Hausdorff limits of submanifolds of symplectic and contact manifolds

We study sequences of immersions respecting bounds coming from Riemannian geometry and apply the ensuing results to the study of sequences of submanifolds of symplectic and contact manifolds. This allows us to study the subtle interaction between the Hausdorﬀ metric and the Lagrangian Hofer and spectral metrics. In the process, we get proofs of metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral norm. We also get C 0 -rigidity results in the vein of the Gromov-Eliashberg theorem for a vast class of important submanifolds of symplectic and contact manifolds — even when no Riemannian bounds are present.


I. Introduction
The main purpose of the present paper is to explore the relation between the Hausdorff metric and many of the metrics appearing in symplectic topology, e.g. the Lagrangian Hofer metric, the spectral metric and the recent shadow metrics of Biran, Cornea and Shelukhin [BCS21].More precisely, we want to explore the topology that these metrics induce on a given collection of Lagrangian submanifolds of a fixed symplectic manifold.In other words, we are interested in studying how convergence of a sequence in one metric affects the behavior of this sequence in the other ones.The reasoning behind the introduction of the Hausdorff metric into the list of considered metrics is twofold: (1) Contrary to the other metrics, its properties are well-known and easy to derive.
(2) It is defined on any choice of collection of (closed) Lagrangian submanifolds.
The second point is of particular interest to us, as the shadow metrics are for example defined on Lagrangian submanifolds which might not even be of the same homotopy type.
This exploration was started in the author's previous work [Cha21].As noted in said work, there is however an obvious problem with introducing the Hausdorff metric: in full generality, there is no relation between the Hausdorff metric and the metrics coming from symplectic topology.Nonetheless, when one only consider sequences respecting certain bounds coming from an auxiliary Riemannian metric, the behavior on each side become intimately related.This is because such bounds essentially stop sequences from Hausdorff-converging to pathological spaces.It is however the hope that it does not greatly restrict the possible limits in the other metrics.
In our previous work, we studied sequences of Lagrangian submanifolds converging in some nice metrics coming from symplectic topology, and we proved that they must also converge in the Hausdorff metric.We now turn to the opposite problem: if we have a Hausdorff-converging sequence of Lagrangian submanifolds, what can we say of its behavior in those nice metrics coming from symplectic topology.
Theorem A If {L i } is a Riemannianly-bounded sequence of exact Lagrangian submanifolds in T * L which Hausdorff-converges to the image of the zero section, then L i is the graph of a 1-form for i large enough.
In fact, we prove the statement for a slightly weaker hypothesis than convergence in the Hausdorff metric.This gives as a corollary metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral metric, as we shall see below.
Furthermore, through a direct computation, this implies that {L i } also converges to L in the Lagrangian Hofer metric.Therefore, in the exact case, convergence in the Hausdorff metric is the same thing as convergence in a nice metric coming from symplectic topology when Riemannian bounds are present.Indeed, all known such metrics are bounded from above by the Lagrangian Hofer metric, and thus convergence in the later metric implies convergence in the other ones.This thus give a complete characterization of a small-enough neighborhood of an exact Lagrangian submanifold in any of these nice metrics when Riemannian bounds are present: they are just graph deformations of the submanifold.
As we shall see below, the exactness condition on {L i } is necessary, as L i could be a nontrivial covering over L without this condition.Note however that even without the exactness condition, there are still rigidity phenomena which are not present for non-Lagrangian submanifolds.Therefore, the present result is truly in the realm of symplectic topology.
We also analyze the type of possible limits that a Hausdorff-converging sequence of Lagrangian submanifolds might have.In particular, this allows us to even better understand limits in those nice metrics coming from symplectic topology when Riemannian bounds are imposed.We show that when Riemannian bounds are present, the limit must be the image of a Lagrangian immersion, although some regularity might be lost in the process.In fact, the techniques that we use there apply to not just Lagrangian submanifolds, but also a large class of important submanifolds of symplectic and contact manifolds.
Theorem B Hausdorff limits of sequences of (co)isotropic submanifolds of a given symplectic or contact manifold -respecting appropriate Riemannian boundsare C 1 -immersed (co)isotropic submanifolds.Furthermore, in the Lagrangian case, exactness, weak exactness and monotonicity are preserved in the limit when said limit is smoothly embedded.We will also explore the possible Hausdorff limits with lighter Riemannian bounds, and even no Riemannian bounds at all.The later exploration is of particular interest to C 0 -symplectic topology, as it proposes a new possible definition of what a C 0 -Lagrangian submanifold should be.

I.a. Precise statements
Throughout the paper, we fix a complete Riemannian manifold (M, g).We will consider immersions f : N M , where N is closed and connected.We denote by B f its associated second fundamental form and by Vol(f ) the volume of N with respect to the metric f * g.For k ∈ N, Λ ∈ [0, ∞), and V ∈ (0, ∞), we consider I k (Λ, V ), the space of such immersions f : N M , where dim N = k, In what follows, we will often take M = T * L, where L is a closed connected Riemannian n-manifold, and T * L is equipped with the associated Sasaki metric.Using this notation, we can give a precise statement for Theorem A.
Theorem 1 Let Λ ≥ 0 and V > 0. Let {f i : L i ֒→ T * L} ⊆ I n (Λ, V ) be a sequence of exact Lagrangian embeddings.Suppose that f i (L i ) sits in the codisk bundle D * r i L of radius r i and that {r i } tends to 0. Then, f i (L i ) is the graph of a 1-form for i large enough.
