Quantitative h-principle in symplectic geometry
aa r X i v : . [ m a t h . S G ] J a n Quantitative h -principle in symplectic geometry Lev Buhovsky and Emmanuel Opshtein.February 2, 2021
Dedicated to Claude Viterbo, on the occasion of his 60th birthday
Abstract
We prove a quantitative h -principle statement for subcritical isotropic embeddings.As an application, we construct a symplectic homeomorphism that takes a symplecticdisc into an isotropic one in dimension at least 6. Gromov’s h -principle lies at the core of symplectic topology, by reducing many questions onthe existence of embeddings or immersions to verifying their compatibility with algebraictopology. Symplectic topology focuses mainly on the other problems, that do not abide byan h -principle : Lagrangian embeddings, existence of symplectic hypersurfaces in specifichomology classes etc. In [BO16], we have proved a refined version of h -principle, which inturn yielded applications to C -symplectic geometry. For instance, we proved in [BO16] thatin dimension at least 6, C -close symplectic 2-discs of the same area are isotopic by a smallsymplectic isotopy, while in dimension 4, this does no longer hold. A similar quantitative h -principle was also used in [BHS18] in order to show that the symplectic rigidity manifestedin the Arnold conjecture for the the number of fixed points of a Hamiltonian diffeomorphismcompletely disappears for Hamiltonian homeomorphisms in dimension at least 4.The goal of this note is to prove a quantitative h -principle for isotropic embeddings andto derive some flexibility statements on symplectic homeomorphisms. Theorem 1 (Quantitative h -principle for subcritical isotropic embeddings) . Let V be anopen subset of C n , k < n , u , u : D k ֒ → V be isotropic embeddings of closed discs. Weassume that there exists a homotopy F : D k × [0 , → V between u and u (so F ( · ,
0) = u , F ( · ,
1) = u ) of size less than ε (Diam F ( { z } × [0 , < ε for all z ∈ D k ).Then there exists a Hamiltonian isotopy (Ψ t ) t ∈ [0 , such that Ψ ◦ u = u , of size ε . The proof shows that the theorem holds in the relative case, provided u , u are sym-plectically isotopic, relative to the boundary. The method of the proof of theorem 1 followsa very similar track as the quantitative h -principle for symplectic discs that we established1n [BO16]. Paralleling the construction of a symplectic homeomorphism whose restrictionto a symplectic disc is a contraction in dimension 6, we can deduce from theorem 1 thefollowing statement: Theorem 2.
There exists a symplectic homeomorphism with compact support in C whichtakes a symplectic disc to an isotropic one. Of course, by considering products, we infer that there exists symplectic homeomor-phisms that take some codimension 4 symplectic submanifolds to submanifolds which arenowhere symplectic.The note is organized as follows. We prove theorem 1 in the next section. The con-struction of a symplectic homeomorphism that takes a symplectic disc to an isotropic oneis explained in section 3, where we also explain a relation to relative Eliashberg-Gromovtype questions, as posed in [BO16].
