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Coisotropic rigidity and C^0-symplectic geometry

We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C^0-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C^0-Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.


Introduction and main results
A submanifold C of a symplectic manifold (M, ω) is called coisotropic if for all p ∈ C, (T p C) ω ⊂ T p C where (T p C) ω denotes the symplectic orthogonal of T p C. For instance, hypersurfaces and Lagrangians are coisotropic. A coisotropic submanifold carries a natural foliation F which integrates the distribution (T C) ω ; F is called the characteristic foliation of C. Coisotropic submanifolds and their characteristic foliations have been studied extensively in symplectic topology. The various rigidity properties that they exhibit have been of particular interest. For example, in [7] Ginzburg initiated a program for studying rigidity of coisotropic intersections. In this paper, we prove that coisotropic submanifolds, along with their characterisitic foliations, are C 0 -rigid in the spirit of the Gromov-Eliashberg Theorem.
The Gromov-Eliashberg Theorem states that a diffeomorphism which is a C 0 -limit of symplectomorphisms is itself symplectic. Motivated by this, symplectic homeomorphisms are defined as C 0 -limits of symplectomorphisms (see Definition 11). Area preserving homeomorphisms, and their products, are examples of symplectic homeomorphisms. Here is our main result. Theorem 1. Let C be a smooth coisotropic submanifold of a symplectic manifold (M, ω). Let U be an open subset of M and θ : U → V be a symplectic homeomorphism. If θ(C ∩ U ) is smooth, then it is coisotropic. Furthermore, θ maps the characteristic foliation of C ∩ U to that of θ(C ∩ U ).
An important feature of the above theorem is its locality: C is not assumed to be necessarily closed and θ is not necessarily globally defined. Here is an immediate, but surprising, consequence of Theorem 1.
Corollary 2. If the image of a coisotropic submanifold via a symplectic homeomorphism is smooth, then so is the image of its characteristic foliation.
Theorem 1 uncovers a link between two previous rigidity results and demonstrates that they are in fact extreme cases of a single rigidity phenomenon.
One extreme case, where C is a hypersurface, was established by Opshtein [22]. Clearly, in this case, the interesting part is the assertion on rigidity of characteristics, as the first assertion is trivially true.
Lagrangians constitute the other extreme case. When C is Lagrangian, its characteristic foliation consists of one leaf, C itself. In this case the theorem reads: If θ is a symplectic homeomorphism and θ(C) is smooth, then θ(C) is Lagrangian. In [11], Laudenbach-Sikorav proved a similar result: Let L be a closed manifold and ι k denote a sequence of Lagrangian embeddings L → (M, ω) which C 0 -converges to an embedding ι. If ι(L) is smooth, then (under some technical assumptions) ι(L) is Lagrangian. On one hand, their result only requires convergence of embeddings while Theorem 1 requires convergence of symplectomorphisms. On the other hand, Theorem 1 is local: It does not require the Lagrangian nor the symplectic manifold to be closed.
The above discussion raises the following question.
Question. What can one say about C 0 -limits of coisotropic embeddings and their characteristic foliations?
We would like to point out that Theorem 1 is a coisotropic generalization of the Gromov-Eliashberg Theorem. Indeed, it implies that if the graph of a symplectic homeomorphism is smooth, then it is Lagrangian.
Definition 3. A codimension-k C 0 -submanifold C of a symplectic manifold (M, ω) is C 0 -coisotropic if around each point p ∈ C there exists a C 0coisotropic chart, that is, a pair (U, θ) with U an open neighborhood of p and θ : U → V ⊂ R 2n a symplectic homeomorphism, such that θ(C ∩U ) = C 0 ∩V .
A codimension-n C 0 -coisotropic submanifold is called a C 0 -Lagrangian.
Example. Graphs of symplectic homeomorphisms are C 0 -Lagrangians. Graphs of differentials of C 1 functions and, more generally, graphs of C 0 1-forms, closed in the sense of distributions, provide a family of non trivial examples, see Appendix B for a proof. Conversely, we could ask whether every continuous 1-form whose graph is a C 0 -Lagrangian is closed in the sense of distributions. An affirmative answer in a particular case appears in Viterbo [25,Corollary 22].
As a consequence of Theorem 1, C 0 -coisotropic submanifolds carry a (C 0 -)characteristic foliation in the following sense.
Corollary 4. Any C 0 -coisotropic submanifold C admits a unique C 0 -foliation F which is mapped to F 0 by any C 0 -coisotropic chart.