The main tool in the proof of this result is a theorem of Shen [She95] proving some sort of precompacity result for I k (Λ, V ).In fact, Shen's result proves that I k (Λ, V ) can be naturally compactified using C 1,α -immersions, for any α ∈ (0, 1).Together with the fact that the projection f i (L i ) → L must be a homotopy equivalence [AK18], this gives Theorem 1.
The main application of this result is in proving metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral norm.
Corollary 1 Let L be a closed connected Riemannian n-manifold.There exist constants A = A(L) > 0 and R = R(L, Λ, V ) > 0 with the following property.Let L ′ be an exact Lagrangian submanifold of the codisk bundle Then, (i) L ′ is Hamiltonian isotopic to the zero section; (ii) the spectral norm respects the inequality Remark 1: We briefly review the advances made in proving both conjectures in full generality, i.e. without any Riemannian bounds.
(1) The nearby Lagrangian conjecture, i.e. (i) in Corollary 1, is only known when L is S 1 (folklore), S 2 (follows from work of Hind [Hin04]), RP 2 (follows from work of Hind, Pinsonnault, and Wu [HPW13]), and T 2 (proved by Dimitroglou Rizell, Goodman and Ivrii [RGI16]).However, great advancement has been made in proving the conjecture in full generality.As noted before, it is known that the canonical projection π : . Likewise, L ′ and L must be isomorphic objects in the Fukaya category of T * L when L is spin [FSS08a,FSS08b,Nad09].
(2) The Viterbo conjecture on the spectral norm, i.e. (ii) in Corollary 1, has recently been proven for Zoll symmetric spaces and so-called string-invertible spaces by Shelukhin [She18,She19], and for a large class of manifolds -which includes homogeneous spaces of compact Lie groups -by Viterbo [Vit22].
Another consequence of Theorem 1 is in completing the author's previous work on the equivalence of the topologies induced by various metrics on an appropriate space of Lagrangian submanifolds.
Corollary 2 Let λ be a Liouville form of the exact symplectic manifold M , and let {L i ⊆ M } be a sequence of λ-exact Lagrangian submanifolds such that the inclusions are in some fixed I n (Λ, V ).If {L i } Hausdorff-converges to a (smooth) λ-exact Lagrangian submanifold L 0 , then it also converges to L 0 in the Lagrangian Hofer metric.In particular, it also converges to L 0 in the spectral norm and in any shadow metric.
In fact, as we shall see below, the exactness requirement on the limit is superfluous, as the Hausdorff limit of a sequence of λ-exact Lagrangian immersions is automatically itself exact for the same Liouville form.
In fact, Shen's result allows us to explore sequences of other important type of submanifolds in symplectic and contact topology, not just Lagrangian ones.Indeed, since the result gives some sort of compacity in C 1,α -topology, 0 < α < 1, many of these properties are preserved in the limit.This leads to rigidity results in the vein of the celebrated theorem of Gromov and Eliashberg on the C 0 -closedness of the group of symplectomorphisms in the group of diffeomorphisms [Gro86,Eli87].
Theorem 2 Let {f i : L i M } ⊆ I k (Λ, V ) be a sequence of Lagrangian (resp.isotropic, coisotropic, Legendrian, contact isotropic, or contact coisotropic) immersions.Suppose that {f i (L i )} Hausdorff-converges to a closed subset N .Then, N is the image of a Lagrangian (resp.isotropic, coisotropic, Legendrian, contact isotropic, or contact coisotropic) C 1,α -immersion f 0 : L 0 M , where L 0 is closed and connected (and of dimension k).Furthermore, if the f i 's are exact for some Liouville form λ, then there is a Likewise, if the f i 's are weakly exact (resp.monotone with monotonicity constant ρ > 0), and f 0 is an embedding, then f 0 is weakly exact (resp.monotone with monotonicity constant ρ).
Remark 2: This result is an extension of previously known result on sequences of Lagrangian submanifolds and of symplectomorphisms.
(1) It can be seen as a generalization of Laudenbach and Sikorav's result [LS94] on the C 0 -rigidity of Lagrangian embeddings (under some technical assumptions).This result was recently upgraded to general Lagrangian submanifolds and to a class of nice Legendrian submanifolds by Nakamura [Nak20].The great improvement here is that we allow embeddings -and in fact, even immersions -of varying domains; the price to pay are bounds coming from Riemannian geometry.We will however later explain how we can partially get rid on the volume bound, and we will show some C 0 -rigidity result without any type of Riemannian bounds.
(2) Likewise, note that when f i is the inclusion of the graph of some symplectomorphism ψ i : Furthermore, uniform C 2 -bounds on {ψ i } implies the existence of a C 1 -converging subsequence by the Arzela-Ascoli theorem, and thus the limit ψ is a C 1 -symplectomorphism.However, such bounds also implies that {f i } is in I n (Λ, V ) for some Λ ≥ 0 and V > 0. Therefore, Theorem 2 can also be seen as a generalization of that simple fact.

