PROPERTIES OF EMBEDDED LAGRANGE SUMBANIFOLDS

CLAUDE VITERBO

To the Memory of Andreas FLOER.

1. INTRODUCTION, EXAMPLES

A symplectic manifold is a pair (M^{2n}, ω) of a manifold and a symplectic form ω, that is:

(1) ω is closed (i.e. dω = 0).

(2) ω is non-degenerate (i.e. if ∀ξ ω(ξ, η) = 0 ⟹ η = 0.

It follows from elementary linear algebra that such a manifold always has even dimension (denoted by 2n). A Lagrange submanifold is a sumbanifold L^{n} on which the symplectic form restricts to 0 (i.e. ω_{|T L} = 0), and such that it has half the dimension of the symplectic manifold.

A diffeomorphism ϕ of (M^{2n}, ω) is said to be symplectic if it preserves the symplectic form (i.e. ϕ^{∗}ω = ω).

(M^{2n}, ω) is said to be exact if ω is an exact form. In the sequel, we assume that if a manifold is exact, we are also given a one form λ such that its differential is ω, and by abuse of notation shall denote such a manifold by (M, dλ). Note that since ω^{n} is a volume form, an exact manifold is never compact.

On such a manifold, we have the notion of exact Lagrange sumbanifold, that is a Lagrange submanifold on which λ restricts to an exact form, and an exact diffeomorphism is a symplectic diffeomorphism such that the closed form ϕ^{∗}λ − λ is exact.

In particular let H(t, x) be a C^{∞} function on ℝ × M. The time dependent vector field X_{H} defined by ω(X_{H}, ξ) = d_{x}H, has a flow made of symplectic maps. Moreover in the exact case it is actually made of exact maps.

Examples

a. (ℝ^{2n}, d(∑_{i=1}^{n} p_{i} ∧ dq^{i})). Locally this is the universal example, any maniofold is locally symplectomorphic to this one.

b. (n = 1) M is an orientable surface endowed with a volume form ω. It is exact if and only if it is non compact. Any curve is a Lagrange submanifold.

b’.M = ℝ^{2}, λ = ydx, (resp. M = S^{1} × ℝ, λ = ydθ). A curve is exact if its ”algebraic area” (resp. the ”algebraic area enclosed between the curve and S^{1} × {0}”) is zero (since this area is equal to ∫_{S^{1}} λ).

Exact curve in T^{∗}S^{1}

Exact curve in ℝ^{2} c. (T^{∗}N, dλ) where in local coordinates λ = ∑_{i=1}^{n} p_{i}dq^{i}, where

q^{1},…, q^{n} are local cordinates on N and p_{1},…, p_{n} are dual coordinates. The form λ is defined by the property of being the unique one form such that if α is a one form on N, and s_{α} the obvious map N → T^{∗}N it defines, we have s_{α}^{∗}λ = α. Then s_{α}(N) is Lagrange if and only if α is closed , it is exact if and only if α is.

d. If M is a Kähler manifold, the imaginary part of its Hermitian metric is a symplectic form by definition. In particular if M is a projective manifold then it is symplectic. In the case where it is defined by real equations, its real part (i.e. the intersection of M with ℝP^{n} is a Lagrange submanifold.

The aim of symplectic topology is to understand qualitative properties of symplectic manifolds and maps. The type of question one deals with is quite well illustrated by two of the most famous problems that were solved in the last ten years. First we shall mention many cases of the ”Arnold connjecture” which essentially states that a symplectic map should have more fixed points than a smooth one. For instance for the 2n dimensional torus, endowed with its standard structure, the number of fixed points for an (exact) symplectic map is at least 2n + 1 , while it may be zero for a smooth map. This result has been greatly generalized in particular by Gromov and Floer. Another important question concerns the notion of size for subsets in a symplectic manifold. In other words, are there other obstructions to embedding one set into another one through a symplectic map than just volume. That this turns out to be the case due to new invariant due to Gromov, the width (or more generally capacities). Here we shall focus onn properties of Lagrange submanifolds. That this is an important class of manifolds is already clear from the fact that the graph of a symplectic map is a Lagrange submanifold in the product. In fact Lagrange submanifolds may sometimes be considered as ”symplectic relations” and thus a reasonable generalization of symplectic maps. Here we shall mostly focus on properties of Lagrange sumbanifolds of a cotangent bundle.That most results in symplectic topology may be proved by studying such Lagrange submanifolds has been considered in [V3].