Example. If C = θ(C ′ ), with C ′ a smooth coisotropic submanifold and θ a symplectic homeomorphism, then One may wonder if every topological hypersurface is C 0 -coisotropic. It is possible to show, via an application of Corollary 4, that the boundary of the standard cube in R 4 does not possess a C 0 -characteristic foliation, and hence, it is not C 0 -coisotropic. This, in particular, implies that it is not possible to map the boundary of the sphere to that of the cube by symplectic homeomorphisms.

Corollary 5.
A smooth C 0 -coisotropic submanifold is coisotropic and the natural C 0 -foliation it carries coincides with its characteristic foliation.
As we shall see, the proof of Theorem 1 relies on dynamical properties of coisotropic submanifolds. In particular, we use C 0 -Hamiltonian dynamics as defined by Müller and Oh [21]. To the best of our knowledge, this is the first extrinsic application of this recent, yet promising, theory.
Following [21], we call a path of homeomorphisms φ t a hameotopy if there exists a sequence of smooth Hamiltonian functions H k such that the isotopies φ t H k C 0 -converge to φ and the Hamiltonians H k C 0 -converge to a continuous function H (see Definition 12). Then, H is said to generate the hameotopy φ t , and to emphasize this we write φ t H ; the set of such generators will be denoted C 0 Ham . A foundational result of C 0 -Hamiltonian dynamics is the uniqueness of generators Theorem (see [25,3]) which states that the trivial hameotopy, φ t = Id, can only be generated by those functions in C 0 Ham which solely depend on time (see also Corollary 7 below).
Recall the following dynamical property of smooth coisotropic submanifolds: Let H ∈ C ∞ (S 1 × M ), H| C is a function of time if and only if φ H (preserves C and) flows along the characteristic foliation of C. By flowing along characteristics we mean that for any point p ∈ C and any time t 0, φ t H (p) ∈ F(p), where F(p) stands for the characteristic leaf through p. The following result, which plays an important role in the proof of Theorem 1, is a C 0 -analog of the above. This result answers a question raised by Buhovsky and Opshtein who asked if the above holds in the particular case where C is a smooth hypersurface. It also drastically generalizes the aforementioned uniqueness of generators Theorem. Indeed, if C is taken to be M , then the characteristic foliation consists of the points of M and the theorem follows immediately: Ham is a function of time if and only if φ t H = Id. When C is a C 0 -Lagrangian, Theorem 6 states that: The restriction of H to L is a function of time if and only if φ t H (L) = L for all t. Interestingly enough, the general case of both Theorems 1 and 6 will be essentially deduced from the a priori particular case of Lagrangians. This is, most probably, a C 0 manifestation of Weinstein's creed: "Everything is a C 0 -Lagrangian submanifold!" The results of this paper establish C 0 -rigidity of coisotropic submanifolds together with their characteristic foliations. It would be interesting to see if isotropic or symplectic submanifolds exhibit similar rigidity properties: If a smooth submanifold is the image of an isotropic (respectively symplectic) submanifold under a symplectic homeomorphism, is it isotropic (respectively symplectic)? Note that if in these questions one considers, instead of symplectic homeomorphisms, C 0 -limits of isotropic (respectively symplectic) embeddings then Gromov's results on the h-principle provide negative answers in general. In short, isotropic and symplectic embeddings are not Main tools: Lagrangian spectral invariants. In order to prove the main results, we use the theory of Lagrangian spectral invariants. One consequence of this theory is the existence of the spectral distance γ on the space of Lagrangians Hamiltonian isotopic to the 0-section in cotangent bundles introduced by Viterbo in [24]. 1 More precisely, we establish inequalities comparing γ to a capacity recently defined by Lisi-Rieser [15]. This capacity, which we denote by c LR , is a relative (to a fixed Lagrangian) version of the Hofer-Zehnder capacity. We 1 One of the main features of γ is that it is bounded from above by Hofer's distance on Lagrangians. In particular, Lemmas 8 and 9 also hold with γ replaced by Hofer's distance.
will now define c LR . Fix a Lagrangian L. Recall that a Hamiltonian chord of a Hamiltonian H, of length T , is a path x : [0, T ] → M such that x(0), x(T ) ∈ L and for all t ∈ [0, T ],ẋ(t) = X t H (x(t)). A Hamiltonian is said to be L-slow if all of its Hamiltonian chords of length at most 1 are constant. We denote by H(U ) the set of admissible Hamiltonians, that is, smooth time-independent functions with compact support included in U , which are non-negative and reach their maximum at a point of L. For an open set U which intersects L, the relative capacity of U with respect to L is defined as For instance, if B is the ball of radius r in R 2n and L 0 = R n × {0}, then c LR (B; L 0 ) = πr 2 2 .