I.b. Organization of the paper
The rest of the paper is divided in two parts.The first one is mainly concerned with proving Theorem 1 and exploring the rigidity phenomenon underlying it.More precisely, we begin by studying sequences of not-necessarily-Lagrangian immersions which behave well in the codomain (Subsection II.a), then we prove Theorem 1 (Subsection II.b), and we end by showing that this is truly a Lagrangian phenomenon (Subsection II.c).The second part is not only concerned with proving Theorem 2, but also with relating it to the author's previous work (Subsection III.b), extending parts of it to the case V = ∞ (Subsection III.c), and exploring rigidity phenomena beyond Riemannian bounds (Subsection III.d).
Acknowledgements: This research is part of my PhD thesis and was financed by a NSERC and a FRQNT scholarship.I would like to thank my advisor, Octav Cornea, for his continued interest in my research and for the many insightful discussions that we have had.I would also like to thank Sobhan Seyfaddini for his observations regarding C 0 -converging sequences of Hamiltonian diffeomorphisms.Finally, I am indebted to Dominique Rathel-Fournier for sharing his insight on coverings and pointing out to me Polterovitch's construction of nontrivial Lagrangian coverings in the cotangent bundle of flat manifold.

II. Sequences of immersions
The focus of this section is the proof of Theorem 1 and the study of the rigidity of Lagrangian embeddings in general.We thus begin with a general study of sequences of immersions, then we apply this new knowledge to Lagrangian embeddings specifically, and finally explore how this is truly a Lagrangian phenomenon.
As mentioned above, we will make great use of Shen's result on sequences of immersions.Therefore, we thought that it could be useful for the reader to simply write the explicit statement here before moving on.
Then, we have the following: (i) a subsequence, still denoted {f i }; Essentially, this theorem is the appropriate generalization to (immersed) submanifolds of the Gromov-Hausdorff compactness theorem [GLP81,Kat85] for Riemannian manifolds with uniformly bounded sectional curvature and injectivity radiusthe latter bound comes from the bound on the second fundamental form in the submanifold case.

II.a. Immersions with converging images
In this subsection, we use Shen's theorem to study sequences of immersions in I k (Λ, V ) which have images behaving well with respect to the Hausdorff metric of M .
For a closed submanifold N of M and r > 0, denote by B r (N ) its tubular neighborhood of size r.In this section, we will suppose that there is a sequence {r i } ⊆ R >0 converging to 0 such that In other words, if we define for subsets Remember that the Hausdorff metric of M is given by δ H (A, B) = max{s(A, B), s(B, A)}.Thus, this condition is a priori strictly weaker than Hausdorff convergence to N .
Lemma 1 If Property ( †) holds, then {f i (N i )} converges to N in the Hausdorff metric.
Proof: Property ( †) implies that there exists a Hausdorff-converging subsequence {f i (N i )} and that such a sequence must have as limit a subset E ⊆ N .On the other hand, there exists by Shen's theorem yet another subsequence {f i } which C 1,α ′ -converge to a C 1,α -immersion f 0 : N 0 M .We must then have f 0 (N 0 ) = N .Indeed, we may see f 0 as an immersion into N .But by the inverse function theorem, this immersion is open.Since N 0 is compact, f 0 is also closed.Therefore, E = f 0 (N 0 ) is clopen; it must thus be equal to N .Suppose now that {f i (N i )} did not converge to N .Then we would have a subsequence {f i (N i )} such that s(N, f i (N i )) ≥ ε for some ε > 0. We then get a contradiction by passing to a converging subsequence {f i (N i )}, since we have just proven that its Hausdorff limit must be N .
Lemma 1 allows us to prove the main technical result of this subsection.
Denote by ι : N ֒→ M the inclusion of the limit manifold.For all ε > 0, there exists I ∈ N such that if i ≥ I, then there exists a covering p i : Proof: Take a C 1,α ′ -converging subsequence {f i } -which exists by Shen's result -and denote by f 0 : N 0 M its limit.By Lemma 1, the image of f 0 is N .We thus have the commutative diagram by inverting ι on the image of f 0 .
Note that p is a surjective submersion between compact spaces.Therefore, by Ehresmann's fibration theorem, p must be a locally trivial fibration.For dimensional reasons, p must thus be a covering.We can then take This thus implies the result for any converging subsequence.Suppose that the statement is not true for the sequence {f i } itself.Then, we get a subsequence {f i } such that d C 1,α ′ (f i , ι • p) ≥ ε for some ε > 0 and for all coverings p : N i → N .Passing to a converging subsequence, we clearly get a contradiction.
Remark 3: Of course, p i must be a diffeomorphism whenever N is simply connected.However, in full generality, it is entirely possible for p i to be a covering -even in the nicest of cases.For example, one can consider the sequence of embeddings Clearly, this is a sequence having Property ( †) for N = T 2 × {0}.Furthermore, a direct computation gives that this sequence is in ).However, the associated map p :

II.b. Proof of Theorem 1
We now apply the above results to Lagrangian submanifolds and prove metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral norm.Therefore, from now on, we suppose that M = T * L for some ndimensional closed connected Riemannian manifold L. We equip T * L with the standard symplectic form, almost complex structure and metric.
We need to prove the Theorem 1.Note that it gives us a proof of Corollary 1 right away; the proof of Corollary 2 will only be given in Subsection III.a.
Proof of Corollary 1: Let {f i } be as in Theorem 1.Since f i (L i ) is an exact Lagrangian graph, it must be the graph of an exact 1-form dh i .We take the vector field X i defined via ι X i ω = βπ * dh i , where β is a compactly supported bump function which is identically 1 on B r 1 (L) = D * r 1 L. Here, π : T * L → L denotes the canonical projection.Then, X i generates a compactly supported Hamiltonian isotopy sending the zero section to f i (L i ).