Acknowledgments:
We are very much indebted to Claude Viterbo for his support andinterest in our research. Claude’s fundamental contributions to symplectic geometry andtopology, and in particular to the field C symplectic geometry, are widely recognized. Wewish Claude all the best, and to continue enjoying math and delighting us with his creativeand inspiring mathematical works.This paper is a result of the work done during visits of the first author at StrasbourgUniversity, and a visit of the second author at Tel Aviv University. We thank both uni-versities and their symplectic teams for a warm hospitality. The first author was partiallysupported by ERC Starting Grant 757585 and ISF Grant 2026/17. Conventions and Notations
We convene the following in the course of this paper: • All our homotopies and isotopies have parameter space [0 , g t ) denotesan isotopy ( g t ) t ∈ [0 , . • Similarly, by concatenation of homotopies we always mean reparametrized concatena-tion. • If F : [0 , × X → Y is a homotopy with value in a metric space, Size ( F ) :=max { Diam (cid:0) F ([0 , × { x } ) (cid:1) , x ∈ X } . • For A ⊂ B , Op ( A, B ) stands for an arbitrarily small neighbourhood of A in B . Tokeep light notation, we omit B whenever there is no possible ambiguity. • A homotopy F : [0 , × N → M is said relative to A ⊂ N if it is constant on A . • A homotopy G : [0 , × N → M between F , F : [0 , × N → M (that is acontinuous map such that G ( i, t, z ) = F i ( t, z ) for i = 0 ,
1) is said relative to A and { , } if G ( s, t, z ) = F ( t, z ) = F ( t, z ) for all z ∈ A and if G ( s, i, z ) = F ( i, z ) for all s ∈ [0 , Quantitative h -principle for isotropic discs The aim of this section is to prove theorem 1. h -principle for subcritical isotropic embeddings We recall in this section the main properties of the action of the Hamiltonian group onisotropic embeddings, as described in [Gro86, EM02]. To this purpose, we first fix somenotations. In the current note, a disk D k is always assumed to be closed, unless explicitlystated (hence an embedding of D inside an open set is always compactly embedded). Sincewe only deal with isotropic embeddings, it is enough to prove theorem 1 for subcriticalisotropic embeddings of [ − , k rather than of a closed disc. By abuse of notation, in thissection we denote D k = [ − , k . The set of isotropic framings G iso ( k, n ) is the space of( k, n )-matrices of rank k whose columns span an isotropic vector space in ( R n , ω st ).The following statement is a specialization to C n of the h -principle for subcriticalisotropic embeddings: Recall that the h -principle for subcritical isotropic embeddings pro-vides existence of isotropic embeddings or homotopies whose derivatives realize homotopyclasses of maps to G iso ( k, n ). In the following, if A ⊂ D k , a homotopy of f : D k → G iso ( k, n )rel Op ( A ) is a continuous map F : [0 , × D k → G iso ( k, n ) such that F ( t, z ) = f ( z ) forall z ∈ Op ( A ). A homotopy G : [0 , × D k → G iso ( k, n ) between F , F : [0 , × D k → G iso ( k, n ) (that is a continuous map such that G ( i, t, z ) = F i ( t, z ) for i = 0 ,
1) is saidrelative to Op ( A ) and { , } if G ( s, t, z ) = F ( t, z ) = F ( t, z ) for all z ∈ Op ( A ) and if G ( s, i, z ) = F ( i, z ) for all s ∈ [0 ,
1] and i ∈ { , } . Theorem 2.1 (Parametric C -dense relative h -principle for isotropic embeddings [EM02]) . Let k < n :a) Let ρ : D k → C n be a continuous map whose restriction to a neighbourhood of aclosed subset A ⊂ D k is an isotropic embedding. Assume that dρ is homotopic to amap G : D k → G iso ( k, n ) relative to Op ( A ) . Then, for any ε > , there exists anisotropic embedding u : D k ֒ → C n which coincides with ρ on Op ( A ) , d C ( ρ, u ) < ε and such that du : D k → G iso ( k, n ) is homotopic to G rel Op (A).b) Let u , u : D k ֒ → C n be isotropic embeddings, which coincide on a neighbourhoodof a closed subset A ⊂ D k . Let G : [0 , × D k → G iso ( k, n ) be a homotopy between du , du rel Op ( A ) and ρ t : D k → C n a homotopy between u , u rel Op ( A ) . For any ε > , there exists an isotropic isotopy u t : D k ֒ → C n ( t ∈ [0 , ) relative to Op ( A ) such that d C ( ρ t , u t ) < ε and { du t } is homotopic to G rel Op ( A ) and { , } . We now state a related statement in a proper situation, when the disc D k is open ,hence not necessarily compactly embedded into V . The proof is a rather straightforwardapplication of theorem 2.1, and goes exactly along the lines of the proof of lemma A.3 b)in [BO16]. We leave the details to the reader.3 roposition 2.2. Let V ⊂ R n be a bounded open set, u , u : ◦ D k ֒ → V be subcriticalisotropic embeddings which coincide on Op ( ∂D k ) , are homotopic relative to Op ( ∂D k ) in V , and whose differentials are homotopic in G iso ( k, n ) relative to Op ( ∂D k ) . We fix such arelative homotopy G : [0 , × D k → G iso ( k, n ) between du and du . If k = 1 , we furtherassume that for a -form λ which is a primitive of ω in V , Z D ×{ } u ∗ λ = Z D ×{ } u ∗ λ. Then there exists a Hamiltonian isotopy ( ψ t ) with compact support in V such that ψ ◦ u = u and for the induced isotropic isotopy u t = ψ t ◦ u , { du t } is homotopic to G rel Op ( ∂D k ) and { , } . The next lemma will be also used in the proof of theorem 1.