In what follows, we denote by L 0 the 0-section of T * L. The first energycapacity inequality used in this paper is the following: and let C ± be real numbers such that C ± > c LR (U ± ; L 0 ). If a compactly supported Hamiltonian H satisfies This is the Lagrangian analog of the energy-capacity inequality proven for the Hamiltonian spectral distance in [10,Corollary 12]. Then, as in [10], we will derive a similar inequality for Hamiltonians (not necessarily constant but) with controlled oscillations on U ± , see Corollary 15.
Lemma 8 can also be established on compact manifolds for weakly exact Lagrangians via Leclercq [12] and for monotone Lagrangians via Leclercq-Zapolsky [13].
The second energy-capacity inequality is due to Lisi-Rieser [15]. This is a relative version of the standard energy-capacity inequality, see for example Viterbo [24].
A special case of this specific inequality appears in Barraud-Cornea [1] and Charette [4]. A similar inequality is worked out in Borman-McLean [2].
Finally, we will need an inequality which provides an upper bound for the spectral distance. Let g denote a Riemannian metric on a closed manifold L and denote by T * r L = {(q, p) ∈ T * L | p g r} the cotangent ball bundle of radius r. Suppose that φ t H (L 0 ) ⊂ T * r L for all t ∈ [0, 1]. Viterbo has conjectured [26] that there exists a constant C > 0, depending on g, such that γ(φ 1 H (L 0 ), L 0 ) Cr. This conjecture has many important ramifications; see [17,26]. Lemma 10 below is a special case of Viterbo's conjecture; a more precise version of the lemma appears in [20,Theorem 9.7].
There exists C > 0, depending on the set V , such that: For all r > 0, if H is a smooth, compactly supported Hamiltonian on T * L such that H| V = 0, Cr. Organization of the paper. In Section 2, we review the preliminaries on C 0 -Hamiltonian dynamics and Lagrangian spectral invariants. In Section 3, we prove energy-capacity inequalities (Lemmas 8 and 9) as well as the upper bound on the spectral distance (Lemma 10). In Section 4, we use these inequalities in order to prove localized versions of Theorem 6 in the special case of Lagrangians. In Section 5, we prove Theorems 1 and 6 and Corollaries 4 and 5 using the results of Section 4.
In Appendix A, we provide relatively simple, and hopefully enlightening, proofs of Theorems 1 and 6 in the special case of closed Lagrangians in cotangent bundles. We hope that this appendix will give the reader an idea of the proofs of the main results while avoiding the technicalities of Sections 4 and 5. In Appendix B, we prove that the graph of a closed C 0 1-form is a C 0 -Lagrangian.
Aknowledgements. We thank Samuel Lisi and Tony Rieser for sharing their work with us before it was completed and for related discussions. The inspiration for this paper came partly from an unpublished work by Lev Buhovsky and Emmanuel Opshtein, whom we also thank. We are especially grateful to Emmanuel Opshtein for generously sharing his ideas and insights with us through many stimulating discussions. We are also grateful to Alan Weinstein for interesting questions and suggestions.

Symplectic and Hamiltonian homeomorphisms.
In this section we give precise definitions for symplectic and Hamiltonian homeomorphisms and recall a few basic properties of the theory of continuous Hamiltonian dynamics developed by Müller and Oh [21]. Given two manifolds M 1 , M 2 , a compact subset K ⊂ M 1 , a Riemannian distance d on M 2 , and two maps f, g : We say that a sequence of maps f i : converges to 0. This notion does not depend on the choice of the Riemannian metric.
Clearly, if θ is a symplectic homeomorphism, so is θ −1 . By the Gromov-Eliashberg Theorem a symplectic homeomorphism which is smooth is a symplectic diffeomorphism. We now turn to the definition of Hamiltonian homeomorphisms (called hameomorphisms) introduced by Müller and Oh [21].