2

Immersion theory

Let i : L → (M, ω) be a Lagrange immersion. Then ϕ = di : T L → T M is a vector bundle morphism satisfying the following properties:

(1) ϕ is injective on each fiber

(2) for each x ∈ L, ϕ(x)T_{x}L is a linear Lagrange subspace of T_{ϕ̄(x)}M endowed with the symplectic structure defined by ω.

(3) ϕ̄^{∗}(ω) is exact.

A map ϕ satisfying the above two properties will be called a formal Lagrange immersion. We now have

Theorem1. (Gromov, Lees) The map i → di is a homotopy equivalence between the space of Lagrange (resp. exact) immersions to the space of formal Lagrange immersions..

This implies in particular that any formal Lagrange immersion may be deformed (in the space of formal Lagrange immersion) into the differential of a Lagrange immersion, and that if two Lagrange immersions have homotopic differentials (in the space of formal Lagrange immersion), then they are homotopic in the set of Lagrange immersions (one may add exact everywhere).

Using this result, one may already exclude in many cases that a given manifold may be Lagrangianly embedded in a given symplectic manifold. For example, no unorientable surface of Euler characteristic ≡ 2[mod 4] may be embedded in ℝ^{4}. Also one may use this to prove that no sphere except maybe the three sphere may be embedded in ℝ^{2n}. We will see that there is in fact no exception. For more consequences of Gromov’s theorem, we refer to [Audin], [Lalonde-Sikorav].

3. Obstructions to embeddings

From now on all Lagrange sumbanifolds will be compact and embedded, unless otherwise claimed. Also, we shall restrict ourselves to the case (M, ω) = (T^{∗}N, d(pdq)) (N is allowed to be noncompact). Let us see in case dim(N) = 1 what are the obstructions to embeddings.

(1) N = ℝ There is no exact (closed) curve in T^{∗}ℝ, neither an embedded curve with winding number ≠ ±1.

(2) N = S^{1} Then the projection of an embedded curve on S^{1} must have degree ±1, and its winding number is zero.

Moreover, the only examples of an exact Lagrange manifold in T^{∗}N that we know of may always be obtained as ϕ_{1}(O_{N}) where ϕ_{t} is a Hamiltonian flow, and 0_{N} the zero section of T^{∗}N. One may thus conjecture that there are no other examples. A more modest but more reasonable claim was made by V.I. Arnold:

Conjecture1. Let L be an exact submanifold in T^{∗}N. Then the projection of L on N has degree ±1. .

This is of course a generalization of the fact that in T^{∗}S^{1} an exact curve projects on S^{1} with degree ±1. We now generalize the remark on the vanishing of the winding number. Since the generalization of the winding number to Lagrange submanifolds is the Maslov class (read further for a definition or see [Ar1]).

Conjecture2 (M.Audin ?). If L is exact in T^{∗}N, then its Maslov class vanishes. .

Even though we do not have real evidence for such a claim, it is the author feeling that Conjecture 2 should be easier to prove if we already know that conjecture 1 holds.

After these conjectures let us state our main results.

TheoremA. There is a set S of compact manifolds, containing the spheres, complex projective spaces and their products, as well as their codimension zero submanifols, such that for W ∈ S we have :

(1) If L → T^{∗}W is an exact embedding, then p : L → N has nonzero degree (2) If L → T^{∗}(N × W) is exact then the projection p_{2} : L → W induces a nonzero map in cohomology. .

For the proof we refer to [V 4].

3.Remarks.