Lemma 2.3.
Let
A, B be two closed subsets of D k . Let u , u : D k ֒ → C n be subcriticalisotropic embeddings that coincide on Op ( A ) . Assume that we are given a homotopy G t : D k → G iso ( k, n ) between du and du rel Op ( A ) . Let v t : D k ֒ → C n be an isotropicisotopy between u and v rel Op ( A ) , such that v | Op ( B ) = u , and such that { dv t | Op ( B ) } ishomotopic to { G t | Op ( B ) } relative to Op ( A ) and { , } . Then dv and du are homotopicrel Op ( A ∪ B ) among maps D k → G iso ( k, n ) . Remark 2.4.
In the setting of lemma 2.3, since v and u are homotopic rel Op ( A ∪ B ) (just consider the linear homotopy between them), the lemma and theorem 2.1 immediatelyimply that v is in fact isotropic isotopic to u rel Op ( A ∪ B ) .Proof of lemma 2.3: Consider the homotopy K t := dv t : D k → G iso ( k, n ) between du and dv relative to Op ( A ), and the homotopy G t : D k → G iso ( k, n ) between du and du rel Op ( A ), provided by the assumption. Letting K t := K − t , we now consider theconcatenation H t := K t ⋆ G t . Since { dv t | Op ( B ) } is homotopic to { G t | Op ( B ) } relative toOp ( A ) and { , } (as assumed by the lemma), there exists a homotopy H s,t ( s ∈ [0 , H t | Op ( B ) and I t relative to Op ( A ) and { , } , where I t ≡ du | Op ( B ) = dv | Op ( B ) isa constant homotopy. Let χ : D k → [0 ,
1] be a continuous function such that χ ( x ) = 0 ona complement of a sufficiently small neighborhood of B in D k , and χ ( x ) = 1 on a (smaller)neighborhood of B . Now define a homotopy ˜ G t : D k → G iso ( k, n ) ( t ∈ [0 , G t ( z ) := ( H χ ( z ) ,t ( z ) when z ∈ Op ( B ) ,G t ( z ) otherwise.Then ˜ G t is a desired homotopy between du and dv rel Op ( A ∪ B ). (cid:3) We will also need the following lemma, which allows to achieve general positions byHamiltonian perturbations. Recall that this means there exists a continuous map G : [0 , × Op ( B ) → G iso ( k, n ) such that G (0 , t, z ) = G t ( z ) and G (1 , t, z ) = dv t ( z ) ∀ ( t, z ) ∈ [0 , × Op ( B ), G ( s, t, z ) = du ( z ) ∀ ( s, t, z ) ∈ [0 , × Op ( A ∩ B ), G ( s, , z ) = G ( z ) = du ( z ) and G ( s, , z ) = G ( z ) = dv ( z ) ∀ ( s, z ) ∈ [0 , × Op ( B )). emma 2.5. Let V ⊂ C n be an open set, Σ , Σ be two smooth submanifolds of V , whichare transverse in a neighbourhood of ∂V . Then there exists an arbitrarily small Hamiltonianflow ( ϕ t ) t ∈ [0 , with compact support in V , such that ϕ (Σ ) ⋔ Σ . Let k < n , D k := [ − , k , D k ( µ ) := [ − − µ, µ ] k , u , u : D k ֒ → V ⊂ C n be smoothisotropic embeddings, and F : D k × [0 , → V a homotopy between u , u with Size F < ε .We need to prove that there exists a
Hamiltonian isotopy of size 2 ε , which takes u to u on D k .Before passing to the proof, we need to modify slightly the framework. First, extendthe isotropic embeddings and the homotopy to slightly larger isotropic embeddings: u , u : D k ( µ ) ֒ → V , F : D k ( µ ) × [0 , → V , where D k ( µ ) = [ − µ, µ ] k . By lemma 2.5, we do notlose generality if we assume that the images of u and u are disjoint (since k < n ), whichwe do henceforth. Next, the homotopy F can be turned into a more convenient object: Lemma 2.6 (see [BO16, lemma A.1]) . There exists a smooth embedding ˜ F : D k ( µ ) × [0 , ֒ → V , with ˜ F ( x,