Definition 12. Let (M, ω) be a symplectic manifold and I ⊂ R an interval. An isotopy (φ t ) t∈I is called a hameotopy if there exist a compact subset K ⊂ M and a sequence of smooth Hamiltonians H i supported in K such that: (1) The sequence of flows φ t H i C 0 -converges to φ t uniformly in t on every compact subset of I, uniformly in t on every compact subset of I. We say that H generates φ t , denote φ t = φ t H , and call H a continuous Hamiltonian. We denote by C 0 Ham (M, ω) (or just C 0 Ham ) the set of all continuous functions H : S 1 × M → R which, seen as functions defined on R × M , generate hameotopies parametrized by R. A homeomorphism is called a hameomorphism if it is the time-1 map of some hameotopy parametrized by [0, 1].
Ham generates a unique hameotopy [21]. Conversely, Viterbo [25] and Buhovsky-Seyfaddini [3] proved that a hameotopy has a unique (up to addition of a function of time) continuous generator.
One can easily check that generators of hameotopies satisfy the same composition formulas as their smooth counterparts. Namely, if φ t H is a hameotopy, then (φ t H ) −1 is a hameotopy generated by −H(t, φ t H (x)); given another hameotopy φ t K , the isotopy φ t H φ t K is also a hameotopy, generated by H(t, x) + K(t, (φ t H ) −1 (x)). Moreover, we will repeatedly use the following simple fact: Ham (U ) and generates the hameotopy θ −1 φ t H θ. This, in particular, holds for smooth H :

2.2.
Lagrangian spectral invariants. In [24], Viterbo defined Lagrangian spectral invariants on R 2n and cotangent bundles via generating functions. Then Oh [18] defined similar invariants via Lagrangian Floer homology in cotangent bundles which have been proven to coincide with Viterbo's invariants by Milinković [16]. They have been adapted to the compact case by Leclercq [12] for weakly exact Lagrangians and Leclercq-Zapolsky [13] for monotone Lagrangians. However, for the type of problems which we consider here (C 0 -convergence of Lagrangians), we can restrict ourselves to Weinstein neighborhoods and thus work only in cotangent bundles. We briefly outline below the construction of these invariants via Lagrangian Floer homology and collect their main properties in this situation. We refer to Monzner-Vichery-Zapolsky [17] which gives a very nice exposition of the theory.
Let L be a smooth compact manifold, L 0 denote the 0-section in T * L, and λ the Liouville 1-form. To a compactly supported smooth time-dependent whose generators are the critical chords and whose differential counts the elements of the 0-dimensional component of moduli spaces of Floer trajectories (i.e pseudo-holomorphic curves perturbed by H) which run between the critical chords (with boundary conditions on L 0 ). The differential relies on the additional data of a generic enough pseudo-complex structure, J.
This complex is filtered by the values of the action, that is, for a ∈ R a regular value of A H , one can consider only chords of action less than a. Such chords generate a subcomplex of the total complex CF a * (H) (because the action decreases along Floer trajectories). We denote by i a * the inclusion CF a * (H) → CF * (H). By considering homotopies between pairs (H, J) and (H ′ , J ′ ), one can canonically identify the homology induced by the respective Floer complexes H * (CF (H), ∂ H,J ) and H * (CF (H ′ ), ∂ H ′ ,J ′ ) and by considering C 2 -small enough Hamiltonian functions, one can see that the resulting object HF * (L 0 ) is canonically isomorphic to the singular homology of L.
Thus, one can consider spectral invariants associated to any non-zero homology class α of L, defined as the smallest action level which detects α: In what follows we will only be interested in the spectral invariants associated to the class of a point and the fundamental class which will be respectively These invariants were proven to be continuous with respect to the C 0norm on Hamiltonian functions so that they are defined for any (not necessarily generic) Hamiltonian. Moreover, they only depend on the time-1 map φ 1 H induced by the flow of H; hence they are well-defined on Ham c (T * L, dλ).
Their main properties are collected in the following theorem, which corresponds to [17,Theorem 2.20], except for (7) which we prove below. Note that, except for (6), these properties already appear in Viterbo [24].
(3) For any two Hamiltonian functions H and K, Proof of item (7). Since f is L 0 -slow, Spec(φ 1 f ) consists of critical values of f corresponding to critical points lying in L 0 . Now for all s ∈ [0, 1], since f is autonomous, Spec(φ s f ) = Spec(φ 1 sf ) = s · Spec(φ 1 f ). Since in cotangent bundles spectral invariants lie in the spectrum regardless of degeneracy of f , by continuity of ℓ ± there exist p ± ∈ crit(f )∩L 0 such that ℓ ± (φ s f ) = s·f (p ± ). Now for small times s, sf is C 2 -small and thus ℓ + (φ 1 sf ) (respectively ℓ − (φ 1 sf )) is the maximum (respectively the minimum) of sf so that f (p + ) = max(f ) (respectively f (p − ) = min(f )).