The fact that f i (L i ) is a graph also implies that the Floer complex CF (f i (L i ), T * x L) has only one generator for any x ∈ L. In particular, its boundary depth is zero.Therefore, by work of Biran and Cornea [BC21], we also get γ(L, f i (L i )) ≤ A. Here, A is the constant appearing in the work of Biran and Cornea associated to The proof concludes by contradiction: if a R ≤ 1 as in the theorem did not exist, we would then have a sequence of exact Lagrangian embeddings respecting Property ( †), but not respecting the conclusions of the theorem.This would be a contradiction with the above paragraph.
We now give a proof of Theorem 1.
Proof of Theorem 1: By work of Abouzaid and Kragh [AK18], the composition L i ֒→ T * L → L is a (simple) homotopy equivalence.In particular, it is an isomorphism on the fundamental group.
On the other hand, by Proposition 1, f i must be transverse to every fiber for i large enough.Therefore, π| f i (L i ) must be a covering onto L. However, by the above paragraph, the covering is trivial, i.e. π| f i (L i ) is a diffeomorphism.Therefore, f i (L i ) must be the graph of a 1-form for i large enough.
Remark 4: (1) The proof of Theorem 1 applies for any simply connected complete Riemannian manifold N .We then get that f i (N i ) is the graph of a section of the normal bundle of N in M .More generally, without any topological assumption on N , f i (N i ) admits a lift N i in the normal bundle of N in f * 0 T M , and this lift is the graph of a section of that normal bundle.Note that this is true even when neither f i nor the limit f 0 is an embedding.
(2) As we have seen in Remark 3 however, Theorem 1 is not true for non-Lagrangian embeddings.In fact, Theorem 1 is typically not even true for nonexact Lagrangian submanifolds.Indeed, Polterovich [Pol90] constructed for any closed flat manifold W = T n totally geodesic Lagrangian tori in T * W having the property that the composition T n ֒→ T * W → W is a nontrivial cover.These tori may be taken to be arbitrary close to the zero section.We will explore this kind of phenomenon in more details in the next subsection.

II.c. Rigidity of Lagrangian embeddings
As we are studying Riemannian and symplectic phenomena at the same time, it can be hard to parse what comes from the Riemannian bounds and what comes from the Lagrangian condition.In fact, it could a priori be the case that a result analogous to Theorem 1 exists for an appropriate class of non-Lagrangian submanifolds.Indeed, as noted in Remark 4, it seems that the exactness condition -not just the Lagrangian condition -is required for the result.In this subsection, we thus want to dispel the idea that this could be a non-Lagrangian phenomenon.
We begin by exploring some basic properties of Lagrangian embeddings.Proof: One implication is of course trivial.Suppose therefore that b 1 By the condition on the Betti numbers, it is thus an isomorphism.
Let σ ′ := f * λ, where λ is the canonical 1-form on T * L. By the above paragraph, there exists a 1-form σ on L such that (π Let {ψ t } be the symplectic isotopy generated by the vector field X defined by ι X ω = −βπ * σ, where β is an appropriate bump function.We then have L is an exact Lagrangian embedding.As previously noted, π • ψ 1 • f must then be an isomorphism on the fundamental group.Therefore, the same holds for π Combining Propositions 1 and 2, we directly get the following result. Corollary 3 Let L be a closed connected Riemannian manifold such that any finite covering L ′ → L is such that b 1 (L ′ ) = b 1 (L), e.g.π 1 (L) is free, abelian free, or finite.Let Λ ≥ 0 and V > 0. There exists R > 0 such that whenever f ∈ I n (Λ, V ) is a Lagrangian embedding with image in D * R L, then said image is symplectomorphic to the zero section.
Note that the equivalent result in the smooth category is entirely false, as we have seen in Remark 3 with the 2-torus.Therefore, the introduction of Riemannian bounds truly allows to capture some symplectic rigidity, even when just considering Hausdorff-converging sequences.
However, the rigidity goes further than this.Indeed, the main motivation behind the study of Hausdorff-converging sequences is its importance when studying sequences converging in metrics coming from symplectic topology (c.f.[Cha21]).Therefore, there is another rigidity question that crops up: does Theorem A of [Cha21] holds for non-Lagrangian submanifolds.Of course, such a question makes no sense for most metrics coming from symplectic topology.One exception to this rule is however the Hofer pseudometric [Che00], which makes sense for any submanifolds.
In other words, for a n-dimensional submanifold N of a 2n-dimensional symplectic manifold M , we can define whenever N ′ is Hamiltonian isotopic to N .Here, || • || H denotes the Hofer norm.Let {N i } is a sequence of non-Lagrangian submanifolds converging in d H to N 0 and such that the inclusion N n ֒→ M is in I n (Λ, V ) for some Λ and V .Does {N i } behave like in the Lagrangian case, i.e. must N i → N 0 in δ H ?An obvious obstruction to that being the case is if d H is degenerate on the Hamiltonian orbit of N .However, by work of Usher [Ush14], this is precisely the case whenever N is non-Lagrangian.In fact, d H (N, •) ≡ 0 whenever N is nowhere Lagrangian, i.e. ω| TxN = 0 for all x ∈ N .Furthermore, the set of nowhere Lagrangian embeddings N ֒→ M is residual in the C ∞ topology and open in the C 2 topology whenever n ≥ 2, i.e. whenever there are non-Lagrangian n-dimensional submanifolds.Therefore, d H is generically very much degenerate in the non-Lagrangian case.
