{−}|^{2}, so that Σ_{1} = B × {0}.

Now theorem 4.1.10 of [La], (or [Ce] prop 4 p.168), states that there is a fiber preserving diffeomorphism φ : U_{1} → U_{2}, where U_{1}, U_{2} are neighbourhoods of Σ_{1}, Σ_{2}, such that

S_{2} ∘ φ = S_{1} + C

(provided S_{1} and S_{2} have been suitably stabilized).

Remark:. In [La], only generating functions on trivial bundles are considered, and stabilization is only done with a quadratic form independent on the base variable. Then the quoted theorem of [La] actually states that the set of equivalence classes of germs of generating functions under fiber preserving diffeomorphisms is isomorphic to an affine space on the vector space [X, BO] the space of stable vector bundles over X. Then in 4.4.3, (loc. cit.) the action of [X, BO] is explicited; if α ∈ [X, BO] is represented by the vector bundle A, and Q_{A} is a quadratic form on each fiber of a trivial vector bundle with negative bundle V, then our action is given by S → S +Q_{A}. Thus Latour’s theorem really implies the local equivalence of generating functions with our definition of equivalence.

We now show here φ can be extended to a fiber preserving diffeomorphism Φ : E_{1} → E_{2}, such that S_{2} ∘ Φ = S_{1} + C.

Consider on E_{1} the metric induced by the Euclidean metric |ξ^{+}|^{2} + |ξ^{−}|^{2} on each fiber, and endow E_{2} with a metric such that φ is an isometry where defined.

Now φ sends the stable (resp. unstable) manifold of Σ_{1} to the stable (resp. unstable) manifold of Σ_{2}.

Upon replacing U_{0} by a subset, we may assume that U_{0} is contained in S_{1}^{−1}([−ε, ε]) for ε small enough, and that U_{0} is stable by the gradient flow of S_{1} (restricted to S_{1}^{−1}([−ε, ε])).

We first extend φ to U_{0} ∪ S_{1}^{−1}(−ε). Since U_{0} ∩ S_{1}^{−1}(−ε) is a neighbourhood of the “negative sphere bundle in S_{1}^{−1}(−ε)” that is the intersection of the unstable manifold of Σ_{0} with S_{1}^{−1}(ε), and since S_{1}^{−1}(−ε) is diffeomorphic to the product of the “negative sphere bundle” with E_{1}^{+}, existence of an extension of φ follows from the contractibility of the fibre of the map

(*)  Diff_{+}(S^{k} × ^{q}) ⟶ Emb_{0}(S^{k} × D^{q}, S^{k} × ^{q})

Here Diff_{+}(S^{k} × ^{q}) is the space of orientation preserving diffeomorphisms, and Emb_{0} is the set of embeddings inducing a degree one map on the spheres. That the above map is indeed a Serre fibration follows easily from the extension of isotopies principle, while the contractibility of the fibre is proved using Alexander’s trick as follows: the fibre over the ”standard” embedding is the space

{ψ ∈ Diff_{+}(S^{k} × ^{q})∥ψ_{|S^{k}×D^{q}} = i}

where i is the standard embedding of S^{k}×D^{q} into S^{k}× ^{q}. Set ψ(x, α) = (X(x, α), A(x, α)) and ψ_{t}(x, α) = (X(x, tα), \frac{1}{t}A(x, tα)). Then as t goes to 0, ψ_{t}(x, α) goes to the identity map and we thus get our retraction of the fibre to a point. Now, this means that the map (∗) is a weak homotopy equivalence, which implies, using the connexity of Emb_{0}(S^{k} × D^{q}, S^{k} × ^{q}) that any family of embeddings of S^{k} × D^{q} into S^{k} × ^{q} (inducing a map of degree one on the spheres) extends to a family of diffeomorphisms of S^{k} × ^{q}. This is exactly what we need to be able to extend φ, since one can show that the embedding of the neighbourhood of the negative sphere (which is diffeomorphic to S^{k} × D^{q}) into S^{−1}(−ε) ≃ S^{k} × ^{q} induces a map of degree one on S^{k}.

We thus found our extension of φ to a map

φ̃ : U_{0} ∪ S_{1}^{−1}(−ε) → U_{1} ∪ S_{2}^{−1}(−ε) .

The next step is to extend φ̃ to \overline{φ} defined on all S_{1}^{−1}([−ε, ε]). Let η_{t}^{1} be the flow of the normalized fiber gradient of S_{1}, that is S_{1}(η_{t}^{2}(b, ξ)) = S_{1}(b, ξ) + t, which is well defined in S_{1}^{−1}([−ε, ε]) − U_{0}. Let η_{t}^{2} be the analogous flow for S_{2}. We set for (b, ξ) ∈ S_{1}^{−1}(−ε)

\overline{φ}(η_{t}^{1}(b, ξ)) = η_{t}^{2}(φ(b, ξ)) .

This definition coincides with φ on ∂U_{0}, since by assumption φ maps the fiber gradient flow of S_{1} to the fiber gradient flow of S_{2}.

Now that \overline{φ} is defined on S_{1}^{−1}([−ε, ε]) we may extend it to Φ : E_{1} → E_{2} by again following the flow of η_{t}^{1}, η_{t}^{2}. Our proof is now complete.

We may now conclude the proof of 1.5. This follows mainly from the construction of Laudenbach and Sikorav (cf [L-S], [Si 1], [Si 2])

Let L be a Lagrange submanifold of T^{∗}B having a g.f.q.i.. Then, according to [Si 1], if ψ_{t} is a Hamiltonian isotopy of T^{∗}B, then φ_{1}(L) has a g.f.q.i. given by

S_{1}(b, ξ, η) = A(b, ξ) + S_{0}(β(b, ξ), η)

Here A = A_{φ_{t}} is a function depending on φ_{t} only, and β : E → B is \underline{not} the canonical projection. We shall write S_{1} = A ∗ S_{0}. now it is clear that if S_{0} is equivalent to S′_{0}, then S′_{1} = A ∗ S′_{0} will be equivalent to S_{1}.

Consider now the case where L = φ_{1}(O_{B}) and set A_{t} = A_{φ_{t}^{−1}}, A_{−t} = A_{φ_{t}}, then if S_{0}, S′_{0} are g.f.q.i. for L, then

A_{1} ∗ S_{0} and A_{1} ∗ S′_{0} are g.f.q.i.

for the zero section, and are thus equivalent after stabilization. Therefore A_{−1}∗A_{1}∗S_{0} and A_{−1} ∗ A_{1} ∗ S′_{0} are equivalent after stabilization.

Now consider the family A_{−t} ∗A_{t} ∗S_{0}. It is a g.f.q.i. of φ_{t}^{−1} ∘φ_{t} ∘φ_{1}(O_{B}) = φ_{1}(O_{B}), that we denote by R_{t}. Similarly R′_{t} = A_{−t} ∗ A_{t} ∗ S′_{0} is also a g.f.q.i. for φ_{1}(O_{B}), and we just proved that R_{1} and R′_{1} are equivalent after stabilization.

But R_{1} is equivalent to R_{0} (and R′_{1} to R′_{0}) because R_{t} is a continuous family of g.f.q.i. for φ_{1}(O_{B}). It is then easy to find a vector field generating a family of fiber preserving diffeomorphisms ψ_{t} such that R_{t}(ψ_{t}(b, ξ) = R_{0}(b, ξ).

Thus R_{0} is equivalent to R′_{0} after stabilization, but R_{0} = A_{0} ∗ A_{0} ∗ S_{0} is easily seen to be a stabilization of S_{0}, and similarly R′_{0} is a stabilization of S′_{0}. This concludes our proof.

2. Invariants for Lagrange submanifolds.

In the next two sections, we suppose that generating functions are normalized in such a way that the indeterminacy of S by a constant is removed. In most applications, we shall consider Lagrange submanifolds of T^{∗}B coinciding with the zero section in a certain subset of B. It is then natural to normalize our generating functions so that the critical value corresponding to this region is zero.

In this section L will be a compact connected manifold, H^{∗}, H_{∗} will be homology and cohomology with coefficients in some field , so that the Künneth formula can be written as

\begin{aligned}H^{∗}(X × Y) & ≃ H^{∗}(X) ⊗ H^{∗}(Y) \\ H_{∗}(X × Y) & ≃ H_{∗}(X) ⊗ H_{∗}(Y) \end{aligned}

Given B, we consider the following sets

L = {Compact Lagrange submanifolds of T^{∗}B
isotopic to the zero section}

and for L in L

G(L) = { g.f.q.i. of L such that E_{∞}^{−} the negative bundle
of q_{∞} is trivial}

Note that if S coïncides with q_{∞} at infinity, then for λ large enough, the homotopy type of the pairs (E^{λ}, E^{μ}), (E^{μ}, E^{−λ}) does not depend on λ. We may thus write E^{∞}, E^{−∞} to denote E^{λ}, E^{−λ} for λ large enough. We remind the reader that, if we denote by D(E_{∞}^{−}), S(E_{∞}^{−}) the disk and sphere bundles associated to E_{∞}^{−}, we have

H^{∗}(E^{∞}, E^{−∞}) ≃ H^{∗}(D(E_{∞}^{−}), S(E_{∞}^{−})) →^{T^{−1}}_{≃} H^{∗}(B)

where T is the Thom isomorphism, shifting the grading by d^{−} =^{def} dim E_{∞}^{−}. We are now ready to set

Definition 2.1.

Let (u, L) ∈ H^{∗}(B) × L (u ≠ 0), we define a real number c(u, L) as follows. Let S ∈ G(L), and consider T u ∈ H^{∗}(E^{∞}, E^{−∞}). Then

c(u, L) = inf {λ | the image of T u by the natural map
H^{∗}(E^{∞}, E^{−∞}) → H^{∗}(E^{λ}, E^{−∞}) is non zero}

In view of the results of section 1, c(u, L) is indeed independent of S. It is a classical result of Lusternik Schnirelman’s theory that c(u, L) is a critical value of S. Moreover we have

Proposition 2.2.

If u = v ∪ w then

c(u, L) ≥ c(v, L)

with equality only if w induces a non zero class in a neighbourhood of

K_{c} = {critical points on the level c(u, L)}.