In view of Property (8), Viterbo (followed by Oh) derived an invariant γ of Lagrangians Hamiltonian isotopic to the 0-section, defined as follows.
The main property of γ which will be used in what follows is the fact that . This inequality can be directly derived from Property (6) of Theorem 13. Note that this yields (osc(H t | L 0 )).

Energy-capacity inequalities
3.1. The energy-capacity inequality for Hamiltonians constant on open sets. We prove Lemma 8 by mimicking the proof of the corresponding inequality in [10] (here for Lagrangians, in the easier world of aspherical objects). Then we prove a corollary which will be used in the proof of the main result.
Proof of Lemma 8. First, assume that ℓ − (φ 1 H ) 0. Then for any admissible L 0 -slow function with support in U + , f ∈ H(U + ), we define the 1-parameter By triangle inequality and duality (i.e Properties (4) and (5)) of spectral invariants, we get Then notice that Since for all p ∈ crit(f ) ∩ U + , sf  (2), Property (7) of spectral invariants immediately leads to

Since this holds for any
We choose disjoint open subsets V ± such that U ± ⋐ V ± (with ⋐ denoting compact containment) and osc V ± (H) < osc U ± (H) + ε. We also choose cut-off functions ρ ± with support in V ± , such that 0 ρ ± 1 and ρ ± | U ± = 1. Then we define and we now bound the right-hand side terms.

Thus we immediately get that
The energy-capacity inequality for Lagrangians displaced from an open set. We give a proof of Lemma 9 from Lisi-Rieser [15], for the reader's convenience. The method of proof is now classical and goes back to Viterbo [24] (see also Usher's proof of the analogous result [23] for compact manifolds, itself heavily influenced by Frauenfelder-Ginzburg-Schlenk [6]). Recall that in cotangent bundles the spectral invariants only depend on the endpoint of Hamiltonian isotopies; this drastically simplifies the proof.

3.3.
An upper bound for the spectral distance. In this section we prove Lemma 10 which establishes Viterbo's conjecture in a special case. The fact that Viterbo's conjecture holds under the additional assumptions of Lemma 10 seems to be well known to experts; we provide a proof here for the sake of completeness.
, and hence γ(φ 1 H (L 0 ), L 0 ), unchanged. Therefore, by cutting H off outside of T * r L and replacing r with 2r we may assume that H is supported inside T * r L \ V . Pick f : L → R to be a Morse function on L whose critical points are all contained in the open set V. Because f has no critical points inside L \ V, we may assume, by rescaling, that (3) df | L\V g 1.
Let β : T * L → R denote a non-negative cutoff function such that β = 1 on T * R L, where R is picked so that R ≫ r. Let F = βπ * f : T * L → R. By picking R to be sufficiently large, we can ensure that, for t ∈ [0, 1] and (q, p) in a neighborhood of T * r L, the Hamiltonian flow of F is given by the formula φ t F (q, p) = (q, p + tdf (q)). This, combined with (3), implies that φ t F (q, 0) / ∈ T * r L for any t > r and any point q ∈ L \ V. Hence, we see that φ t F (L 0 ) is outside the support of H for any t > r. Therefore, ). Using Property (3) from Theorem 13, we obtain that
The main goal of this section is to establish a suitable localized version of Theorem 6 for C 0 -Lagrangians; since we seek localized statements we do not assume that the C 0 -Lagrangians in question are necessarily closed. Not surprisingly, in this new setting Theorem 6 does not hold as stated. The localized results of this section, which have more complicated statements and proofs, are more powerful and they constitute the main technical steps towards proving Theorems 1 and 6.
We prove the analog of the direct implication of Theorem 6 in Section 4.1. The analog of the converse implication is proven in Section 4.2. Since L is a C 0 -Lagrangian, its C 0 -characteristic foliation has a single leaf, which is L itself, and thus the results of this section make no mention of C 0characteristic foliations. Our proof of the above proposition will use the following lemma on the local structure of C 0 -Lagrangians. Lemma 17. Let L ⊂ M denote a C 0 -Lagrangian. Around each point p ∈ L there exists a C 0 -Lagrangian L p ⊂ L and a C 0 -Lagrangian chart (U, θ) such that L p is compactly contained in U , and θ(L p ) ⊂ T ⊂ θ(U ), where T denotes a Lagrangian torus in R 2n .
x i a 2 , y i = 0}). Then, for all k, the projection of θ(L p ) onto the x k coordinate is the interval [− a 2 , a 2 ] which can be completed to a smooth embedded loop, say T k , in the x k , y k plane. Furthermore, we can ensure that T k is contained in the projection of θ(U ) onto the x k , y k plane. We set T = T 1 × · · · × T n .