This may also be seen as a generalization of another theorem by Gromov, Indeed he proved that there is no exact submanifold in T^{∗}N × ℝ^{2}. This is equivalent to (2) of our theorem applied to W = ℝ that we identify to a submanifold of S^{1}, since all maps to ℝ are homotopically trivial. As a result of Gromov’s theorem, the three sphere has no Lagrange embedding into ℝ^{6}.

Our method also answers the following question. It follows from Gromov’s result that there is no exact embedding of T^{n} into ℝ^{2n} = T^{∗}ℝ^{n}. It is natural to ask whether such an embedding exists if we add to T^{∗}ℝ^{n} a space over infinity to get T^{∗}S^{n}. The answer is still negative (for n = 2 this question is due to [La-Si]).

We now discuss conjecture 2, and begin by reminding the reader about the Maslov class. This is a cohomology class μ(L) ∈ H^{1}(L, BbbZ) associated to an immersed Lagrange submanifold. In other words it is a map μ(L) : π_{1}(L) → ℤ that associates to a loop the winding of the tangent space to L as we go around the loop. This means that we must associate to a loop of Lagrange subspaces in T T^{∗}N, an integer depending only on the homotopy class of the loop. This is easily done as follows: let V (x) = T_{x}(T_{q}^{∗}N) be the vertical distribution on T^{∗}N, and Σ = {x ∈ L | T_{x}L ∩ V (x) ≠ {0}}. Then provided L is in general position, Σ is a regular hypersurface, with codimension 3 singularities. Moreover Σ is coorientable, which means that there is an ”outward normal direction” defined continuously. In other words, given a loop on L avoiding the singular set of Σ, and having transverse intersecting it, the difference between the number of points where γ intesects Σ positively and negatively is by definition the Maslov class of γ, i.e.

μ(L).γ = #_{alg}(γ ∩ Σ)

Theorem3. If L = N = T^{n}, and L is exact in T^{∗}N then μ(L) = 0.

This is proved in [V2] (see also [La-Si], [Po 1]) and was generalized to other pairs (with L = N) by Polterovich (see [Po 2]).

3.Remarks.

In the situation of non-exact embeddings, the situation is more complicated, because it is not even clear what one should expect. It is easy to see that a manifold M × S^{1} with T M ⊗ ℂ trivial, has always a Lagrange embedding into ℝ^{2n}. Also, it was proved by Givental that except for the Klein bottle all orientable surfaces,as well as the non orientable ones with Euler characteristic divisible by four have a Lagrange embedding in ℝ^{4}. The case of the Klein bottle is still open.

4. RELATED QUESTIONS AND APPLICATIONS 1. Invariant Lagrange submanifolds for Hamiltonian flows

It is remarkable that the question of existence of Lagrange invariant submanifolds with some natural properties is related to the question of the nullity of the Maslov class. Since the recent developments by M. Herman and Byalyi, Polterovich are very beautiful, we cannot resist explaining some of its aspects.

The original problem was to find a higher dimensional generalization of Birkhoff’s theory, that we first summarize. Let H(t, q, p) be a time dependent Hamiltonian on T^{∗}S^{1} convex in the p variable (for all t), and ϕ the time one map of its flow. These map coïncide with the so called area preserving twist maps. Let γ be an embedded curve with degree one projection on the circle, invariant by the flow. Then Birkhoff’s theorem asserts that γ is in fact the graph of a Lipschitz function over S^{1}, and the Lipschitz norm is bounded a priori by a constant depending only on the norm of the twist map. Note that the assumption on the degree of the projection cannot be removed. Indeed, there are twist maps with an elliptic fixed point, and near such a point there are many invariant circles. In higher dimensions this has been generalized as follows by M. Herman (see also Byalyi and Polterovich’s [B-P]).

Theorem4. Assume H is time independent Hamiltonian on T^{∗}N such that \frac{∂^{2}H}{∂p^{2}} > 0, and let L be an invariant Lagrange submanifold for the associated flow such that:

(1) the restriction of the flow to L is chain recurrent1.