Note also that there is a one to one correspondence between critical points of S and points of L ∩ B (where B is identified with the zero section of T^{∗}B). We shall elaborate on this later, meanwhile we get

Corollary 2.3.

Let μ ∈ H^{n}(B) be the orientation class of B. Then c(μ, L) = c(1, L) if and only if L = B.

Proof: According to the proposition, c(μ, L) = c(1, L) implies that π^{∗}μ induces a non zero cohomology class on a neighbourhood of K_{c}, or that μ induces a non zero cohomology class in a neighbourhood of π(K_{c}) ⊂ L ∩ B. Because μ induces zero on B − {pt}, we must have L ∩ B = B that is B ⊂ L hence B = L.

We now explain more precisely the relationship between the c(μ, L) and the points of L∩B. Re call from [V 1] that we may associate to a pair of Lagrange submanifolds L_{1}, L_{2} of T^{∗}B and a pair x, y of transverse intersection points of L_{1} ∩ L_{2} an integer m(x, y; L_{1}, L_{2}) and a real number l(x, y; L_{1}, L_{2}) (denoted by m(x, y) and l(x, y) if there is no ambiguity). The number l(x, y) has the following simple definition : let γ_{1} (resp. γ_{2}) be a path in L_{1} (resp. L_{2}) connecting x to y. Let γ be the loop γ_{1}γ_{2}^{−1}, then l(x, y) = ∫_{γ} p dq. It is easy to see that our definition does not depend on the choices of γ_{1}, γ_{2}.

The definition of m(x, y) can be found in [V 1], we however point out that according to [V 1] m(x, y) is an integer modulo μ(L_{1}) · H_{1}(L_{1}) + μ(L_{2}) · H_{1}(L_{2}) where μ(L_{1}), μ(L_{2}) are the Maslov classes of L_{1}, L_{2}. Since we are only interested in these Lagrange submanifolds given by a generating function, their Maslov class vanishes, so m(x, y) is indeed an integer. Now proposition 5.2 of [V 1] (p. 370) and the above proposition 2.2 yield Proposition 2.4.

To each u ∈ H^{∗}(B) we can associate a point x_{u} of L ∩ B such that

(1)  l(x_{u}, x_{v}) = c(u, L) − c(v, L)

(2) if L is transverse to B, then m(x_{u}, x_{v}) = deg u − deg v

This result is very useful, especially when combined with

Proposition 2.5.

If L_{1}, L_{2} have generating functions S_{1}, S_{2} defined on the same vector bundle, and such that ∥S_{1} − S_{2}∥_{C^{0}} ≤ ε, then for all u in H^{∗}(B) we have |c(u, L_{1}) − c(u, L_{2})| ≤ ε.

Proof: The result is obvious, set E_{1}^{λ} (resp. E_{2}^{λ}) to be {x ∈ E | S_{1}(x) ≤ λ} (resp. {x ∈ E | S_{2}(x) ≤ λ} and remark that E_{2}^{λ−ε} ⊂ E_{1}^{λ} ⊂ E_{2}^{λ+ε} and for |λ| large enough all the inclusions are homotopy equivalences. Using the definition of c(u, L) yields the result.

We are now ready to prove

Proposition 2.6.

Let ψ_{t} be flow of a Hamiltonian with support in T^{∗}B − B. Then for all u in H^{∗}(L) we have

c(u, ψ_{t}L) = c(u, L) .

Proof: Note that for all t, ψ_{t}L ∩ B = L ∩ B, and for any two points x, y in L ∩ B, l_{t}(x, y) (that we define to be l(x, y; ψ_{t}L, 0_{B})) is really independent of t. If we assume that the g.f.q.i. of L has finitely many critical values, then the set of their differences is finite. Thus l(x, y) has values in a finite set.

On the other hand c(u, ψ_{t}L) changes continuously with t, and according to 2.4 has its values in the set of l(x, y). This implies that if the g.f.q.i. has only finitely many critical values, c(u, ψ_{t}L) is a constant. The general case is obtained by perturbing a g.f.q.i. of L so that the above argument applies, and use prop. 2.5 to conclude.

Before we define the “composition” of Lagrange manifolds, and study the capacities of this “composition”, we would like to state and prove some more elementary results. Let α be a homology class on B, then we can define c(α, L) by

c(α, L) = inf {λ | T̃α ∈ H_{∗}(E^{∞}, E^{−∞}) is in the image of
the map H_{∗}(E^{λ}, E^{∞}) → H_{∗}(E^{∞}, E^{−∞})}

Here α → T̃α is the Thom isomorphism in homology.

Finally if L is an immersed Lagrange submanifold of T^{∗}B, \overline{L} denotes the image of L by (q, p) → (q, −p). Obviously, \overline{L} is again a Lagrange submanifold. We now have Proposition 2.7.

If u ∈ H^{q}(B) and α ∈ H_{n−q}(B) are Poincaré dual to each other, then c(u, \overline{L}) = −c(α, L).

Proof: If S is a g.f.q.i. for L, then −S is a g.f.q.i. for \overline{L}. Then \overline{E}^{λ} = \{x ∈ E | − S(x) ≤ λ\} = E − E^{−λ}. Thus H^{∗}(\overline{E}^{λ}, \overline{E}^{−∞}) = H^{∗}(E − E^{−λ} , E − E^{∞}). This last group is isomorphic, by Alexander duality, to H_{∗}(E^{∞}, E^{−λ}).

\begin{aligned}H_{∗}(E^{−λ}, E^{−∞})_{↓} & →^{∼} H^{∗}(\overline{E}^{∞}, \overline{E}^{λ})_{↓} \\ T̃α ∈ H_{∗}(E^{∞}, E^{−∞})_{↓} & →^{∼} H^{∗}(\overline{E}^{∞}, \overline{E}^{−∞}) ∋ T u_{↓} \\ H_{∗}(E^{∞}, E^{−λ}) & →^{∼} H^{∗}(\overline{E}^{λ}, \overline{E}^{−∞}) \end{aligned}

\underline{Fig. 1}

Consider the diagram of figure 1, where the horizontal isomorphisms are given by Alexander duality, and the vertical ones from the homology (resp. cohomology) exact sequence of the triple (E^{∞}, E^{−λ}, E^{−∞}) (resp. (\overline{E}^{∞}, \overline{E}^{λ}, \overline{E}^{−∞})).

Note that the Alexander duality sends T̃α to T u if u is the Poincaré dual of α. We can then conclude our argument as follows :

If λ < c(u, \overline{L}) then the class T u goes to zero in H^{∗}(\overline{E}^{λ}, \overline{E}^{−∞}). This is equivalent to the fact that T̃α goes to zero in H_{∗}(E^{∞}, E^{−λ}), or else by exactness of the homology sequence of a triple, that T̃α is in the image of H_{∗}(E^{−λ}, E^{−∞}), which implies −λ ≥ c(u, L) or λ ≤ −c(α, L). We just proved that c(u, \overline{L}) ≤ −c(α, L). The same argument yields the reversed inequality, hence our result.

Corollary 2.8.

If μ is the orientation class of B, then

c(μ, \overline{L}) = −c(1, L)

Proof: Let α ∈ H_{0}(L) be the Poincaré dual of μ. Let N = dim E^{−}, so that T̃α ∈ H_{N}(E^{∞}, E^{−∞}). Because dim H_{N}(E^{∞}, E^{−∞}) = 1 (due to dim H_{0}(L) = 1), c(α, L) can also be defined as

inf {λ | H_{N}(E^{λ}, E^{−∞}) → H_{N}(E^{∞}, E^{−∞}) is non zero}

Since we are dealing with coefficients in a field, ,

H_{N}(E^{λ}, E^{−∞}) ≃ Hom(H^{N}(E^{λ}, E^{−∞}), )

and the map H_{N}(E^{λ}, E^{−∞}) ⇀ H_{N}(E^{∞}, E^{−∞}) is the transpose of the map H^{N}(E^{∞}, E^{−∞}) ⇀ H^{N}(E^{λ}, E^{−∞}) which is non zero if and only λ ≥ c(1, L). Thus c(α, L) = c(1, L), and the conclusion follows from 2.7.

3. Composition of Lagrange submanifolds.

In the coming sections we shall deal with Lagrange submanifolds of T^{∗}B not necessarily isotopic to the zero section, but still possesing a g.f.q.i., S.

We shall first define for a pair of Lagrange submanifolds L_{1}, L_{2} of T^{∗}B the Lagrange manifold L_{1}♯L_{2}.

Definition 3.1.

Let L_{1}, L_{2} be two Lagrange submanifold of T^{∗}B, and assume L_{1} × L_{2} to be transverse to T_{Δ}^{∗}(B × B) (= the restriction to the diagonal Δ of T^{∗}(B × B)).

Then, since T_{Δ}^{∗}(B × B) is a coisotropic manifold the reduction of which can be identified with T^{∗}B, L_{1} × L_{2} has a reduction to T^{∗}B that we denote by L_{1}♯L_{2}.

Note that L_{1}♯L_{2} could be written as L_{1} + L_{2}, since

L_{1}♯L_{2} = {(q, p) ∈ T^{∗}L | p = p_{1} + p_{2} , (q, p_{1}) ∈ L_{1} , (q, p_{2}) ∈ L_{2}}

and L_{1}♯\overline{L}_{2} could be written L_{1} − L_{2}. Note that L_{1} − L_{1} is not the zero section O_{B} unless the projection L_{1} → B is one to one.

The proof of the following result is left to the reader.

Proposition 3.2.

Let S_{1}, S_{2} be g.f.q.i. for L_{1} and L_{2}. Set

S_{1}♯S_{2}(x, ξ, η) = S_{1}(x, ξ) + S_{2}(x, η) .

Then S_{1}♯S_{2} is a g.f.q.i. for L_{1}♯L_{2}.

The main result of this section can now be stated as

Proposition 3.3.

c(u · v , S_{1}♯S_{2}) ≥ c(u, S_{1}) + c(v, S_{2})

for any u, v ∈ H^{∗}(L).