This thus shows that at every step of the process, introducing Riemannian bounds does not reduce the Lagrangian case to the general one, but rather shows some new type of symplectic rigidity.
Remark 5: The submanifold N ′ such that d H (N, N ′ ) = 0 that we find is in I n (Λ ′ , V ′ ) for some Λ ′ ≥ Λ and V ′ ≥ V , but not necessarily in I n (Λ, V ).This flexibility in the choice of a constant is however necessary to study symplecticand not Riemannian -rigidity phenomena.For example, I n (0, V ) is made out of totally geodesic submanifolds, and we should expect some very strong rigidity, whether N is Lagrangian or not.

III. Hausdorff limits of sequences of certain submanifolds
In this section, we prove rigidity results for sequences of certain submanifolds of symplectic and contact manifolds.These results are shown mostly in the presence of Riemannian bounds, but some still hold without their presence.We also use this opportunity to prove Corollary 2 and relate it to the author's previous work.

III.a. Proof of Theorem 2
In this subsection, we prove the various parts of Theorem 2. In order to make the presentation smoother, we however instead present it as a series of simpler results.
Lemma 2 Let {f i : L i M } ⊆ I k (Λ, V ) be a sequence of isotropic immersions of a symplectic manifold (M, ω) or of a contact manifold (M, ξ).Suppose that {f i (L i )} Hausdorff-converges to a closed subset N .Then, N is the image of a k-dimensional isotropic C 1,α -immersion f 0 : L 0 M , where L 0 is closed and connected.
Note that we recover the Lagrangian case when M is symplectic and k = 1 2 dim M , and the Legendrian case when M is contact and k = 1 2 (dim M − 1) Proof: Suppose that M is symplectic.Passing to a subsequence, we have diffeomorphisms , where L 0 is closed and connected.Since the sequence {f i (L i )} must also converge to f 0 (L 0 ).Therefore, we must have f 0 (L 0 ) = N since f 0 (L 0 ) is compact, and thus closed.Finally, 0 = f * i ω converges in the C 0,α ′ -topology to f * 0 ω.Therefore, f * 0 ω = 0, and f 0 is isotropic.
Suppose now that M is contact with contact form α.Then, the proof is analogous to the symplectic case: it suffices to replace ω by α in the proof above.If M does not have a contact form, i.e. if ξ is not coorientable, every point p still has a neighborhood U p onto which ξ = Ker α.Since {f i • ϕ i } and its first order derivatives uniformly converge to f 0 on all compact subsets of f −1 0 (U p ), the same argument still works.

Lemma 3 Let {f
) be a sequence of coisotropic immersions of a 2n-dimensional symplectic manifold (M, ω) or of a co-oriented (2n + 1)dimensional contact manifold (M, ξ = Ker α).Suppose that {f i (L i )} Hausdorffconverges to a closed subset N .Then, N is the image of a (n + k)-dimensional coisotropic C 1,α -immersion f 0 : L 0 M , where L 0 is closed and connected.
We recall that f is the symplectic complement of a vector space V ⊆ T y M .Following Huang [Hua15], we then say that f : for all x ∈ L. Note that this definition depends only on ξ, not on the precise contact form α chosen.
Proof: Suppose that M is symplectic.As in Lemma 2, we have {ϕ i } and As before, we have C 0,α ′ -convergence of {σ i } to the C 0,α -form σ 0 = f * 0 ω.Since the rank of matrices is lower semicontinuous, we must have dim Ker σ 0,x ≥ dim Ker σ i,x = n − k for all x ∈ L 0 .However, since f 0 is an immersion and ω is nondegenerate, dim Ker σ 0,x ≤ n − k.Therefore, f 0 is coisotropic.
When M is contact, the proof is analogous, but the condition is instead that Ker . Therefore, the proof goes through as before.
Remark 6: The proof of Lemma 3 showcases well why the equivalent statement for symplectic submanifolds -or contact submanifolds -does not hold: the limit immersion might develop some degeneracy.For example, a generic nonsymplectic perturbation of the zero section of T * L will be symplectic, even though the zero section itself is of course Lagrangian.
We now go back to the symplectic isotropic case and show some additional rigidity when additional conditions are imposed on the immersions.Lemma 4 If the f i 's of Lemma 2 are exact for some Liouville form λ on a symplectic manifold (M, ω = dλ), then the immersion f 0 : L 0 M may be chosen so that there is a C 1,α function h 0 : L 0 → R with f * 0 λ = dh 0 .
Proof: By hypothesis, each L i has a function h i : L i → R such that f * i λ = dh i .These functions are unique up to a constant.Therefore, we may fix x ∈ L 0 and take h i such that h i (ϕ i (x)) = 0, where the ϕ i 's are the diffeomorphisms of Shen's theorem.Take In particular, the first order derivatives of h ′ i are uniformly C 0,α ′ -bounded.Furthermore, for any y ∈ L 0 , we have that where γ is a unit-speed minimizing geodesic segment from x to y, and || • || denotes here the supremum over L 0 of the operator norm.Since ||dh ′ i || is uniformly bounded, the image of all h ′ i 's is contained in some compact interval I. Therefore, {h i } is contained in a finite closed ball in C 1,α ′ (L 0 , I) for α ′ < α.By compactness of this ball in the C 1,α ′ topology, we may pass to another subsequence which C 1,α ′ -converges to a C 1,α -function h 0 : L 0 → R. Taking the limit of (f ′ i ) * λ = dh ′ i on both sides, we have the relation f * 0 λ = dh 0 .