(2) The Maslov class of L vanishes. Then L is a graph over N. .

If L = N = T^{n} we proved that (ii) is implied by deg(p) = 0 Now this is a much more reasonable assumption, since it mainly excludes the case of a torus located in a small region. As in the 1-dimensional case, this cannot be avoided.

Let us point out that if g is a Riemannian metric on N, then H(q, p) =< p, p >_{g} is a Hamiltonian generating the geodesic flow on N. It is easy to construct on T^{n} a geodesic flow without invariant Lagrange graph. Indeed to say that the Hamiltonian given by some metric has a Lagrange invariant graph, is equivalent to saying that the manifold has a 1-dimensional foliation by so called A-geodesics. An A-geodesic is such that for any two points on the curve the A geodesic realizes the minimum of the arclenght the two points in its homotopy class.

For the metric represented on figure 3, no such foliation can exist, because if the neck is small enough, no A-geodesic can go through z_{0} (it

1 We refer to [H] for the precise definition, let us just mention that this is satisfied for instance if there is a dense orbit. is always ”cheaper” to go around the neck). Thus there is no Lagange invariant graph for this metric, and according to the above theorem, no invariant Lagrange manifold satisfying conditions (i) and (ii) above.

2. Symplectic topology

Let U, V be subset of ℝ^{2n}. Is there a symplectic diffeomorphism of ℝ^{2n} sending U into V. Because a symplectic map is volume preserving, one must have vol(U) ≤ vol(V). That this is far from being sufficient is a deep result by Gromov: he proved for instance that there is no symplectic embedding from the ball of radius r, B^{2n}(r) into the cylinder B^{2}(r′) × ℝ^{2n−2}, unless r ≤ r′. One may wonder what happens if B^{2n}(r) is replaced by a set of dimension less than 2n. Can we embed it in an arbitrarily small cylinder. The following result gives an answer in a very special but interesting case:

Proposition1. Let L be a Lagrange torus in ℝ^{2n}. Then ther is a positive number r(L), such that

∀ψ ∈ Diff_{ω}(ℝ^{2n}), ψ(L) ⊄ B^{2}(r(L)) × ℝ^{2n−2}

.

This is a consequence of a more precise version of theorem ?

Proposition 2. Let L be a Lagrange torus in ℝ^{2n}. Then, there is a loop γ on L such that : (1) c(L) = ∫_{γ} pdq > 0, where c(L) is the Ekeland-Hofer capacity of L.

(2) < μ(L), γ >∈ [2, n + 1]. .

The Ekeland-Hofer capacity is a symplectic invariant, monotone (i.e if U ⊂ V then c(U) ≤ c(V)) and equal to πr^{2} on the cylinder B^{2}(r) × ℝ^{2n−2}. It is clear that Prop. 2 (1) implies Prop. 1

Now since L has a neighbourhood that may be identified to a neighbourhood of the zero section in T^{∗}L, a sufficiently small one-form on L may be identified to a deformation of L. Moreover if the one form is closed, the deformation is an isotopy of Lagrange sumbmanifolds. During such an isotopy, the Maslov class is unchanged, while the Liouville class pdq changes exactly by the cohomology class of the one-form. Let us then consider a deformation of L induced by a representative of −tμ(L) (t ∈ [0, 1]). During such a deformation, we may select the value of ∫_{γ_{t}} pdq to vary continuously with t. Then generically the loop γ_{t} varies continuously, except for a finite number of values of t. Now except for these values, we have that \frac{d}{dt} ∫_{γ_{t}} pdq = −μ(L) < −2. But since ∫_{γ_{t}} pdq must remain positive according to (1), this implies that t has to be smaller than 1/2 ∫_{γ_{0}} pdq. Thus the quantity in (1) yields a bound on the size of a tubular neighbourhood of L, ”measured by the Maslov class”. Thus we see that for all Lagrange tori contained in a cylinder, the size of a tubular neighbourhood of these tori is uniformly bounded. This phenomenon is reminiscent of Riemannian geometry where a manifold contained in a small ball has somewhere large curvature hence its tubular neighbourhood (wich has size of the order of the inverse of the largest radius of curvature) cannot be too large.