Proof: Let S_{1} be defined on E_{1}, S_{2} on E_{2} and S_{3} = S_{1}♯S_{2} on E_{3} = E_{1} ⊕ E_{2} = E_{1} ×_{B} E_{2}. Then

E_{3}^{ν} = ⋃_{λ+μ=ν} E_{1}^{λ} ×_{B} E_{2}^{μ} .

H^{∗}(E_{3}^{∞}, E_{3}^{−∞})_{↓}  ≃  H^{∗}(E_{1}^{∞}, E_{1}^{−∞}) ⊗_{H^{∗}(B)} H^{∗}(E_{2}^{∞}, E_{2}^{−∞})_{↓}
H^{∗}(E_{3}^{ν}, E_{3}^{−∞})_{↘}  H^{∗}(E_{1}^{−λ}, E_{1}^{−∞}) ⊗_{H^{∗}(B)} H^{∗}(E_{2}^{μ}, E_{2}^{−∞})_{↙}
H^{∗}(E_{1}^{λ} ×_{B} E_{2}^{μ} , E_{1}^{−∞} ×_{B} E_{2}^{−∞})

\underline{Fig. 2}

\begin{matrix}X ×_{B} Y_{↓} → & X_{↓} \\ Y → & B \end{matrix}

\underline{Fig. 3}

X ×_{B} Y
↗  ↘
Z^{ν} = ⋃_{λ+μ=ν} X^{λ} ×_{B} Y^{μ}  X × Y
↘  ↗
⋃_{λ+μ=ν} X^{λ} × Y^{μ}

\underline{Fig. 4}

Consider the commutative diagram of figure 2, where all arrows are induced by natural maps, except for the lower right one

\begin{matrix}H^{∗}(E_{1}^{λ}, E_{1}^{−∞}) ⊗_{H^{∗}(B)} H^{∗}(E_{2}^{μ}, E_{2}^{−∞}) \\ → H^{∗}(E_{1}^{λ} ×_{B} E_{2}μ , E_{1}^{−∞} ×_{B} E_{2}^{−∞}) . \end{matrix}

This last map is defined as follows. Let p_{1}, p_{2} be the projections of E_{3} on E_{1}, E_{2}. Then we have a map p_{1}^{∗} ⊗ p_{2}^{∗}

H^{∗}(E_{1}^{λ}, E_{1}^{−∞}) ⊗ H^{∗}(E_{2}^{μ}, E_{2}^{−∞}) → H^{∗}(E_{1}^{λ} ×_{B} E_{2}^{μ} , E_{1}^{−∞} ×_{B} E_{2}^{−∞}) .

and it is clear that for β ∈ H^{∗}(B), x_{1} ∈ H^{∗}(E_{1}^{λ}, E_{1}^{−∞}), x_{2} ∈ H^{∗}(E_{2}^{μ}, E_{2}^{−∞}), βx_{1} ⊗ x_{2} − x_{1} ⊗ βx_{2} is in the kernel of p_{1}^{∗} ⊗ p_{2}^{∗}. Thus p_{1}^{∗} ⊗ p_{2}^{∗} can in fact be defined on H^{∗}(E_{1}^{λ}, E_{1}^{−∞}) ⊗_{H^{∗}(B)} H^{∗}(E_{2}^{μ}, E_{2}^{−∞}).

We now need the Lemma 3.4.

Let X, Y be total spaces of fiber bundles over B. Then

H^{∗}(X ×_{B} Y) ≃ H^{∗}(X) ⊗_{H^{∗}(B)} H^{∗}(Y) .

Consider now two increasing filtrations (X^{λ}), (Y^{μ}) of X and Y, and let Z^{ν} = ⋃_{λ+μ=ν} X^{λ} ×_{B} Y^{μ}. If a cohomology class x ⊗ y in H^{∗}(X ×_{B} Y) induces a non zero class on H^{∗}(Z^{ν}), then there exists λ, μ with λ + μ = ν such that x ⊗ y is non zero in H^{∗}(X^{λ}) ⊗ H^{∗}(Y^{μ}).

The Thom isomorphism just tells us that H^{∗}(E^{∞}, E^{−∞}) is a free H^{∗}(B) module. By applying the lemma to our situation, we see that there exist λ, μ with λ + μ = ν such that if the image of T (uv) ∈ H^{∗}(E_{3}^{∞}, E_{3}^{−∞}) in H^{∗}(E_{3}^{ν}, E_{3}^{−∞}) is non zero, then its image in H^{∗}(E_{1}^{λ} ×_{B} E_{2}^{μ} , E_{1}^{−∞} ×_{B} E_{2}^{−∞}) is also non zero. Using the diagram of figure 2, this implies that the image of T u⊗T v in H^{∗}(E_{1}^{λ}, E_{1}^{−∞})⊗_{H^{∗}(B)}H^{∗}(E_{2}^{μ}, E_{2}^{−∞}) is non zero, thus λ ≥ c(u, L_{1}), μ ≥ c(v, L_{2}). This shows that

c(uv , L_{1}♯L_{2}) ≥ c(u, L_{1}) + c(v, L_{2}) .

We now complete our proof by proving lemma 3.4. The first statement is proved by either using the Eilenberg-Moore spectral sequence associated to the diagram of figure 3, or in our case, the fact that if T_{E} denotes the Thom class associated to (E^{∞}, E^{−∞}), then we have T_{E_{1}⊕E_{2}} = T_{E_{1}} ⊗ T_{E_{2}}. For the second part, we use the commutative diagram of figure 4 andthe first part of the lemma. As a consequence of this, the map X ×_{B} Y → X × Y induces the map H^{∗}(X) ⊗ H^{∗}(Y) → H^{∗}(X) ⊗_{H^{∗}(B)} H^{∗}(Y) which is surjective. Thus if a given class in H^{∗}(X ×_{B} Y) induces a non zero class in H^{∗}(Z), the same will be true for some class x ⊗ y in H^{∗}(X × Y). Using the commutativity of the diagram of figure 4, this implies that x ⊗ y has non zero image in H^{∗}(⋃_{λ+μ=ν} X^{λ} × Y^{μ}), we now show that this implies that x ⊗ y has non zero image in H^{∗}(X^{λ} × Y^{μ}) for some λ, μ with λ + μ = ν.

This is in fact readily checked, let

λ(x) = inf {λ | x is non zero in H^{∗}(X^{λ})}
μ(x) = inf {μ | x is non zero in H^{∗}(X^{μ})}

We claim that λ(x) + μ(y) ≥ ν. Assume this is not so, we can then find cocycles a and b with support in X − X^{λ(x)}, Y − Y^{μ(y)}, and cocycles α, β such that

x = [a + dα] y = [b + dβ]

hence

x ⊗ y = a ⊗ b + d(−)

Now the cocycle a ⊗ b vanishes on X^{λ} × Y^{μ} if either λ < λ(x) or μ < μ(y), in particular if λ + μ = ν we are in one of the two cases, and a ⊗ b vanishes. Since Z = ⋃_{λ+μ=ν} X^{λ} × Y^{μ}, a ⊗ b vanishes on Z, a contradiction.

Proposition 3.5.

c(u, ψ(L)) = c(u , L − ψ^{−1}(O_{B}))

Proof: We now consider c(u, ψ_{t}(L) − ψ_{t}ψ_{1}^{−1}(0_{B})). This takes values in the set of ∫_{γ_{1,t}} pdq − ∫_{γ_{2,t}} pdq, where γ_{1,t} and γ_{2,t} are paths connecting two points x_{t}, y_{t} in ψ_{t}(L) ∩ ψ_{t}ψ_{1}^{−1}(0_{B}) = ψ_{t}(L ∩ ψ_{1}^{−1}(0_{B})), and γ_{1,t} (resp. γ_{2,t}) is in ψ_{t}(L) − ψ_{t}ψ_{1}^{−1}(0_{B}) (resp. 0_{B}).

Now we would like to take for γ_{1,t}, ψ_{t}ψ_{1}^{−1}(γ_{1,1}) − ψ_{t}ψ_{1}^{−1}(γ_{2,1}) but this expression is meaningless since the two paths we are trying to substract do not necessarily have the same projection. However it is easy to get around thid difficulty as follows.

Let S_{1,t}(b, ξ) (resp. S_{2,t}(b, ξ) ) be a g.f.q.i. for ψ_{t}ψ_{1}^{−1}(ψ_{1}(L)) (resp. ψ_{t}ψ_{1}^{−1}(0_{B})), and x_{t} = (q_{t}, p_{t}), y_{t} = (q′_{t}, p′_{t}). Then (q_{t}, 0) and (q′_{t}, 0) are in

(ψ_{t}ψ_{1}^{−1}(ψ_{1}(L)) − ψ_{t}ψ_{1}^{−1}(0_{B})) ∩ 0_{B}.

Now l((q_{t}, 0), (q′_{t}, 0)) is given by

[S_{1,t}(q_{t}, ξ_{t}) − S_{2,t}(q_{t}, η_{t})] − [S_{1,t}(q′_{t}, ξ′_{t}) − S_{2,t}(q′_{t}, η′_{t})]

where (q_{t}, ξ_{t}, η_{t}) and (q′_{t}, ξ′_{t}, η′_{t}) correspond to (q_{t}, 0) and (q′_{t}, 0). But the above expression can be reordered as

[S_{1,t}(q_{t}, ξ_{t}) − S_{1,t}(q′_{t}, ξ′_{t})] − [S_{2,t}(q_{t}, η_{t}) − S_{2,t}(q′_{t}, η′_{t})].

This can be read as the difference of ∫_{ψ_{t}ψ_{1}^{−1}(γ_{1,1})} pdq, and ∫_{ψ_{t}ψ_{1}^{−1}(γ_{2,1})} pdq, and it is now clear that l((q_{t}, 0), (q′_{t}, 0)) = ∫_{γ_{2,1}∘γ_{1,1}^{−1}} pdq is indeed independent of t.

corollary 3.6. c(uv, ψ(L)) ≥ c(u, L) + c(v, ψ(0_{B}))

Proof: According to the proposition we just proved, we have:

c(uv, ψ(L)) = c(uv, L − ψ(0_{B})).

We now apply prop. 3.3 to get

c(uv, ψ(L)) ≥ c(u, L) + c(v, \overline{ψ^{−1}(0_{B})}).