Proof of Proposition 16. By replacing H by H − c(t) we can suppose that H| L = 0. Apply Lemma 17 to obtain L p as described in the lemma and note that by replacing L with L p we may make the following simplifying assumption: L is a C 0 -Lagrangian covered by a C 0 -Lagrangian chart (U, θ) such that θ(L) ⊂ T ⊂ θ(U ) where T is a (standard) Lagrangian torus in R 2n .
We will now prove the proposition under this simplifying assumption. Let H i : [0, 1] × M → R denote a sequence of smooth Hamiltonians such that H i converges uniformly to H and φ H i converges to φ H in C 0 -topology. Take W ⋐ V to be proper open subsets of U such that Recall that the symbol ⋐ denotes compact containment and V denotes the closure of V . Denote by β : M → R a cutoff function such that β is supported in U, β| V = 1, and β| θ −1 (T )\L = 0. By shrinking V , if needed, we may assume that θ maps the support of β into a Weinstein neighborhood of T . Let G i = βH i and G = βH. Observe that G i converges uniformly to G, G| θ −1 (T ) = 0. We pick ε > 0 such that (4) and (5) we conclude that φ t 0 H (p) / ∈ θ −1 (T ). Hence, we can find a small ball B ⊂ W around p which intersects θ −1 (T ) non-trivially and such that φ t 0 H (B) ∩ θ −1 (T ) = ∅. Hence, for i large enough, we have We picked β such that the Hamiltonians F i all have support in a Weinstein neighborhood of T . Hence, we can pass to T * T , and work with the Lagrangian spectral invariants of the 0-section T 0 , associated to these Hamiltonians; see Section 2.2. From (7) and Lemma 9, we get that Since F i converges uniformly to F , the same inequality must hold for F but this contradicts the fact that F | T = 0.

C 0 -Hamiltonians preserving a Lagrangian are constant on it.
In this subsection, we show that if H ∈ C 0 Ham generates an hameotopy φ H which (locally) preserves a C 0 -Lagrangian L then, the restriction of H to L is (locally) constant. More precisely, We were recently informed by Y.-G. Oh that it is possible to extract the above proposition from [19,Theorem 4.9]; the techniques of [19] are different than ours.
Our proof of the proposition uses the following consequence of Corollary 15. This result can be viewed as a Lagrangian analog of the uniqueness of generators Theorem [10, Theorem 2]. The argument presented here is similar to the proof of the mentioned uniqueness theorem.
Proposition 19. Let L be a smooth closed manifold and {H k } k a sequence of smooth, uniformly compactly supported-that is, there exists a compact K such that ∪ k supp(H k ) ⊂ K-, Hamiltonian functions on T * L so that (1) for all t ∈ [0, 1], γ(φ t H k (L 0 ), L 0 ) converges to 0, and (2) {H k } k uniformly converges to a continuous function H. Then, H restricted to L 0 is a function of time.