5. METHOD OF PROOF

Here we shall explain some of the ideas in the proof of theorem A. However the same ideas, simplified because of the context, appear in the proof of theorem B. We first choose a Hamiltonian which will reflect the geometry of the problem as follows. Let U be a tubular neighbourhood of the L. We identify it to a neghbourhood of the zero section in T^{∗}L, we denote by q, p the induced coordinates. Now our Hamiltonian will be such that:

H(q, p) = |p|^{2} for |p| ≤ ε

Here |p| is induced by some Riemannian metric on L (in the proof of theorem B, since L = T^{n} this metric may be chosen to be flat).

Thus in some V ⊂ U, the flow associated to H is given by the geodesic flow of the metric. From the variational point of view, the closed geodesic reconstruct the topology of the free loop space of L. Now if we extend H to the whole space, so that all periodic solutions of ẋ = X_{H}(x) with positive action(A_{H}(z) = ∫_{0}^{2π} \frac{1}{2}(Jz, ż) − H(t, z)) must be contained in U. Now if a geodesic γ has lenght c, its action will be ∫_{γ} pdq + c. So when the manifold is exact itis exactly c. Now c is also the critical value of the Energy functional E(γ) = ∫_{S^{1}} |γ̇|^{2}. Thus in this case, the Action functional and the Energy functional have the same critical values. In fact the relationship between these functionals goes deeper. While one of the main problems when dealing with such a functional as the action functional comes from the fact that all critical points are of infinite Morse index (and coindex) so that there occurs no change in the topology of the level set as we go through a critical point, we shall argue as if our functional were defined on a finite dimensional space. The argument could be made correct either by using Floer’s homology (this behaves like a middle dimensional homology theory) but this would require more work than we actually did, or, as we acually did in [V1] use a finite dimensional reduction. Let us now explain more precisely how E appears in A_{H}. Note that the first functional is defined on ΛL while the second is defined in a neighbourhood of ΛL in ΛT^{∗}L. We claim that for some pseudo gradient vector field for the acion functional, there is a submanifold of ΛT^{∗}L that is a graph over ΛL which is invariant and a repeller (this means that any point in a neighbourhood of the set which is not on the set, eventually exits from this neighbourhood). This graph is constructed as follows, using Legendre duality.

Figure 3

For each q ∈ ΛL the map p → A_{H}(q, p) is concave. Hence we define a function q → p(q) by p(q)achievesthemaximumofA_{H}(q, p). Now we have E(q) = A_{H}(q, p(q)), and we have a picture as in figure 3. Then it is not hard to show, using the ideas of Conley’s theory, that the relative topology of A_{H} ≤ c divided by A_{H} ≤ b is the same up to a generalized suspension as that of E ≤ c divided by E ≤ b, provided all periodic orbits with action in [b, c] are in the neighbourhood of L (usually we take b negative so we do not have to divide in the second case). This variational picture applies in two circumstances. The first case is when we are in T^{∗}N and H = |p|^{2} globally. Then we see that if we set A_{H}^{c} = {A_{H} ≤ c}, the relative topology of A_{H}^{c} divided by A_{H}^{a} (a < 0) is given by the (generalized) suspension of the space Λ^{c}N = {γ ∈ ΛN | E(γ) ≤ c}. Note that once we have this, we may perturb our function (that is the Hamiltonian) in a bounded region, we this will always hold for a < 0 < c, |a|, |c| large enough. Secondly if we are in the case of L exact Lagrange submanifold as before, then A_{H}^{c} divided by A_{H}^{b} has the topology of Λ^{d}L = {γ ∈ ΛL | E(γ) ≤ d}. To summarize the situation, we have A_{H}^{c}/A_{H}^{b} ≃ Λ^{d}L and A_{H}^{c}/A_{H}^{a} ≃ Λ^{c}N. The obvious map from A_{H}^{c}/A_{H}^{a} to A_{H}^{c}/A_{H}^{b} thus induces a map from Λ^{c}N to Λ^{d}L, hence a map in cohomology H^{∗}(ΛL) → H^{∗}(ΛN) (obtained by letting c and thus d go to +∞. This map has many of the properties of a ”push forward” or ”transfer”. A map f between two compact manifolds X and Y induces a map f^{!} : H^{∗}(X) → H^{∗}(Y) as follows. If f is a fibration, f^{!} corresponds to integration over the fibers, if it is an embedding, it corresponds to the composition of the Thom isomorphism T : H^{∗}(X) → H_{c}^{∗}(U), where U is a tubular neighbourhood of X, and the obvious injection H_{c}^{∗}(U) → H^{∗}(Y). Since any map is homotopic to the composition of a fibration and an embedding, this defines f^{!} in general. One of the main properties of f^{!} is that it is an H^{∗}(Y) module homomorphism that is we have f^{!}(x∪f^{∗}(y)) = f^{!}(x)∪y.