Applying 3.5 once more shows that c(v, ψ(0_{B})) = c(v, 0_{B} − ψ^{−1}(0_{B})) = c(v, \overline{ψ^{−1}(0_{B})}, thus concluding our proof.

4. Applications to Hamiltonian diffeomorphisms.

In this section we consider the following subgroups of Diff_{ω}( ^{2n}), the group of symplectic diffeomorphisms of ^{2n}:

H^{0}( ^{2n}) : the group of time one maps of the Hamiltonian flow associated to a time

dependent, compact supported Hamiltonian H(x, t)

H(U) : the subgroup of H^{0}( ^{2n}) obtained by imposing to the

support of H(x, t) to be in U × [0, 1].

Remark. If H(t, x) is a Hamiltonian, generating a flow ψ_{t} we will denote by s̃upp(ψ_{t}) the support of H. We point out that it is generally bigger than the support of the isotopy (i.e. {x|∃ t , ψ_{t}(x) ≠ x}). For instance take H to be one in a neighbourhood of the origin, and 0 outside a compact set. Then the neigbourhood of the origin is in s̃upp(ψ_{t}) − supp(ψ_{t})

The group H(B^{2n}(r)) has obvious C^{k} (k ≥ 0) topologies, and H^{0}( ^{2n}) will have the obvious topology that makes it a limit of H(B^{2n}(r)) as r goes to infinity. Let us remind the reader that as a result, a continuous map from a compact set into H^{0}( ^{2n}) can be factored through H(B^{2n}(r)) for r large enough.

We now consider, for ψ ∈ H^{0}( ^{2n}), its graph Γ_{ψ}, a Lagrange submanifold of \overline{^{2n}} × 2n. For r large enough, Γ_{ψ} coincides with the diagonal Δ outside B^{2n}(r) × B^{2n}(r). We can identify \overline{^{2n}} × ^{2n} with T^{∗}Δ through the map

(q, p, Q, P) → (\frac{q + Q}{2} , \frac{p + P}{2} , P − p , q − Q)

Now the image of Γ_{ψ} in T^{∗}Δ, Γ_{ψ} coïncides with the zero section outside a compact set. If we consider Δ ≃ ^{2n} as S^{2n} − {P }, where P is the “north pole”,we get an embedding T^{∗}Δ ⇀ T^{∗}S^{2n}, and the image of Γ_{ψ} coincides with the zero section in a neighbourhood of P, so that we can make this image closed by adding the point (P, 0). We thus get a Lagrange submanifold Γ̃_{ψ} of T^{∗}S^{2n}, such that there is a neighbourhood U of P in S^{2n} with Γ̃_{ψ} ∩ T^{∗}U = U. We shall normalize generating functions by assuming that S(P, 0) = 0.

We may now set

Definition 4.1.

Let H^{∗}(S^{2n}) = · 1 ⊕ · μ, we set

c_{−}(ψ) = −c(μ, Γ̃_{ψ})
c_{+}(ψ) = −c(1, Γ̃_{ψ})
γ(ψ) = c_{+}(ψ) − c_{−}(ψ)

The results of section 2 have obvious translations to c_{±}(ψ), γ(ψ), however we summarize the main facts together with some improvements, and new results.

Proposition 4.2.

(1)  c_{−}(ψ) ≤ 0 ≤ c_{+}(ψ) , with c_{+}(ψ) = c_{−}(ψ) = 0

if and only if ψ = Id

(2)  c_{+}(ψ^{−1}) = −c_{−}(ψ)

(3)

Let U be a compact set in ^{2n}, ψ_{t} be an isotopy in H(U), generated

by H, and z be any point in ^{2n} − U.

There are fixed points of ψ, x_{+}, x_{−} in U such that if γ_{+}, γ_{−} are paths connecting x_{+} to z and x_{−} to z, and g_{+} = γ_{+} ∘ ψ(γ_{+}^{−1}), g_{−} = γ_{−} ∘ ψ(γ_{−}^{−1}) then

c_{+}(ψ) = l(x_{+}, z; ψ) = − ∫_{g_{+}} p dq = − ∫_{ψ_{t}(x_{+})} pdq − Hdt

c_{−}(ψ) = l(x_{−}, z; ψ) − ∫_{g_{−}} p dq = − ∫_{ψ_{t}(x_{−})} pdq − Hdt

Moreover if the fixed points x_{+}, x_{−} are nondegenerate, we have with the notations of [V2],

i_{cz}(x_{+}) = 2n , i_{cz}(x_{−}) = 0

Remark. l(x, y; ψ) is obviously defined as l((x, x), (y, y); Γ_{ψ}, Δ) (cf. section 2).

Proof: (1) Let S be a g.f.q.i. for Γ̃_{ψ} obtained from one for Γ_{ψ}, then S(∞, ξ) = |ξ_{+}^{∗}|^{2} − |ξ_{−}^{∗}|^{2}. Let E_{{P }} be the fiber of E over the north pole identified to the point at infinity, then we have inclusions

(E_{{P }}^{0}, E_{{P }}^{−∞}) ↪ (E^{0}; E^{−∞})

hence a commutative diagramm

\begin{aligned}H^{∗}(E_{\{P \}}^{0}, E_{\{P \}}^{−∞})_{j^{∗}T ↑} & ← H^{∗}(E^{0}, E^{−∞})_{↑ j^{∗}T} \\ H^{∗}(\{P \}) & ← H^{∗}(S^{2n}) \end{aligned}

where T is the Thom isomoprhism

H^{∗}(S^{2n}) → H^{∗}(E^{∞}, E^{−∞})

(resp. H^{∗}({∞}) → H^{∗}(E_{{P }}^{∞}, E_{{∞}}^{−∞})) and j^{∗} is induced by the inclusion map j : (E^{0}, E^{−∞}) → (E^{∞}, E^{−∞}) (resp. ...).

Now it is easy to see that j^{∗}T is an isomorphism, and since the image of 1 ∈ H^{∗}(S^{2n}) is 1 in H^{∗}({∞}) we see that j^{∗}T (1_{S^{2n}}) ≠ 0 hence c_{−}(ψ) = c(1, Γ̃_{ψ}) ≤ 0.

The inequality c_{+}(ψ) ≥ 0 now follows from (2) which is a translation of corollary 2.8 since \overline{Γ̃}_{ψ} = Γ̃_{ψ^{−1}}. Finally the left side equality in (3) is a restatement of proposition 2.4. The equality

∫_{g} p dq = ∫_{ψ_{t}(x)} pdq − Hdt

is proved as follows. Let Ψ : K = [0, 1]^{2} → ^{2n} be the map given by Ψ(t, θ) = ψ_{t}(γ(θ)). Then ∫_{Ψ(K)} d(pdq − Hdt) = 0 since for each θ , t → Ψ(t, θ) is an tangent to the kernel of d(pdq − Hdt), and Ψ(K) is foliated by such curves. Now apply Stoke’s theorem, and decomposing ∂K as {0, 1} × [0, 1] ∪ [0, 1] × {0, 1}, we get

∫_{g} p dq = ∫_{Ψ({0,1}×[0,1])} pdq − Hdt = ∫_{Ψ([0,1]×{0,1})} pdq − Hdt =

∫_{ψ_{t}(x)} pdq − Hdt − ∫_{ψ_{t}(z)} pdq − Hdt = ∫_{ψ_{t}(x)} pdq − Hdt

The last equality holds because ψ_{t}(z) ≡ z and H(t, z) ≡ 0 .

Corollary 4.3.

Let φ be a conformal map isotopic to the identity in the set of conformal maps, i.e. φ = φ_{1}, φ_{0} = id with φ_{t}^{∗}ω = λ(t)ω. Then, for any symplectic (not necessarily compact supported) diffeomorphism of ^{2n}, we have

c_{+}(φψφ^{−1}) = λ^{2}(1)c_{+}(ψ) , c_{−}(φψφ^{−1}) = λ(1)^{2}c_{−}(ψ)

and

γ(φψφ^{−1}) = λ^{2}(1)γ(ψ)

Proof: If x, y are fixed points of ψ, then φ_{t}(x), φ_{t}(y) are fixed points of φ_{t}ψφ_{t}^{−1}, and

l(φ_{t}(x), φ_{t}(y)) = λ(t)^{2} l(x, y)

so \frac{1}{λ(t)^{2}} c_{+}(φ_{t}ψφ_{t}^{−1}) has values in a totally discontinuous set, hence is independent of t. The same holds for c_{−} and γ.

One of the main tools in the applications of c_{−}, c_{+}, γ is given by Proposition 4.4.

c_{+}(φψ) ≤ c_{+}(φ) + c_{+}(ψ)
c_{−}(φψ) ≥ c_{−}(φ) + c_{−}(ψ)
γ(φψ) ≤ γ(φ) + γ(ψ)

Proof: It follows from corollary 3.6 which states

c(uv, Ψ(L)) ≤ c(u, L) + c(v, \overline{Ψ^{−1}(0_{B})})

applied to the extension of Ψ = ψ × Id to T^{∗}S^{2n} and L = Γ̃_{φ}.

Corollary 4.5.

Let φ_{t} be an isotopy in H^{0}(U), and ψ an element in H^{0}( ^{2n}) such that ψ(U)∩U = ∅. Then we have

c_{+}(ψφ_{1}) = c_{+}(ψ) , c_{−}(ψφ_{1}) = c_{−}(ψ)

As a result we get c_{+}(φ) ≤ γ(ψ) and −c_{−}(φ) ≤ γ(ψ).