0) = u ( x ) , ˜ F ( x,
1) = u ( x ) , with Diam ( ˜ F ( { x } × [0 , < ε for all x ∈ D k ( µ ) . In other words, ˜ F has size ε when considered as a homotopy between u , u . Now ˜ F can be further extended to an embedding, still denoted ˜ F ,˜ F : D k ( µ ) × [ − µ, µ ] × [ − µ, µ ] n − k − ֒ → V. Consider now a regular grid Γ := ν Z k ∩ D k in D k ⊂ D k ( µ ), of step ν ≪ ν − ∈ N . This grid generates a cellular decomposition of D k , whose l -skeletonΓ l is the union of the l -faces. The set of k -faces has a natural integer-valued distance, wherethe distance between k -faces x and x ′ is the minimal m such that there exists a sequence x = x , x , . . . , x m = x ′ of k -faces and x j ∩ x j +1 = ∅ for each j ∈ [0 , m −
1] (note thatthose intersections are not required to be along full k − η < ν/
2, andfor each x ∈ Γ , let U x be the η -neighbourhood of { x } × [0 , × { } n − k − in C n , and thendenote W x := ˜ F ( U x ). Similarly, for each k -face x k , denote by U x k the η -neighbourhood of x k × [0 , × { } n − k − in C n , and then put W x k := ˜ F ( U x k ). For a k -face x and m > W mx := ∪ W x ′ , where the union is over all the k -faces x ′ which are at distance atmost m from x . Note that W x = W x , and that W mx is a topological ball. Finally, we put W := ∪ x W x ⊂ V , where the union is over all the k -faces. Hence, W = ˜ F ( U ) where U isthe η -neighborhood of D k × [0 , × { } n − k − in C n .We will prove theorem 1 by successively isotopying the l -skeleton with a control on eachisotopy. Precisely, arguing by induction on l , we prove the following: Proposition 2.7.
There exist Hamiltonian isotopies (Ψ tl ) , l ∈ [0 , k ] with support in W ,and modified embeddings v := Ψ ◦ u , v l := Ψ l ◦ v l − , such that( I ) v l ≡ u on a neighbourhood of the l -skeleton Γ l , for every l ∈ [0 , k ] . I ) v l ( x ) ⊂ W l − x for each k -face x and every l ∈ [0 , k − .( I ) Ψ tl ( W x ) ⊂ W · l − x for each k -face x and l ∈ [1 , k − ,and Ψ t ( W x ) ⊂ W x , Ψ tk ( W x ) ⊂ W k ( k +1) x , for every k -face x .( I ) v l ( ◦ x l +1 ) ∩ u ( ◦ x ′ l +1 ) = ∅ for every pair of distinct ( l + 1) -faces, ∀ l ∈ [0 , k − .( I ) dv l and du are homotopic rel Op (Γ l ) among maps D k ( µ ) → G iso ( k, n ) , for each l ∈ [0 , k − . Proposition 2.7 readily implies theorem 1. Indeed, denoting by (Ψ t ) t ∈ [0 , the (reparamet-rized) concatenation { Ψ tk } ⋆ · · · ⋆ { Ψ t } of the flows, from ( I
3) we conclude that for each k -face x and each t we have Ψ t ( W x ) ⊂ W k k +1 x since (cid:16)P k − j =1 · j (cid:17) + 3 k ( k +1) < k + k +1 .The flow (Ψ t ) is supported in W = ∪ x ∈ Γ k W x ⊂ V , and if the step ν of the grid is chosento be sufficiently small, then for each k -face x , the diameter of W k k +1 x is less than 2 ε .Consequently, the size of the flow (Ψ t ) t ∈ [0 , is less than 2 ε . Moreover, by ( I