Now loop spaces are not finite dimensional manifolds. Thus one cannot hope that for any map between loops spaces, a transfer be defined. However, most of the maps we are interested in come from a map f : L → N. One may then wonder if it is possible to define Λf^{!} for the map Λf : ΛL → ΛN induced by f. It is easy to see that there are still obstructions, at least if we require that

(1) Λ(f ∘ g)^{!} = Λf^{!} ∘ Λg^{!}

(2) Λf^{!}(x ∪ Λf^{∗}(y)) = Λf^{!}(x) ∪ y for x ∈ H^{∗}(ΛL) y ∈ H^{∗}(ΛN)

(3) Λf^{!} restricts to f^{!} on the image of H^{∗}(X) by the inclusion of constants. In other words, the following diagram is commutative:

H^{∗}(ΛL) ⟶^{(Λj)!} H^{∗}(ΛN)
↓_{c^{∗}}↑_{e^{∗}}  ↓_{c^{∗}}↑_{e^{∗}}
H^{∗}(L) ⟶^{j!} H^{∗}(N)

It is not hard to see that there are obstructions to do this. If we take cohomology with rational coefficients, the cohomology of the loop space is easy to compute, using Sullivan’s minimal model theory. These obstructions will depend on L, N and f^{∗}, and for f with degree zero N ∈ S one may show that there cannot be a map such as Λf.

Now one may prove that the map we obtained from H^{∗}(ΛL) → H^{∗}(ΛN) using our variational problem satisfies the above properties, with f the projection of L ⊂ T^{∗}N on N. Then obstructions to the existence of such a map yield obstructions to existence of an exact embedding of L into N.

6. OTHER PROPERTIES OF LAGRANGE SUBMANIFOLDS

Even though we cannot say very much about it, there are beautiful results, mainly due to J. Duval, concerning the relationship between Lagrange surfaces and complex geometry in ℂ^{4}. In fact we have

Theorem5. A Lagrange surface in ℂ^{2} is rationally convex (i.e. any function on L is approximated by the restriction of a meromorphic function on ℂ^{4}). .

Theorem5’. A totally real rationally convex n-dimensional submanifold in ℂ^{n} is Lagrange for some symplectic form isotopic to the standard one.

In particular in the case of the torus, its Maslov class is non-zero (the Maslov class is still defineed for a totally real submanifold).

We can only refer to the beatiful papers [Du 1], [Du 2] since the proof uses a totally different method from what we explained sofar.

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[V 3] Viterbo,C. , Symplectic topology as the geometry of generating functions; Math. Annalen 692(1992) , pp. 685–710.

[V 4] Viterbo,C. , Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces, (in preparation).

U.R.A. 1169 ”Topologie et dynamique”, Université de Paris XI, F-91405 Orsay Cedex, FRANCE.