Proof: Let x be a fixed point of ψφ_{t}. If x is such a point , then x is not in U for otherwise φ_{t}(x) would be in U thus contradicting the assumption that ψ(U) ∩ U = ∅. Since φ_{t} ≡ id outside U, we have that x must be a fixed point of ψ. Now let x, y be two fixed points of ψ ∘ φ_{t} (and thus of ψ). Then the number l(x, y; ψϕ_{t}) is independent of t. We choose a path γ_{1} joining x to y, then γ_{1} ∘ ψφ_{t}(γ_{1}^{−1}) = γ_{1} ∘ ψ(γ_{1}^{−1}) ∘ ψ(γ_{1}) ∘ ψϕ_{t}(γ_{1}^{−1}) = γ_{1} ∘ ψ(γ_{1}^{−1}) ∘ ψ(γ_{1} ∘ ϕ_{t}(γ_{1}^{−1})) First observe that the area of the closed loop ψ(γ_{1} ∘ ϕ_{t}(γ_{1}^{−1})) is the same as that of γ_{1} ∘ ϕ_{t}(γ_{1}^{−1}) which is equal to l(x, y; ϕ_{1}), it is enough to show that this last quantity is zero. But by the proof of 4.2, l(x, y; ϕ_{t}) is also given by ∫_{ϕ_{t}(x)} pdq − Hdt − ∫_{ϕ_{t}(y)} pdq − Hdt. Since ϕ_{t}(x) = x; ϕ_{t}(y) = y and H(t, x) = H(t, y) = 0, the area of γ_{1} ∘ ϕ_{t}(γ_{1}^{−1}) vanishes. The area we are looking for is therefore the area of γ_{1} ∘ ψ(γ_{1}^{−1}), that is l(x, y; ψ), and this proves our claim.

Our now classical argument shows that c_{+}(ψ ∘φ_{t}) is independent of t, and the same holds for c_{−}, γ. Now we apply 4.4

c_{+}(ψφ) + c_{+}(ψ^{−1}) ≥ c_{+}(φ)

since c_{+}(ψφ) = c_{+}(ψ), c_{+}(ψ^{−1}) = −c_{−}(ψ) we get

c_{+}(φ) ≤ γ(ψ) .

The other inequality follows similarly.

Proposition 4.6.

Let H_{1}(t, x), H_{2}(t, x) be two compact supported Hamiltonians, ψ_{1} and ψ_{2} the time one maps of the associated flow. Assume H_{1}(t, x) ≤ H_{2}(t, x) for all x and t. Then c_{+}(ψ_{1}) ≤ c_{+}(ψ_{2}) and c_{−}(ψ_{1}) ≤ c_{−}(ψ_{2}).

Proof: Let K_{1}(x, y, t) be defined on T^{∗} ^{n} × T^{∗} ^{n} by K_{1}(x, y, t) = H_{1}(y, t), and the same for K_{2}. Then the time one flow of the Hamiltonian vector field associated to K_{1} is (Id, ψ_{1}) and so sends the diagonal to the graph of ψ_{1}. K_{1} has an obvious extension to T^{∗}S^{2n} and his time one flow sends the zero action to Γ̃_{ψ_{1}}. The same holds for K_{2}, and of course K_{1} ≤ K_{2}. We consider H_{u}(t, x) to be a linear homotopy from H_{1} to H_{2} and K_{u} the corresponding function on T^{∗} ^{n} × T^{∗} ^{n}. We first prove:

Lemma 4.7. Let S_{λ} be a continuous one parameter family of functions, and let c(λ) = c(u, S_{λ}) be a critical value obtained by minmax. We assume that: dS_{λ}(x) = 0 implies \frac{∂}{∂λ}S_{λ}(x) ≥ 0. Then c(λ) is non decreasing

Proof: We first make the stronger assumption that for x a crtical point of S_{λ}, we have \frac{∂}{∂λ}S_{λ}(x) > 0. We also assume that the family S_{λ} is generic, i.e. each S_{λ} has Morse critical points with distinct critical values, except for a finite number of values of λ, that we shall denote by λ_{1} < λ_{2} < ... < λ_{k}. Now if λ ∈]λ_{j}, λ_{j+1}[ then we may follow the unique critical point whith critical value c(λ), let x_{λ} be this critical point. Then \frac{d}{dλ}c(λ) = \frac{d}{dλ}(S_{λ}(x_{λ})) = \frac{∂}{∂λ}S_{λ}(x_{λ})+dS_{λ}(x_{λ})\frac{d}{dλ}x_{λ}. Because x_{λ} is a critical point of S_{λ} the second term is zero while the first one is positive by assumption. Thus c is a continuous function with strictly positive derivative except at finitely many points. It is thus increasing. It is now easy to conclude in the general case. First the genericity assumption on the family is easily removed by approximation in the C^{1} topology. This will not destroy the stronger hypothesis we made at the beginning of the proof, since the critical points of the perturbed family are in a neighbourhood of the critical point of the old one. If we now consider a sequence of generic families converging to the original one, since c(u, S) depends continuously on S in the C^{1} topology, we get that c(t) is non decreasing. Finally the stronger assumption we made is easily removed by replacing S_{λ} by S_{λ} + ϵλ for ϵ arbitrarily small.

We may now prove the proposition. Let ψ_{t}^{λ}(x_{λ}) be the flow of H_{λ}(t, x), and ϕ_{λ}(t) = ψ_{t}^{λ}(x_{λ}) . According to 4.2, we have for a critical point x_{λ} of a g.f.q.i. S_{λ} of L_{λ}, the graph of ψ_{1}^{λ}, that S_{λ}(x_{λ}) = − ∫_{S^{1}} ϕ_{λ}^{∗}[pdq − H_{λ}dt]. It is easy to check that \frac{∂}{∂λ}S_{λ}(x_{λ}) = ∫_{S^{1}} \frac{∂}{∂λ}H_{λ}(t, ϕ_{λ}(t))dt ≥ 0.

The above result has two applications. The first is to generalize the notion of time one flow of a nonnegative Hamiltonianin terms of the invariants that we defined. To begin with, we have:

Corollary 4.8.

If ψ is the time one map of the flow generated by a nonnegative Hamiltonian H(t, x), then c_{−}(ψ) = 0. If H(t, x) ≥ ε > 0 for (t, x) in ]t_{0}, t_{1}[×B(η), then c_{+}(ψ) > 0.

Proof: Because H ≥ 0, c_{−}(ψ) ≥ c_{−}(Id) = 0 and since c_{−}(ψ) ≤ 0 we get c_{−}(ψ) = 0.

We now prove that if H(t, x) ≥ ε > 0 for (t, x) in ]t_{0}, t_{1}[×B(η), then c_{+}(ψ) > 0. This will conclude our proof.

For this it is enough to construct some nonnegative Hamiltonian H_{0} such that H_{0} has support in ]t_{0}, T_{1}[×B(η), is bounded by ε, and such that φ, the time one flow associated to H_{0}, is not the identity. Indeed, we will have, according to the first part of our corollary, c_{−}(φ) = 0. Since φ ≠ id, c_{+}(φ) > 0, and because H_{0} ≤ H, 0 < c_{+}(φ) < c_{+}(ψ).

From 4.7 we see that the property of being the time one flow for a nonnegative Hamiltonian implies c_{−}(ψ) = 0. We can thus define a partial order relation, that we denote by ≺ :

Definition 4.9.

For φ, ψ in H^{0}( ^{2n}) we write φ ≺ ψ if an only if c_{−}(ψφ^{−1}) = 0.

Proposition 4.10.

(1)  Id ≺ ψ ⟺ ψ^{−1} ≺ Id

(2)  ψ_{1} ≺ ψ_{2} ⟺ φψ_{1} ≺ φψ_{2}

(3)  φ_{1} ≺ φ_{2} and ψ_{1} ≺ ψ_{2} ⟹ φ_{1}ψ_{1} ≺ φ_{2}ψ_{2}

Proof: Clearly (3) ⟶ (2) ⟶ (1). Now φ_{1}ψ_{1} ≺ φ_{2}ψ_{2} means c_{−}(φ_{2}ψ_{2}ψ_{1}^{−1}φ_{1}^{−1}) = 0. But c_{−}(φ_{2}ψ_{2}ψ_{1}^{−1}φ_{1}^{−1}) = c_{−}(φ_{1}^{−1}φ_{2}ψ_{2}ψ_{1}^{−1}) ≥ c_{−}(φ_{1}^{−1}φ_{2}) + c_{−}(ψ_{2}ψ_{1}^{−1}) since φ_{2} ≻ φ_{1}, ψ_{2} ≻ ψ_{1} the right hand side is zero, and so is the left hand side.

Now let ψ ≻ Id, then ψ^{k} ≻ Id for all positive k, moreover c_{+}(ψ^{k}) ≤ c_{+}(ψ^{k+1}) + c_{+}(ψ^{−1}) since c_{+}(ψ^{−1}) = −c_{−}(ψ) = 0 we get that the sequence c_{+}(ψ^{k}) is increasing. Thus either c_{+}(ψ^{k}) goes to infinity, or it converges to some finite value.

In fact we see that c_{+}(ψ^{k}) is bounded by the following argument. Denote by U the support of ψ_{t}, and let φ be an element of H^{0}( ^{2n}) such that φ(U) ∩ U = ∅. By 4.5, we have γ(φ) ≥ c_{+}(ψ^{k}) hence hence our result. In fact we just showed that

sup {c_{+}(ψ) | support(ψ) ⊂ U} ≤ inf {γ(φ) | φ(U) ∩ U = ∅}

We now set Definition 4.11.

We define c(U) as sup {c_{+}(ψ) | support(ψ) ⊂ U} whenever U is an open bounded set of ^{2n}, and γ(U) as inf {γ(ψ) | ψ(U) ∩ U = ∅}.

We may thus summarize our findings in

Proposition 4.12.

(1) If support(ϕ) ⊂ U , and ψ(U) ∩ U = ∅ c_{+}(ϕ) ≤ c(U) ≤ γ(U) ≤ γ(ψ)

(2) If Id ≺ ψ the increasing sequence c_{+}(ψ^{k})

has a limit, denoted by \overline{c}_{+}(ψ).

We now get a result on periodic points of compactly supported Hamiltonian systems.

Note that if x_{+}(k), x_{−}(k) are fixed points of ψ^{k} given by proposition 4.2, then x_{+}(k) = x_{+}(kl) implies that c_{+}(ψ^{kl}) = lc_{+}(ψ^{k}), since if γ is a path from x_{+}(k) to z and g the loop γ · ψ^{k}(γ^{−1}), then

γ · ψ^{kl}(γ^{−1}) ≃ g · ψ^{k}(g) … ψ^{k(l−1)}(g)

i.e. both sides have the same area. But the area of the right hand side is l times the area of g (because ψ is symplectic) hence our result.