This proof shows the importance of working with Hölder spaces: if we only had uniform C 0 -bounds on dh ′ i , then we would only know that h 0 is continuous, and that {h ′ i } uniformly converges to h 0 .In particular, the relation f * 0 λ = dh 0 would not necessarily hold.
We now turn our attention to sequences of weakly exact or monotone Lagrangian submanifolds.Contrary to what preceded, these results employ results from Section II in an essential way.
Proposition 3 If the f i 's of Lemma 2 are weakly exact Lagrangian embeddings or monotone Lagrangian embeddings with uniform monotonicity constant ρ > 0, then so is f 0 whenever it is an embedding.
Proof: For ease of notation, we will identify f i (L i ) with L i , i ≥ 0, and see f i simply as an inclusion.Suppose that the L i 's are weakly exact.By Proposition 1, there are finite coverings p i : L i → L 0 such that f i may be C 1,α ′ -approximated by f 0 • p i .In fact, by the proof of the proposition, we may pass to a subsequence so that the isomorphism type of p i is constant.Let u : D → M be a disk with boundary along L 0 and symplectic area ω(u).
Suppose that γ := u| ∂D=S 1 admits a lift γi to L i -since the isomorphism type of the covering is constant, this is independent of i.
However, we have that ω(v i ) → 0. To see this, we could for example equip M with a metric which corresponds with the Sasaki metric on the previously-mentioned Weinstein neighborhood of L 0 .Then, we get ω(v i ) ≤ Area(v i ), which obviously tends to 0. Therefore, we must have ω(u) = 0.If γ does not admit a lift, there is some k ≥ 2 such that γ k = u k | S 1 does.Indeed, p i : L i → L 0 is a finite covering, and thus p * (π 1 (L i )) has finite index.But then, 0 = ω(u k ) = kω(u), which gives the result.Suppose now the L i 's are monotone with uniform monotonicity constant ρ > 0, i.e. ω = ρµ, where µ : π 2 (M, L i ) → Z is the Maslov index of L i .The proof then goes through similarly as before.Indeed, when there is a lift γi , then we must have µ(v i #u) = µ(u).This is because µ(v) depends only on the homotopy class of the path t → T v(e it ) L ⊆ R 2n in the Lagrangian Grassmannian.However, v i gives precisely a homotopy from the path associated to u to the one associated to v i #u.Therefore, we have that This again gives the result since ω(v i ) → 0. When there is no lift, the result also follows similarly as before: and k > 0.
We now finally prove Corollary 2 while making use of the notation used so far.
Proof: Using a Weinstein neighborhood, we may assume without loss of generality that L i is an exact Lagrangian submanifold in T * L 0 , and the f i 's are simply inclusions.Then, L i Hausdorff-converges to the zero section, and f 0 is the natural inclusion L 0 ֒→ T * L 0 .By Theorem 1, L i is the graph of some exact 1-form dh i for i large enough.
Consider H i := βπ * h i , where π : T * L 0 → L 0 is the natural projection, and β is a bump function equal to 1 in some codisk bundle D * r L 0 containing all L i for i large and equal to 0 outside some other codisk bundle D * R L 0 .Then, the Hamiltonian diffeomorphism that it generates sends L 0 to L i .Therefore, by the definition of the Lagrangian Hofer metric.By compactness of L 0 , there are points x i , y i ∈ L 0 where h i attains its maximum and minimum, respectively.We may then take a unit-speed minimizing geodesic γ i from x to y.Then, However, ||dh i || = s(L i , L 0 ) when T * L 0 is equipped with the Sasaki metric, because L i = graph dh i .Since convergence in the Hausdorff metric is independent on the distance function, and since we know that δ H (L i , L 0 ) → 0 in some distance function, then ||dh i || → 0. Therefore, d H (L 0 , L i ) → 0.
Remark 7: (1) Following Remark 4, there is an analogous statement for immersions if we instead consider L 0 in its normal bundle in f * 0 T M .Then, a neighborhood of L 0 can be identified with a neighborhood of the zero section of T * L 0 using an ω-compatible almost complex structure.
(2) In light of the rigidity of the Hofer metric for coisotropic submanifolds [Ush14], we expect that a similar result also holds for them under adequate conditions.Said conditions are however unclear for the time being.
(3) Likewise, we expect a similar result for Legendrian submanifolds -or more generally contact coisotropic submanifolds -with the Shelukhin-Hofer pseudometric as defined by Rosen and Zhang [RZ18], based on work of Shelukhin [She17].
(4) Note that that Lemma 2 implies that {f i (L i )} also converges to L 0 in the spectral metric.However, we could have gotten this result directly from Corollary 1 via a rescaling argument à la Shelukhin [She18].

III.b. The tame and bounded volume conditions
In this subsection, we explore the relation between the ε-tameness condition of the author's previous work [Cha21] and the condition of having bounded volume.
We recall that a submanifold N of a Riemannian manifold (M, g) is said to be ε-tame, 0 < ε ≤ 1, if for all x = y ∈ N .Here, d M denotes the metric on M induced by the Riemannian metric g, whilst d N denotes the metric on N induced by the restriction g| T N of g to N .