1) we haveΨ ◦ u = v k = u on D k . (cid:3) Proof of proposition 2.7:
As already explained, the proof goes by induction over the dimen-sion of the skeleton Γ l . Since D k ( µ ) is contractible, there exists a homotopy G t : D k → G iso ( k, n ) between du and du . The -skeleton: Let x ∈ Γ be a 0-face, ρ < η , and D ρ ( x ) the ρ -neighbourhood of x in D k ( µ ). Then u ( D ρ ( x )) , u ( D ρ ( x )) both lie in W x , and ˜ F provides an isotopy between u | D ρ ( x ) and u | D ρ ( x ) in W x . By theorem 2.1.b), there exists a Hamiltonian isotopy ( ψ tx )with support in W x , such that ψ x ◦ u = u on D ρ ( x ) and dψ tx ◦ du is homotopic to G t rel { , } . Since W x ∩ W x ′ = ∅ for different 0-faces x, x ′ , the isotopies ψ x have pairwise disjointsupports.The isotopy ψ t := ◦ ψ tx , where the composition runs over all 0-faces x of Γ, verifies ( I I
3) because it is supported inside thedisjoint union ∪ x ∈ Γ W x , and for every x ∈ Γ and a k -face x ′ we have either W x ⊂ W x ′ or W x ∩ W x ′ = ∅ . However, ψ ◦ u might not verify ( I ψ ◦ u coincides with u ona neighbourhood of Γ , there exist closed balls B x = B ( u ( x ) , r ) ⊂ W x for each 0-face x of Γ, such that ( I
4) is verified inside these balls. Therefore the traces of the submanifolds ψ ◦ u ( x ) and u ( x ′ ) inside ∪ x ∈ Γ (cid:0) W x \ B x (cid:1) verify the hypothesis of lemma 2.5, forevery pair of distinct 1-faces x , x ′ . Thus an arbitrarily C -small Hamiltonian perturbation( ˜ ψ t ) with support in ∪ x ∈ Γ (cid:0) W x \ B x (cid:1) ⊂ ∪ x ∈ Γ W x achieves ˜ ψ ◦ ψ ◦ u ( x ) ⋔ u ( x ′ ),for every pair x , x ′ of different 1-faces of Γ (hence these intersections are empty). NowΨ t := ( ˜ ψ t ) ∗ ( ψ t ) verifies ( I I
1) and ( I I
2) follows immediatelyfrom ( I v = Ψ ◦ u satisfies ( I
5) by direct application of lemma 2.3.
The l -skeleton ( l < n − ): We now assume that Ψ , . . . , Ψ l − have been constructed,and we proceed with the induction step. Recall that v l − = Ψ l − ◦ · · · ◦ Ψ ◦ u coincideswith u on Op (Γ l − ) and that v l − ( x k ) ⊂ W l − − x k for every k -face x k . Recall also that we6ave a homotopy G lt : D k → G iso ( k, n ) between dv l − and du rel Op (Γ l − ). Our aim isnow to find a Hamiltonian flow (Ψ tl ) which in particular isotopes v l − | Op ( x l ) to u | Op ( x l ) , foreach l -face x l .Step I: Adjusting the actions of the edges (case l = 1). When l = 1, beside the formal ob-structions, relative isotopies can be performed via localized Hamiltonians only when theactions of the edges coincide (see proposition 2.2). In [BO16], we show that there exists aHamiltonian isotopy ( ψ t A ), supported in an arbitrarily small neighborhood v (Γ ) = u (Γ ),whose flow is the identity on a (smaller) neighbourhood of the Γ , such that A (cid:0) ψ A ◦ v ( x ) (cid:1) := Z ψ A ◦ v ( x ) λ = Z u ( x ) λ = A (cid:0) u ( x ) (cid:1) for every 1-face x of Γ,and ψ A ◦ v ( ◦ x ) ∩ u ( ◦ x ′ ) = ∅ for each pair of distinct 1-faces x , x ′ of Γ. Since ψ t A ≡ Id nearΓ , ˜Ψ t := ψ t A ◦ Ψ t and v ′ := ˜Ψ ◦ u still verify ( I − t ) by( ˜Ψ t ) and v by v ′ , we can freely assume that A (cid:0) v ( x ) (cid:1) = A (cid:0) u ( x ) (cid:1) for each 1-face x .Step II: Isotopying the l -skeleton. Fix an l -face x l of Γ. By ( I x l ⊂ ◦ x l such that v l − and u coincide on Op ( x l \ ◦ ˆ x l ). Choose a k -face x k which contains x l .Since u (ˆ x l ) and v l − (ˆ x l ) both lie in the topological ball W l − − x k and coincide near theirboundary, there exists a homotopy σ x l : ˆ x l × [0 , → W l − − x k such that σ x l ( · ,
0) = v l − , σ x l ( · ,
1) = u , and σ x l ( z, t ) = u ( z ) ∀ z ∈ Op ( ∂ ˆ x l ) , t ∈ [0 , x l ⋐ x l and l < n , ( I
4) allows to use a general position argument to ensure thatmoreover Im σ x l admits a regular neighbourhood V x l ⊂ W l − − x k (a topological ball), suchthat all these neighbourhoods V x l are pairwise disjoint when x l runs over the l -faces (thisis the only point in the proof where we need that l < n − G tl : [0 , × D k → G iso ( k, n ) between dv l − and du , with G tl | Op (Γ l − ) = du = dv l − . Also, v l − | ˆ x l is clearly homotopic to u | ˆ x l rel Op ( ∂ ˆ x l )in V x l , and when l = 1, A ( v l − ◦ x l ) = A ( u ◦ x l ). Hence by proposition 2.2, there existHamiltonian diffeomorphisms ψ tx l , where x l runs over the l -faces, which have support in V x l ,and are such that ψ x l ◦ v l − | ˆ x l = u , and d ( ψ tx l ◦ v l − ) are homotopic relative to Op ( ∂ ˆ x l )and { , } to G tl | ˆ x l . Let now ψ tl := ◦ ψ tx l and ˆ v l := ψ l ◦ v l − . Since the ( ψ tx l ) have pairwisedisjoint supports, we have ˆ v l | x l = u | x l for each l -face x l of Γ. Hence ˆ v l and u coincide ona neighbourhood of the l -skeleton of Γ, so ˆ v l verifies ( I v l verifies ( I
5) aswell.The flow ( ψ tl ) is supported in the disjoint union ∪ x l ∈ Γ l V x l . Let x be any k -face, andassume that we have an l -face x l such that V x l ∩ W x = ∅ . Let x k ⊃ x l be a k -face as above,so that V x l ⊂ W l − − x k . Then the distance between x and x k is not larger than 3 l − , andwe conclude V x l ⊂ W l − − x k ⊂ W · l − − x . To summarise, for any k -face x , if x l is an l -facewith V x l ∩ W x = ∅ , then V x l ⊂ W · l − − x . As a result, we get ψ tl ( W x ) ⊂ W · l − − x . (2.2.1)7he embedding ˆ v l may fail to satisfy ( I l + 1-faces x l +1 , x ′ l +1 such that ˆ v l ( ◦ x l +1 ) ∩ u ( ◦ x ′ l +1 ) = ∅ . Notice however that since ˆ v l and u coincide on a neighbourhood of Γ l , the set ˆ v l ( x l +1 ) ∩ u ( x ′ l +1 ) is compactly contained in W \ u (Γ l ). By lemma 2.5, there exists an arbitrarilysmall Hamiltonian flow ( ϕ tl ) t ∈ [0 , , with compact support in W \ Γ l such that v l := ϕ l ◦ ˆ v l verifies ( I ϕ tl ) and by (2 . . tl ) := ( ϕ tl ) ∗ ( ψ tl )satisfies Ψ tl ( W x ) ⊂ W · l − x for any k -face x . Hence ( I
3) holds for (Ψ tl ). Since the support of( ϕ tl ) is compactly contained in W \ Γ l , ( I
1) and ( I
5) still holds for v l . Finally, ( I
2) followsas well: if x is any k -face, then by assumption, v l − ( x ) ⊂ W l − − x , hence by (2 . .