We may now state

Proposition 4.13.

(1) Assume Id ≺ ψ. We set Per(N, ψ) = {periodic orbits of ψ in supp(ψ) of period k with 1 ≤ k ≤ N}. Then card Per(N, ψ) ≥ C × N for some C = C(ψ)

(2) In general ψ has infinitely many periodic points.

Proof: We assume ψ ≠ Id otherwise the result is trivial. Let c_{k} = c_{+}(ψ^{k}) then if x_{k} is a critical point associated to c_{k}, we have x_{k} ≠ x_{l} unless \frac{1}{k} c_{k} = \frac{1}{l}c_{l}. Now let k(n) be a sequence defined inductively as follows:

k(n + 1) = inf\{k > k(n) | k > k(n) × \frac{c_{k}}{c_{k(n)}} \}

Because c_{k} ≤ c_{∞} ≤ c(supp(ψ)), the above sequence is well defined. Moreover because the sequence \frac{1}{k(n)}c_{k}(n) is strictly decreasing, the points x_{k(n)} are all distinct.Now we

wish to estimate k(n). Note that \frac{k(n+1)}{k(n)} ≤ \frac{c_{k(n+1)}}{c_{k(n)}} + \frac{1}{k(n)}. Let \frac{c_{k(n+1)}}{c_{k(n)}} = 1 + ϵ_{n} we may write

k(N) ≤ ∏_{i=1}^{N−1}(1 + ϵ_{j} + \frac{1}{k(j)})

≤ ∏_{i=1}^{N−1}(1 + ϵ_{j} + \frac{1}{j}) ≤ ∏_{i=1}^{N−1}(1 + \frac{1}{j}) · ∏_{i=1}^{N−1}(1 + \frac{j}{j + 1}ϵ_{j})

Now ∏_{i=1}^{∞}(1 + ϵ_{j}) = \frac{c_{∞}}{c_{1}} is a convergent product , thus the same is true for

∏_{i=1}^{∞}(1 + \frac{j}{j + 1}ϵ_{j})

since ϵ_{j} > 0. Let γ be its limit, then we have γ ≤ \frac{c_{∞}}{c_{1}} and

k(N) ≤ γ ∏_{i=1}^{N−1}(1 + \frac{1}{j}) = γ · N

and (1) is proved with C = \frac{1}{γ} ≥ \frac{c_{1}}{c_{∞}}.

(2) follows from considering an infinite sequence ψ^{k_{n}} such that c_{+}(ψ^{k_{n}}) > 0 (if k_{n} is free to have any sign, such a sequence always exists). Then because c_{+}(ψ^{k_{n}}) is bounded, the sequence cannot be made of multiples of a finite number of terms, and the above argument applies.

We now apply 4.6 to study the continuity of the invariants that we defined.

Proposition 4.14. Let H_{1}, H_{2} be two compact supported Hamilonians. Let ψ_{1}, ψ_{2}, be the associated time one flows. If |H_{1} − H_{2}| ≤ ϵ, then we have: |γ(ψ_{1}) − γ(ψ_{2})| ≤ ϵ. In other words if C_{0}( ^{2n} × [0, 1], ) is the set of compact supported C^{2} Hamiltonians, the map from C_{0}( ^{2n} × [0, 1], ) to , that associates to H the number γ(ψ), i.e. the capacity of its time one flow is continuous for the C^{0} topology.

Proof: Let H_{R} be a Hamiltonian such that

H_{R}(x) = h_{R}(|x|),

with:

h_{R}(s) = 1 for s ≤ R
h_{R}(s) = 0 for s ≥ 3R
|h′(s)| < 1 for all s.

We set H_{R,ϵ} = ϵH_{R} for ϵ ≤ 1, and ϕ_{R,ϵ} is the time one flow of H_{R,ϵ} Note that there is no nonconstant one periodic orbit for ϕ_{R,ϵ}, hence, using proposition 4.2, we see that γ(ϕ_{R,ϵ}) ≤ 1 Now, our assumtions imply that H_{2} ≤ H_{1} + H_{R,ϵ}, and since H_{R,ϵ} and H_{1} have disjoint support, the flow associated toH_{1} + H_{R,ϵ} is ψ_{1} ∘ ϕ_{R,ϵ}. Using proposition 4.6, we get the inequality

γ(ψ_{2}) ≤ γ(ψ_{1}) + γ(ϕ_{R,ϵ})

i.e.

γ(ψ_{2}) ≤ γ(ψ_{1}) + ϵ.

By exchanging H_{1} and H_{2}, we get the proposition.

We also have another type of C^{0} continuity, which answers a question by J. Moser. We would like to thank him for his suggestion.

Proposition 4.15. There is a constant C, independent of R and ε, such that if ϕ has support in B(0, R), and d(x, φ(x)) ≤ ε for all x, then

γ(φ) ≤ C ε R .

More generally, the functions γ, c_{+}, c_{−} are continuous on H^{0}( ^{2n}) for the C^{0} topology.

Remark.

1) One may explicitely compute C, an unchecked computation yields C = 4π.

2) The above proposition is not symplectically invariant, i.e. if φ satisfies the assumptions of the proposition, ψ φ ψ^{−1} will not in general. As a result, the above proposition does not imply that γ(φ) ≤ C ε \sqrt{c(supp (φ))}, or that γ(φ) ≤ C ε R whenever d(x, φ(x)) ≤ ε and supp (φ) can be embedded in B(0, R).

3) We shall see that we can have γ(φ) = \frac{3}{4} ε R.

Proof of proposition:

Let ψ be a compact supported symplectic map. For z a fixed point of ψ, we consider l(z, ψ) to be the number l(z, ∞) as defined above in this section, i.e. it is the symplectic area bounded by the loop c ∘ ψ(c^{−1}), where c is a path from z to ∞.

We first need

Lemma. For any ε > 0, R > C ε, there exist a symplectic diffeomorphism with compact support such that

(1)  ψ has its support in B(0, 2R)

(2)  d(x, ψ(x)) > ε  in B(0, R)

(3)  0 < l(z, ψ) ≤ \frac{C}{2} ε R  for any z such that ψ(z) = z .

We postpone the proof of the lemma to conclude the proof of the proposition.

According to proposition 4.4 , we may estimate

γ(φ) ≤ γ(ψ^{−1}) + γ(ψφ) .

According to proposition 4.2, c_{±}(ψφ) ≤ 2l(z_{±}, ψφ) for some z_{±} satisfying ψφ(z_{±}) = z_{±}. We take for ψ the map given by the lemma. Then ψφ(z) = z can only happen if φ(z) = z and ψ(z) = z. Indeed,

d(φ(z), ψ(φ(z))) = d(z, φ(z)) ≤ ε

so using (2) φ(z) ∉ B(0, R) hence z ∉ B(0, R) and thus φ(z) = z.

We may conclude that l(z, ψφ) = l(z, ψ) ≤ \frac{C_{0}}{2} ε R so that

c_{+}(ψφ) ≤ \frac{C_{0}}{2} ε R , c_{−}(ψφ) = 0 .

Using 4.2 once more, we see that γ(ψ^{−1}) ≤ \frac{C_{0}}{2} ε R, thus

γ(φ) ≤ γ(ψ^{−1}) + γ(ψφ) ≤ \frac{C_{0}}{2} ε R + \frac{C_{0}}{2} ε R ≤ C_{0} ε R

provided C_{0} ε ≤ R. On the other hand because the support of φ is in B(0, R), we have γ(φ) ≤ C(B^{2n}(R)) = πR^{2} (follows from proposition 4.12 and definition 4.11 ), so that if C_{0} ε ≥ R, γ(φ) ≤ πR^{2} ≤ πC_{0} ε R, thus we may set C = πC_{0}.

Proof of the lemma:

The idea is to extend the translation by a vector of norm ε to a symplectic diffeomorphism with support in B(0, 2R). We first notice that if we constructed ψ_{ε,R} for ε ≤ C_{0}, R = 1, we may set ψ_{ε,R}(z) = R ψ_{ε/R,1}(z).

We now set ψ_{s,1} = ψ_{s}.

ψ_{s} will be the flow of the following Hamiltonian

- H(x, y) = δ + ((x, e^{→}) + 1) - where e^{→} is a unit vector -, if |x|^{2} ≤ 1, |y|^{2} ≤ \frac{9}{4}.

- 0 ≤ H(x, y) ≤ 1 + 2δ for all (x, y) in B(0, 2R).

- H(x, y) = 0 outside B(0, 2R).

Note that for (x, y) ∈ B(0, 1), and s ≤ \frac{1}{2} , we have ψ_{s}(x, y) = (x, y + s e^{→}) thus d(z, ψ_{s}(z)) ≤ s. The last step is to prove that 0 < l(z, ψ_{s}) ≤ Cs if s ≤ \frac{1}{2}. For this we notice that if z is a fixed point of the flow, i.e. H′(z) = 0, then we have that l(z, ψ_{s}) = sH(z).

Now for s ≤ s_{0}, (where s_{0} is less than the smallest period of a non trivial closed orbit of the flow, which is strictly positive), ψ_{s}(z) = z ⇒ H′(z) = 0 and thus

0 ≤ l(z, ψ_{s}(z)) ≤ s(1 + 2δ) ≤ \frac{3}{2} s

for δ ≤ \frac{1}{4} small enough.

Let C_{0} = sup(\frac{1}{s_{0}} , \frac{3}{2}) then we have proved

∀ ε, R R ≥ C_{0}ε we have d(z, ψ_{ε/R}(z)) ≥ ε and

0 ≤ l(z, ψ_{ε/R}(z)) ≤ \frac{C_{0}}{2} ε R

.

One may in fact check that γ(ψ_{s}) ≥ \frac{3}{2}s, and thus γ(ψ_{ε,R}) ≥ \frac{3}{2} εR while d(z, ψ_{ε,R}(z)) ≤ 2ε and supp ψ_{ε,R} = B(0, 2R). Thus our proposition can not be that much improved.

We leave the following easy generalization to the reader

Proposition 4.17.

Given any compact set W in ^{2n}, there is a constant C(W) such that

∀ x d(x, φ(x)) ≤ ε ⇒ γ(φ) ≤ C(W) ε R

if supp (φ) ⊂ R · W.