Proposition 4 Take Λ ≥ 0 and ε ∈ (0, 1].Let K ⊆ M be compact.There exists a constant We leave the proof of the proposition for later and give an application of the result when combined with Theorem 2.
Corollary 4 Let dF,F ′ be a J-adapted metric such that is totally disconnected and which is bounded from above by the Lagrangian Hofer metric d H .Then, for any compact K ⊆ M , dF,F ′ induces the same topology on L e Λ,ε (K) as the Hausdorff metric.If V > 0, then the same result holds on the subset of L e Λ,ε (M ) composed of Lagrangian submanifolds having volume bounded from above by V , whether M is compact or not.
We refer to the author's previous work for the precise definition of what a J-adapted metric is.Note however that the Lagrangian Hofer metric, the spectral metric, and all shadow metrics are J-adapted for all J. Here, L e Λ,ε (K) denotes the collection of all ε-tame exact Lagrangian submanifolds L of M contained in K and such that ||B L || ≤ Λ.
Proof: By the author's previous work [Cha21], every dF,F ′ -converging sequence in L e Λ,ε (M ) also converges in the Hausdorff metric to the same limit.If we are given a volume bound V > 0, then every Hausdorff-converging sequence in the associated subset of L e Λ,ε (M ) also converges in d H to the same limit by Theorem 2. On L e Λ,ε (K), we automatically a volume bound by Proposition 4. Since dF,F ′ ≤ d H by hypothesis, this gives the result.
Remark 8: Note that Corollary 4 implies that d H is bounded on L e Λ,ε (D * L).This is in stark contrast with the behavior of d H on L e (D * L), i.e. without any Riemannian bounds, where it is expected to be unbounded [She18].Therefore, Hamiltonian diffeomorphisms which moves a Lagrangian submanifold a lot in the Lagrangian Hofer metric must also greatly deform it.
Note that γ is however expected to be bounded on L e (D * L) -that is precisely the conjecture of Viterbo.It also follows from work of Biran and Cornea [BC21] that some fragmentation metrics are bounded on L e (D * L).It may thus be that L e Λ,ε (D * L) better capture the topology in these metrics than in the Lagrangian Hofer metric.
We now give the proof of Proposition 4; it relies mostly on the Bishop-Gromov inequalities.
Proof of Proposition 4: Note first that it suffices to bound the diameter of N .Indeed, the bound Λ on the second fundamental form, together with Gauss' equation, gives an upper bound λ = λ(Λ, K) ≥ 0 on the absolute value of the sectional curvature of N .Therefore, by the Bishop-Gromov inequality, we have that for any x ∈ N , any x ′ ∈ M k (−λ), and any r > 0. Here, M k (−λ) is the k-dimensional simply-connected space of constant sectional curvature −λ.In particular, we get When λ = 0, the quotient sinh(t √ λ)/ √ λ should be interpreted as being equal to t.Since sinh t (or t) is increasing and nonnegative, an upper bound on Diam(N ) will thus indeed give an upper bound on Vol(N ).
We now bound the diameter of N .Note that by Shen's work [She95], there exists r 0 = r 0 (Λ, K) > 0 such that the injectivity radius r inj (N ) of N respects r inj (N ) ≥ r 0 .Furthermore, since N is closed and connected, there exist x, y ∈ N such that d(x, y) = Diam(N ) =: T and a unit-speed minimizing geodesic γ of N such that γ(0) = x and γ(T ) = y.Set x i := γ(i) for i ∈ {0, 1, . . ., ⌊T ⌋}.By the construction, we have that d N (x i , x j ) ≥ 1 if i = j.Therefore, d M (x i , x j ) ≥ ε by the tameness condition, i.e.
Proof of Corollary 5: Suppose that {f i : L i M } is a sequence of isotropic or coisotropic immersions such that {f i (L i )} Hausdorff-converges to N .Take z 0 ∈ N .By Hausdorff-convergence, for each i, there is y i ∈ f i (L i ) such that lim y i = z 0 .By Proposition 5, there is a sequence of symplectomorphisms or contactomorphisms {ψ i } which C 2 -converges to the identity and such that ψ i (y i ) = z 0 .
We take The rest of the proofs of the lemmata then goes through as before, except that we instead use the pointed version of Shen's theorem.This works because each point of N 0 is contained in some compact neighborhood of x 0 , and being (exact) (co)isotropic is a local condition about that point.We only get f 0 (L 0 ) = N , because f 0 (L 0 ) might not be closed if L 0 is noncompact.
Remark 10: The proof of Proposition 3 relies in an essential way on the fact that there is a covering L i → L 0 , i.e.Proposition 1.The proof of that relies on applying Ehresmann's fibration theorem to p = ι −1 • f 0 : L 0 → L, which requires p to be proper.When L 0 is compact, this is of course automatic, but not in the noncompact case.For example, we could modify the example in Remark 3 to get This gives in the limit a map f 0 : R × S 1 → T * T 2 such that the corresponding p is not proper.Note that f 0 is nonetheless a covering.We expect that to still be the case whenever the L i 's are closed manifold; one can however easily make counterexamples when they are not.