Proof. If H| [0,1]×L 0 is not a function of time, there exist t 0 and x + , x − ∈ L 0 such that H t 0 (x + ) > H t 0 (x − ). Up to a shift (and cutoff far from K), we can assume that H t 0 (x + ) = −H t 0 (x − ) = ∆ > 0. Now, notice that there exist δ 0 ∈ (0, 1] and r 0 > 0 such that J δ 0 = [t 0 , t 0 + δ 0 ] ⊂ [0, 1] and that there exist symplectically embedded balls, centered at x ± , B r 0 ± = ι ± (B C n (0, r 0 )), with real part mapped to L 0 , which are disjoint and such that To ensure that (8) holds, choose r 0 and δ 0 small enough such that and (8) follows. For r r 0 , we denote by B r ± ⊂ B r 0 ± the embeddings of the smaller balls, ι ± (B C n (0, r)), and by J δ = [t 0 , t 0 + δ] for δ δ 0 . Note that the previous properties continue to hold for any choice of r r 0 and δ δ 0 . Now, assume that δ δ 0 and r r 0 are small enough such that Let B denote an open neighborhood of p, q which is compactly contained of R 2n by setting them to vanish outside of θ(U ). Recall that θ(L ∩ U ) = {(x i , y i ) | − a < x i < a, y i = 0}; hence we can find a Lagrangian torus T in R 2n such that T ∩ θ(U ) = θ(L ∩ U ). The Hamiltonian F is supported in θ(U ) and it generates the hameotopy θφ t G θ −1 , which preserves θ(L ∩ U ), and hence it preserves the Lagrangian torus T . But F is not constant on T : F (0, θ(p)) = 0, but F (t, x) = 0 for all x / ∈ θ(U ). By shrinking the set U we may assume that θ(U ) is contained in a Weinstein neighborhood of T. Hence, we can pass to T * T and work with Lagrangian spectral invariants of the 0section T 0 associated to the Hamiltonians F i . Furthermore, the Hamiltonians F i are all supported in θ(U ) and hence we can apply Lemma 10 and conclude that, for any r > 0, γ(φ 1 Of course, by the same reasoning we obtain that γ(φ t F i (T 0 ), T 0 ) → 0 for all t ∈ [0, 1]. Then, Proposition 19 implies that F | T = c(t), which contradicts the fact that F | T is not a function of time only.

C 0 -Coisotropic submanifolds and their characteristic foliations
This section is devoted to the proofs of Theorems 1 and 6. We begin by proving a local version of the direct implication of Theorem 6 (Lemma 21 below) from which we deduce Theorem 1. We then prove Corollaries 4, 5 and finally Theorem 6.
Before stating the lemma, recall (Definition 3) that given a C 0 -submanifold C, a C 0 -coisotropic chart around a point p ∈ C is a pair (θ, U ) where U is an open neighborhood of p and θ : U → V ⊂ R 2n is a symplectic homeomorphism which maps C to the standard coisotropic linear subspace Ham (M, ω) with induced hameotopy φ H . Assume that the restriction of H to C only depends on time. Let p ∈ C and θ : U → V be a C 0 -coisotropic chart around p such that θ(p) = 0. Then there exists ε > 0 such that for any t ∈ Before going into the details of the proof of Lemma 20, it is interesting to make the following observation. The lemma holds for coisotropic submanifolds of arbitrary codimension but its proof will follow from the particular case of Lagrangians. As mentioned in the introduction, this is not surprising in view of Weinstein's creed: "Everything is a Lagrangian submanifold!" [27].
Clearly, for all i ∈ {1, . . . , n − k}, Λ i ⊂ C 0 and Let H be as in the statement of Lemma 20. Then for any i, the restriction of H to the C 0 -lagrangians In the particular case of the standard coisotropic linear space C 0 ⊂ R 2n , we can prove a similar statement for all times. We will use it to prove Theorem 1.
Ham (R 2n , ω 0 ) be a compactly supported continuous Hamiltonian whose restriction to C 0 is a function of time. Then for every t ∈ [0, +∞), φ t H (0) ∈ F 0 (0). Proof. Assume this does not hold. Then t 0 = inf{t > 0 | φ t H (0) / ∈ F 0 (0)} is a well-defined non-negative real number. Since F 0 (0) is closed, it contains p = φ t 0 H (0). According to Lemma 20 applied to the continuous Hamiltonian K t = H t+t 0 , the point p, and the chart which is just the translation by −p, which therefore contradicts the definition of t 0 .
The proof of Theorem 1 relies on the above lemma and the following characterization of coisotropic submanifolds and their characteristic foliations: A submanifold is coisotropic if and only if the flow of every autonomous Hamiltonian constant on it preserves it. Moreover, the leaf through a point p is locally the union of the orbits of p under the flows of all such Hamiltonians.
The next lemma is based on this characterization.

Lemma 22.
Let C be a submanifold in a symplectic manifold (M, ω). Assume that every point p ∈ C admits an open neighborhood V such that any H ∈ C ∞ c (V ), with H| C ≡ 0, satisfies φ t H (p) ∈ C for every t ∈ [0, +∞). Then C is coisotropic.
Moreover, for such a neighborhood V , there exists a smaller neighborhood W ⋐ V such that, the leaf F(p) of the characteristic foliation of C passing through p satisfies Let p ∈ C and let V be an open subset as in the statement of the lemma. Assume that C coincides locally with f −1 1 (0) ∩ . . . ∩ f −1 k (0) for some smooth functions f 1 , . . . , f k whose differentials are linearly independent at p. By multiplying by an appropriate cutoff function, we can assume that these functions are defined everywhere on M , have compact support in V , and vanish on C.