Remark. Note that the quantity C(W) is not a symplectic invariant of W, it depends also on the metric on ^{2n}. It is however invariant by an isometry of the Kähler structure.

From the above results we get the following

Corollary 4.18. Let H_{n} be a sequence of Hamiltonians such that their time one flow ψ_{n} converges in the C^{0} topology to a symplectic map ψ. Then if H_{n} converges in the C^{0} topology to a Hamiltonian H with time one flow ϕ, then we have ψ = ϕ.

proof: The flow ϕ_{t}^{−1}ψ_{n,t} is generated by the Hamiltonian −H(t, ϕ_{t}(x)0+H_{n}(t, ϕ_{t}(x)). Since this goes to 0 in the C^{0} topology, as n goes to infinity, we infer that γ(ϕ^{−1}ψ_{n}) goes to 0. By the continuity of γ, this implies that γ(ϕ^{−1}ψ) = 0, thus ϕ = ψ.

As a special case, we get the following result by H.Hofer:

Corollary 4.19. (cf. [H]) Let H_{n} be a sequence of compact supported Hamiltonians such that H_{n} →_{C^{0}} 0, and their time one maps ψ_{n} converge C^{0} to some map ψ.

Then ψ = Id.

5. On the effect of symplectic reduction and the camel problem

Consider two compact manifolds V, W, and let L be a Lagrange submanifold of T^{∗}(V × W). For almost every w in W, L is transverse to T^{∗}V × T_{w}^{∗}W, so that we may define the reduction L_{w} of L ∩ T^{∗}V × T_{w}^{∗}W.

Now if L has the g.f.q.i. S : E → defined on the vector bundle E →^{π} V × W, then L_{w} has the g.f.q.i. S_{w} = S_{|π^{−1}(V ×{w})} : E_{w} → .

Note that L_{w} is not necessarily isotopic to the zero section, even if L is. In this case we cannot speak about c(α, L_{w}), since the number in definition 2.1 might depend on the choice of a generating function for L_{w}, however we may define c(α, S_{w}).

We may now state

Proposition 5.1.

For α in H^{∗}(V), we have, if S is a g.f.q.i. for L,

\begin{aligned}c(α ⊗ 1, L) & ≤ inf_{w} c(α, S_{w}) \\ & ≤ sup_{w} c(α, S_{w}) ≤ c(α ⊗ μ_{W} , L) . \end{aligned}

Proof:

Consider the sequences

H^{∗}(V × W)_{↓} →^{T} H^{∗}(E^{∞}, E^{−∞})_{↓} → H^{∗}(E^{λ}, E^{−∞})_{↓}
H^{∗}(V)  →^{T} H^{∗}(E_{w}^{∞}, E_{w}^{−∞}) → H^{∗}(E_{w}^{λ}, E_{w}^{−∞})

where the map H^{∗}(V × W) → H^{∗}(V) is induced by the injection V → V × {w} ⊂ V × W, and coincides with the composition of the projection on H^{∗}(V) ⊗ H^{0}(W) and the obvious identification H^{∗}(V) ⊗ H^{0}(W) → H^{∗}(V).

Now if the image of T (α) is nonzero in H^{∗}(E_{w}^{λ}, E_{w}^{−∞}) that is for λ ≥ c(α, L_{w}), we must have that the image of T (α ⊗ 1) in H^{∗}(E^{λ}, E^{−∞}) is nonzero, that is λ ≥ c(α ⊗ 1, L). Thus we proved that if λ ≥ c(α, S_{w}) then λ ≥ c(α ⊗ 1, L) that is c(α, S_{w}) ≥ c(α ⊗ 1, L). Since this holds for every w ∈ W we get the first inequality. The second inequality is trivial, and the third one is obtained from the first one by changing L to L̄

We now consider an open set U in ^{2(n+m)}, and the family of its reductions, U_{w} = (U ∩ ^{2n} × ^{m} × {0})/{0} × ^{m}. We wish to estimate c(U) using the U_{w}.

First of all we slightly change our familiar setting. If ψ is in H^{0}( ^{2(n+m)}), we consider again Γ_{ψ} in T^{∗}Δ_{^{2n}× ^{2m}} . Now instead of compactifying ^{2n} × ^{2m} to S^{2(n+m)}, we may compactify it to S^{2n}×S^{2m}, and Γ_{ψ} becomes a Lagrange submanifold of T^{∗}(S^{2n} × S^{2m}). We may thus consider four numbers, namely c(1 ⊗ 1, Γ_{ψ}), c(1 ⊗ μ_{S^{2m}}, Γ_{ψ}), c(μ_{S^{2n}⊗1}, Γ_{ψ}) and c(μ_{S^{2n}} ⊗ μ_{S^{2m}}, Γ_{ψ}). Accordingly, as in definition 4.11, we may define c(α, U), γ(α, U) for α one of the above four cohomology classes.

We shall first need to slightly extend several notions that have been introduced in the previous sections.

First of all, we may generalize the capacities defined in section 4 to the case of compact supported Hamiltonian isotopies of ^{2n} × T^{∗}T^{k} (T^{k} is the k dimensional torus).

This is easy if we notice that such isotopy, ψ_{t}, may be lifted to an isotopy ψ̃_{t} of 2n × ^{2m} such that ψ̃_{t}(x + ν) = ψ̃_{t}(x) + ν for ν ∈ {0} × ^{k}.

Thus Γψ̃_{1} is such that if (x, y) ∈ Γψ̃, then (x + ν, y + ν) ∈ Γψ̃, this means that

Γψ̃ ⊂ \overline{^{2n}} × \overline{^{2k}} × ^{2n} × ^{2k} = T^{∗}(Δ_{^{2n}× ^{2k}} )

descends to the quotient T^{∗}(Δ_{^{2n+k}} × T^{k}).

Let Γψ̃ be the Lagrange submanifold thus obtained, it is easy to see that Γ̃_{ψ} coincides with the zero section outside a compact set (because ψ_{t} is compact supported)

Let Γ̃_{ψ} be Lagrange submanifold of T^{∗}( ^{2n+k} × T^{k}) thus obtained. We may again compactify T^{∗}( ^{2n+k} × T^{k}) into T^{∗}(S^{2n+k} × T^{k}), and Γ̃_{ψ} may be compactified into Γ_{ψ} in T^{∗}(S^{2n+k} × T^{k}). Now, if α ⊗ β is a cohomology class in H^{∗}(S^{2n+k}) ⊗ H^{∗}(T^{k}), we may define c(α ⊗ β, Γ_{ψ}).

We may now prove

Proposition 5.2.

c(μ_{S^{2n}} ⊗ 1, U) ≤ inf_{w} γ(U_{w}) .

Proof:

It is clear from 5.1 that the following is true:

c(μ_{S^{2n}} ⊗ 1, ψ) ≤ inf_{w} c(μ, S_{w})

where S is a g.f.q.i. for Γ_{ψ} and ψ is in H^{0}(U).

Now let ψ_{t} be a symplectic isotopy supported in U with ψ_{0} = id, S^{t} a g.f.q.i. for Γ_{ψ_{t}}, and S_{w}^{t} its restriction to π^{−1}(S^{2n}×{w}). We are going to show that c(μ, S_{w}^{1}) ≤ γ(U_{w}).

Let φ be an element in H^{0}( ^{2n}) such that φ(U_{w}) ∩ U_{w} = ∅. We denote by φS_{w}^{t} the generating function of φL_{w}^{t} obtained from S_{w}^{t}. Now the proof of proposition 3.5 applies, and we may state with φ̃ = φ × id

c(α, φ̃S_{w}) = c(α, S_{w} − φ̃^{−1}(O_{S^{2n}}))

where φ̃^{−1}(O_{S^{2n}}) denotes any g.f.q.i. of this manifold.

Because c(α, φ̃S_{w}^{t}) is one of the numbers l(x, y) associated to φ̃L_{w}^{t}, and L_{w}^{t} coincides with Δ_{w} outside U_{w} ×U_{w}, the proof of corollary 4.5 carries over, thus concluding our proof.

The “camel problem” can be stated as follows : there is no symplectic isotopy, ψ_{t}, with support in ( ^{2n} − ^{2n−1}) ∪ B^{2n}(ε), such that ψ_{0} = id and ψ_{1} sends a ball of radius r > ε contained in one component of ^{2n} − ^{2n−1} in the other component.

In other terms a ball of radius r > ε cannot “go through a hole of radius ε” (see also [Ar],[V3] for historical details). In fact this may be generalized as follows : let V ⊂ ^{2n−1} be such that its reduction has capacity γ(V/ ). Then U cannot go through ^{2n−1} − V unless c(U) ≤ γ(V/ ).

To keep our proof simple, we shall only consider the case of balls, but with a little more work, we could easily extend our proof to this more general case.

An alternative (unpublished) proof is due to M. Gromov, and Y. Eliashberg using the technique developed in [Gr1]. I wish to thank them for enlightning discussions.

We now go back to the camel problem. Let ψ_{t} be a compact supported Hamiltonian isotopy such that ψ_{1} sends a ball B_{r} of radius r from one side to the other of the hyperplane ^{2n−1} × {0}. We then have

Proposition 5.3.

Let V be ⋃_{t∈[0,1]} ψ_{t}(B_{r}) ∩ ^{2n−1} × {0}, and V/ be the quotient of V by the characteristic foliation in ^{2n−1}.

Then V/ is a subset of ^{2n−2} such that γ(V/ ) ≥ πr^{2}.

Before we prove the above proposition, we shall make some simplifying assumptions. We take on ^{2n}, the coordinates (q, p, q_{n}, p_{n}) ∈ ^{2n−2} × ^{2}, the hyperplane ^{2n−1} ×

{0} is then given by {p_{n} = 0}.

Without modifying V = ⋃_{t} ψ_{t}(B_{r}) ∩ ^{2n−1} × {0}, we may assume that

(1) For some c ≫ r we have

ψ_{t}(q, p, q_{n}, p_{n} + c) = ψ_{t}(q, p, q_{n}, p_{n}) + c\frac{∂}{∂p_{n}}

ψ_{1}(q, p, q_{n}, p_{n}) = (q, p, q_{n}, p_{n} + c)

(2)  ψ_{t} is defined for all t’s and ψ_{t+1} = ψ_{1} ∘ ψ_{t} .