Note that for i large enough, not only is x i in U , but also ϕ(x i ) ∈ B 2n δ ′ (0).Suppose that we have such i.Let R i be a unitary transformation sending ϕ(x i ) to |ϕ(x i )|u, and define a Hamiltonian H i on M by Let {ψ i t } be the Hamiltonian isotopy that it generates.A direct computation gives that ϕ(ψ 1 is the identity outside U , it also follows from this relation that the sequence {ψ i } converges to 1 M in C 2 -topology.
The case when M is contact is quite similar.Indeed, we can still take a Darboux chart ϕ centered at x 0 , and consider the contact Hamiltonian where β is a bump function with support in B 2n+1 δ (0), α 0 is the standard contact form on R 2n+1 , and u ∈ R 2n × {0} is unitary.The contact isotopy {ψ H t } is quite similar to what we had in the symplectic case: if we write u = (x 0 i , y 0 i , 0) 1≤i≤n , then 0).Therefore, for any z 0 ∈ R, we get that . The rest of the argument is then completely analogous to the symplectic case.
Remark 11: The construction in the symplectic case actually gives a sequence which also converges to the identity in the Hofer norm.Indeed, in the symplectic case, we have that Likewise, if M is a contact manifold admitting a contact form α, then the same argument implies that {ψ i } converges to the identity in the Shelukhin-Hofer norm associated to α [She17].

III.d. Limits in the absence of any Riemannian bounds
We now prove a result which does show that there is some rigidity for sequences of Lagrangian submanifolds, even when no Riemannian bounds are put on such sequence.This proves that there exists some rigidity for the Hausdorff metric between Lagrangian submanifolds in full generality.
Theorem 3 Let {L i } be a sequence of closed connected Lagrangian submanifolds in a 2n-dimensional symplectic manifold M .Suppose the following: (1) The sequence Hausdorff-converges to a closed topological submanifold N of dimension at most n.
(2) The sequence converges to some closed connected Lagrangian submanifold L 0 in a J-adapted metric dF,F ′ such that is totally disconnected, e.g.dF,F ′ is d H or γ.
Seeing this from the other way around, this implies that dF,F ′ -converging sequences either behave like those which are geometrically bounded -although they may not themselves be geometrically bounded (c.f.[Cha21]) -or they Hausdorff-converge to a fairly pathological space.This is thus a good step in the direction of a truly symplectic characterization of "nicely-behaved sequences" of Lagrangian submanifolds.
Furthermore, this indicates that it makes sense to talk of C 0 -Lagrangian submanifolds as n-dimensional topological submanifolds L of M which are the Hausdorff limit of a sequence of smooth Lagrangian submanifolds {L i } such that the sequence {L i } is also Cauchy in a nice J-adapted metric dF,F ′ .Note that a sequence of maps {ψ i } C 0 -converge to a map ψ if and only if the sequence of graphs {graph ψ i } Hausdorffconverge to graph ψ [Wat76].Therefore, graphs of hameomorphisms [OM07] are C 0 -Lagrangians in our sense for dF,F ′ = d H .However, Oh and Müller's definition is a priori stronger than ours in the sense that there may be Lagrangian graphs of homeomorphisms which are not hameomorphisms.Instead, they would be graphs of homeomorphisms obtained as limits of Hamiltonian diffeomorphisms in what they call the weak Hamiltonian topology.On top of that, there could be graphs of C 0 -limits of non-Hamiltonian symplectomorphisms when dF,F ′ can compare non-Hamiltonian diffeomorphic Lagrangian submanifolds.
Likewise, it is unclear how this definition of C 0 -Lagrangian submanifolds relate to the definition of Humilière, Leclercq, and Seyfaddini [HLS15].Indeed, our definition is global whilst theirs is local, which makes a direct comparison hard.
Finally, it was pointed out to us by Seyfaddini that this can be seen as a generalization of results of Hofer [Hof90] and Viterbo [Vit92] saying that if a sequence of Hamiltonian diffeomorphisms {ψ i } is such that (1) ψ i C 0 − − → ϕ; (2) then ψ = ϕ.As noted before, (1) here is equivalent to (1) in Theorem 3 with L i = graph ψ i .However, the two versions of (2) are a priori entirely independent.Indeed, the Hofer and spectral norms on Hamiltonian diffeomorphisms have notoriously different behavior than the Hofer and spectral norms on their graph (c.f.[Ost03]).These differences are however on the large scale geometry of the metrics; this indicates that the local behavior are similar.
Proof of Theorem 3: First of all, note that N is connected since all L i are.By the author's previous work [Cha21], we have a constant C = C(L 0 ) such that for i large.Therefore, L 0 is contained in N ∪ ((∪F ) ∩ (∪F ′ )).Since (∪F ) ∩ (∪F ′ ) is totally disconnected, L 0 must thus be contained in N .Since L 0 is compact, it is a closed subset of N .But since L 0 and N are both submanifolds of M , we must have m = n, and L 0 must be an open subset of N .Therefore, by connectedness, N = L 0 .
Remark 12: Note that the technical upgrade of Sikorav's version of the monotonicty lemma [Sik94] proven in the paper cited above is not necessary here.Since the constant C only needs to depend on L 0 -and not on metric invariants of L 0 -Inequality (⋆) also follows from applying Sikorav's monotonicity lemma on a smallenough metric ball centered at a point of L 0 .The rest of the proof of the inequality is then the same as before.
and only if the first Betti numbers of L and L ′ are the same, i.e. b 1 (L ′ ) = b 1 (L).