The Hamiltonian vector fields at p of f 1 , . . . , f k span (T p C) ω , and by assumption belong to T p C. Thus (T p C) ω ⊂ T p C and C is coisotropic. Now, since the characteristic leaves are preserved by smooth Hamiltonians constant on C, we have the inclusion Since, k i=1 v i f i is constant on C, its flow preserves the characteristics, hence F takes values in the characteristic leaf F(p) through p. The partial derivatives of F at 0 are ∂ v i F (0) = X f i (p) and in particular they are linearly independent and span T p F(p) = (T p C) ω . The inverse function theorem then shows that F is a diffeomorphism from a neighborhood of 0 to a neighborhood W of p in F(p). This shows We believe that the above special cases provide the reader with the opportunity to get an idea of the proofs of Theorems 1 and 6 without having to go through the technical details of Sections 4 and 5.
We will first show that Theorem 23 follows from Theorem 24. In order to do so we will need the following dynamical characterizations of isotropic and coisotropic submanifolds, respectively. We leave the proofs of the above lemmas, which follow from symplectic linear algebra, to the reader. In the proof of Theorem 1, we use Lemma 22 which is a variation of the second of the above two lemmas.
Proof of Theorem 23. Each of Lemmas 25 and 26 gives a different proof. We provide both proofs here. First proof: Suppose that H is any smooth Hamiltonian whose flow φ H preserves L ′ . Then H • θ ∈ C 0 Ham and its flow, θ −1 φ t H θ, preserves L 0 . It follows from Theorem 24 that the restriction of H • θ to L 0 is a function of time only. Therefore, H| L ′ depends on time only, and so using Lemma 25 we conclude that L ′ is isotropic. Second proof: Suppose that H is any smooth Hamiltonian whose restriction to L ′ is a function of time only. Then H • θ ∈ C 0 Ham and its restriction to L 0 depends only on time. It follows from Theorem 24 that the flow of H • θ, which is θ −1 φ t H θ, preserves L 0 and so the flow of H preserves L ′ . Using Lemma 26 we conclude that L ′ is coisotropic.
Proof of Theorem 24. By replacing H with H •θ we may prove the statement for the (smooth) Lagrangian L 0 , rather than for L ′ . Here, we use the fact that φ t H•θ = θ −1 φ t H θ. Hence, we will prove that the restriction of H ∈ C 0 Ham to L 0 is a function of time if and only if φ H preserves L 0 .
To prove the direct implication suppose that H t | L 0 = c(t), where c(t) is a function of time only. By replacing H t with H t − c(t) and then cutting of at infinity, we may assume that H t | L 0 = 0 for all t. For a contradiction assume that φ H does not preserve L 0 and define t 0 = inf{t | φ t H (L 0 ) ⊂ L 0 }.
Proof. Since the statement is local, it is sufficient to prove it when N is an open set of R n . Then α can be written as α = n i=1 p i (x)dx i , where x 1 , . . . , x n are the canonical coordinates in R n and p 1 , . . . , p n continuous functions on N . The fact that α is closed is equivalent to the equations (10) ∀i, j ∈ {1, . . . , n}, ∂ j p i = ∂ i p j , where ∂ i p j is the i-th partial derivative of p j in the sense of distributions. We use convolution to approximate α. To that end, take a compactly supported function ρ such that ρ 0, and N ρ(x)dx = 1 and set ρ ε (x) = 1 ε ρ(εx) for every ε > 0. For any continuous function f on N , the functions f * ρ ε (x) = N f (y)ρ ε (x − y)dy are well-defined on any compact subset of N for ε small enough. Moreover, for any ε, f * ρ ε is smooth, converges locally uniformly to f as ε goes to 0, its differential satisfies d(f * ρ ε ) = (df ) * ρ ε and converges in the sense of distributions to df .
Let U ⋐ N be an open subset of N . Then, for ε small enough, is a well-defined 1-form on U . It satisfies Equations (10) and thus is closed. Moreover, it converges uniformly to α on U . Now let φ ε be the family of symplectic diffeomorphisms of T * U defined by φ ε (x, p) = (x, p + α ε (x)). They converge uniformly on U to the symplectic homeomorphism φ : T * U → T * U , (x, p) → (x, p + α(x)) and graph(α) restricted to T * U is φ(U ). This shows that graph(α) is locally the image of a smooth Lagrangian by a symplectic homeomorphism.