This follows easily from the fact that all symplectic embeddings of B_{r} into the half space ^{2n−1} × _{+} are symplectically isotopic, and from the extension of isotopies.

Now let H(t, q, p, q_{n}, p_{n}) be the Hamiltonian generating ψ_{t}. We shall define a symplectic map \overline{Ψ} : ^{2n+2} → ^{2n+2} by

\overline{Ψ}(q, p, q_{n}, p_{n}, t, h) = (ψ_{t}(q, p, q_{n}, p_{n}), t, h + H(t, q, p, q_{n}, p_{n}))

We infer from (1) and (2) that

\overline{Ψ}(q, p, q_{n}, p_{n} + kc, t + l, h)

= \overline{Ψ}(q, p, q_{n}, p_{n}) + kc\frac{∂}{∂p_{n}} + l \frac{∂}{∂t} .

This means that on ^{2n−2} × T^{∗}T^{2} ≃ ^{2n+2}/( c\frac{∂}{∂p_{n}} + \frac{∂}{∂t}), \overline{Ψ} descends to a map Ψ = Ψ_{1}.

If we denote by Ψ_{0} the map obtained above by taking ψ_{t}(q, p, q_{n}, p_{n}) = (q, p, q_{n}, p_{n}+ tc), it is easy to see that Ψ_{1} is symplectically isotopic to Ψ_{0}.

Now let U(r) be Ψ_{1}(B^{2n}(r) × T^{∗}S^{1}), we have

Lemma 5.4.

c(μ ⊗ d(p_{n} + ct) ⊗ 1, U(r)) ≥ πr^{2}

.

Proof:

For simplicity, we assume c = 1. Clearly

\begin{matrix}c(μ ⊗ d(p_{n} + t) ⊗ 1 , Ψ_{1}(B^{2n}(r) × T^{∗}S^{1})) \\ = c(μ ⊗ d(p_{n} + t) ⊗ 1 , Ψ_{0}(B^{2n}(r) × T^{∗}S^{1})) \end{matrix}

= c(Ψ_{0}^{∗}(ν ⊗ d(p_{n} + t) ⊗ 1) , B^{2n}(r) × T^{∗}S^{1})
= c(μ ⊗ dp_{n} ⊗ 1 , B^{2n}(r) × T^{∗}S^{1})
= c(μ ⊗ dp_{n} , B^{2n}(r)) = πr^{2}

Lemma 5.5.

c(μ ⊗ d(p_{n} + t) ⊗ 1, U(r))

≤ γ(μ, ⋃_{t∈[0,1]} ψ_{t}(B^{2n}(r)) ∩ ^{2n−1} × {0}/ )

= γ(μ, V/ ) .

Proof:

If we remark that the torus T^{2} = {(p_{n}, t) ∈ S^{1} × S^{1}} can also be written as the product of the circles {p_{n} +t = 0} and {p_{n} = 0}, we see that according to proposition 5.2, we may infer that

c(μ ⊗ d(p_{n} + t) ⊗ 1, U(r))

≤ inf_{s} γ(μ ⊗ d(p_{n} + t), U(r) ∩ {p_{n} = s}/ )

≤ γ(μ ⊗ d(p_{n} + t), U(r) ∩ {p_{n} = 0}/ )

Now

U(r) ∩ {p_{n} = 0}/ ⊂ ( ⋃_{t∈[0,1]} ψ_{t}(B^{2n}(r)) ∩ ^{2n−1} × {0}/ )
× T^{∗}S^{1}
= V/ × T^{∗}S^{1} .

Since γ(μ ⊗ d(p_{n} + t) , V/ × T^{∗}S^{1}) = γ(μ, V/ ) according to 5.2, this concludes our proof.

Remark. The same proof yields the following result. Let H^{k}(ϵ) = ( ^{2n} − ^{n} × k) ∪ B^{2n}(ϵ) be the complement of a punctured coisotropic subspace in ^{2n}, and SEmb(B^{2n}(r), H^{k}(ϵ)) be the space of symplectic embeddings of the ball in H^{k}(ϵ), ∂SEmb(B^{2n}(r), H^{k}(ϵ)) be the subspace of those embeddings with their image outside a large ball of ^{2n}. Then one may prove that

π_{k−1}(∂SEmb(B^{2n}(r), H^{k}(ϵ))) → π_{k−1}(SEmb(B^{2n}(r), H^{k}(ϵ)))

is zero if and only if r ≤ ϵ.

6. On some properties of simple manifolds

We consider the hypersurfaces introduced by Y. Eliashberg in [El], which satisfy the following two properties

(i) Σ coincides with Σ_{0} = {(q, q_{n}, p, p_{n}) | p_{n} = 0} outside U × [0, 1] × (U is a compact subset of ^{2n−2})

(ii) a characteristic of Σ which intersects U ×{0}×{0}, also intersects U ×{1}×{0}.

We shall usually summarize (ii) by saying that all characteristics of Σ “go from one side to the other”, and call a hypersurface satisfying the above conditions, simple.

To a simple hypersurface Σ, one may associate

- a symplectic diffeomorphism of ^{2n−2}, ϕ_{Σ}, which coincides with the identity outside U, defined by the property : y = ϕ_{Σ}(x) ⇔ (x, 0, 0) and (y, 1, 0) are on the same characteristic of Σ.

- a Lagrange submanifold of \overline{^{2n−2}} × ^{2n},

L_{Σ} = \{(x, y) ∈ \overline{^{2n−2}} × ^{2n} | y ∈ Σ , and is on the
characteristic of Σ which goes through (x, 0, 0)}

Note that the reduction of L_{Σ} by {q_{n} = 1} is the graph of φ_{Σ}. Note also that if H(t, x) is a time dependent compact supported Hamiltonian on ^{2n−2}, with time one flow φ, then for Σ = {(x, t, h) ∈ ^{2n} |h = H(t, x)} we have φ_{Σ} = φ.

[Remark : We had to assume H(t, x) = 0 for t ∉ [0, 1], but we can always do so by replacing H(t, x) by H(t, x)−a(t), since H(t, x) does not depend on x for t ∉ [0, 1].]

We now, as in section 4, replace \overline{^{2n−2}} × ^{2n} by T^{∗} ^{2n−1} using the map

(q, p, Q, Q_{n}, P, P_{n}) → (\frac{q + Q}{2}, \frac{p + P}{2} , Q_{n}, P − p, q − Q, P_{n})

Note that for Σ_{0} = ^{2n−1} × {0}, L_{0} = L_{Σ_{0}} goes to the zero section. We now compactify L_{Σ} and the zero section L_{0} to S^{2n−2} × S^{1}, the S^{1} direction being given by Q_{n}.

In order to do this we need that L_{Σ} = L_{Σ_{0}} outside a compact set, and this happens if and only if φ_{Σ} = id. This is slightly unpleasant, since we want to relate φ_{Σ} to Σ, so we operate as follows : change Σ on U × [1, 2] × {0} (where L_{Σ} = L_{Σ_{0}}) to the graph of a Hamiltonian K(t, x) such that its time one map is φ_{Σ}^{−1}. We thus get a simple hypersurface Σ′. We may now compactify simultaneously L_{Σ_{0}} and L_{Σ′}, and get a Lagrange submanifold \overline{L_{Σ′}} of T^{∗}\overline{L_{Σ_{0}}} = T^{∗}(S^{2n−2} × S^{1}).

Definition 6.1.

For α ⊗ β ∈ H^{∗}(S^{2n−2} × S^{1}) = H^{∗}(S^{2n−2}) ⊗ H^{∗}(S^{1}), we set c(α ⊗ β, Σ′) for c(α ⊗ β, \overline{L_{Σ′}}).

However if Σ′ is obtained from Σ as explained above, we keep the notation φ_{Σ} for the symplectomorphism (usually not the identity) defined before.

We may prove

Proposition 6.2.

c(1 ⊗ 1, Σ′) ≤ c_{−}(φ_{Σ}) ≤ c(1 ⊗ μ, Σ′)
c(μ ⊗ 1, Σ′) ≤ c_{+}(φ_{Σ}) ≤ c(μ ⊗ μ, Σ′) .

Proof:

Follows immediately from proposition 5.1 and the fact that \overline{Γ}_{φ_{Σ}} is the reduction of L_{Σ′} by {Q_{n} = 1}.

Remember that corollary 4.7 told us that if φ is the flow of a nonnegative Hamiltonian, then c_{−}(φ) = 0. We wish to generalize this statement as follows.

Proposition 6.3.

Assume Σ to be contained in ^{2n−1} × _{+}. Then c_{−}(φ_{Σ}) = 0, i.e. φ_{Σ} ≻ id.

Proof:

Let (x, y, Q_{n}) be coordinates on \overline{L_{Σ_{0}}}, (x, y, Q_{n}, ξ, η, P_{n}) be coordinates in T^{∗}\overline{L_{Σ_{0}}}. Let S(x, y, Q_{n}, ζ) be a g.f.q.i. for \overline{L}_{Σ′}, then S(x, y, 1, ζ) is a g.f.q.i. for \overline{Γ_{φ_{Σ}}}, and S(x, y, Q_{n}, ξ) is a quadratic form in ξ for Q_{n} = 0. For each θ ∈ S^{1}, denote by S_{θ}(x, y, ξ) the restriction of S to {Q_{n} = θ}. Then c(1, S_{θ}) is zero for θ = 0, is equal to c_{−}(φ_{Σ}) for θ = 1 and since \frac{∂S_{θ}}{∂θ} = \frac{∂S}{∂Q_{n}} = P_{n} ≥ 0 for θ ∈ [0, 1] if \frac{∂S_{θ}}{∂ξ} (x, y, S) = 0, we get as in the proof of proposition 4.6 that c(1, S_{θ}) is a nondecreasing function on [−0, 1]. But since c(1, S_{0}) = 0 and c_{−}(φ_{Σ}) ≤ 0, we clearly get c_{−}(φ_{Σ}) = 0.

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