1) and( I
3) we get v l ( x ) = Ψ l ◦ v l − ( x ) ⊂ Ψ l ( W l − − x ) = [ d ( x,y ) l − − Ψ l ( W y ) ⊂⊂ [ d ( x,y ) l − − W · l − y = W l − − · l − x = W l − x . (2.2.2) The k -skeleton: When k < n −
1, the procedure described above works perfectly. However,when k = n −
1, the last step of the induction requires some adjustment. As before, forevery k -face x k , v k − ( x k ) and u ( x k ) both lie in the topological ball W k − − x k and coincidenear the boundary, hence there exist homotopies σ x k : ˆ x k × [0 , → W k − − x k such that σ x k ( · ,
0) = v k − | x k , σ x k ( · ,
1) = u | x k and σ x k ( z, t ) = u ( z ) for all t ∈ [0 , z ∈ Op ( ∂x k ) (as before, ˆ x k ⊂ ◦ x k is a closed box such that u and v k − coincide onOp ( x k \ ◦ ˆ x k )). The difference with the previous steps of the induction is that general positiondoes not make the sets Im σ x k pairwise disjoint. Instead we proceed as follows.By ( I v k − (ˆ x k ) ∩ u ( x ′ k ) = u (ˆ x k ) ∩ u ( x ′ k ) = ∅ for every pair of different k -faces x k , x ′ k . By a standard general position argument, since k < n , we can therefore assume thatIm σ x k ∩ u ( x ′ k ) = ∅ , and that we have a regular neighbourhood V x k ⊂ W k − − x k of Im σ x k ,such that V x k ∩ u ( x ′ k ) = ∅ ∀ x k = x ′ k . (2.2.3)By ( I v k − (ˆ x k ) , u (ˆ x k ) are homotopic relative to ∂ ˆ x k in V x k , there exists aHamiltonian isotopy ( ψ tx k ) with support in V x k such that ψ x k ◦ v k − | x k = u .Consider now a partition of the set of the k -faces into (2 · k − ) k = 2 k · k ( k − subsets F i ( i = 1 , . . . , k · k ( k − ), such that any two faces x k , x ′ k ∈ F i are at distance at least2 · k − from each other. Then for any i and any pair x k , x ′ k ∈ F i of distinct k -faces, we have W k − − x k ∩ W k − − x ′ k = ∅ . Define ( ψ tk,i ) := ◦ x k ∈ F i ψ tx k , which is a composition of Hamiltonianisotopies, compactly supported in the disjoint union ∪ x k ∈ F i W k − − x k . For any k -face x , ifwe have some x k ∈ F i such that W x ∩ W k − − x k = ∅ , then the distance between x and x k isat most 3 k − , and hence W k − − x k ⊂ W · k − − x . We conclude that for any k -face x we have ψ tk,i ( W x ) ⊂ W · k − − x . 8ow, letting (Ψ tk ) := ( ψ tk, k · k ( k − ) ∗ · · · ∗ ( ψ tk, ) and arguing as in (2.2.2), we get for any k -face x Ψ tk ( W x ) ⊂ W N k x ⊂ W k ( k +1) x , where N k = 2 k · k ( k − · (2 · k − − < k ( k +1) . Therefore, ( I
3) holds for (Ψ tk ).Finally, ψ k,i ◦ v k − | Op ( x k ) = u | Op ( x k ) for all x k ∈ F i , and by (2 . . ψ x ′ k ◦ u | Op ( x k ) = u | Op ( x k ) for any pair of k -faces x ′ k = x k . Thus,Ψ k ◦ v k − | Op ( x k ) = u for every k -face x k of Γ , which just means that Ψ k ◦ v k − | Op ( D k ) ≡ u | Op ( D k ) . We have verified ( I
1) for v k :=Ψ k ◦ v k − . (cid:3) We aim now at proving theorem 2. Although it is completely similar to the proof of theflexibility of the disc area provided in [BO16] once theorem 1 is established, we rewritebelow the argument in our situation for the convenience of the reader. Recall that theorem1 holds for symplectic embeddings of discs in C [BO16, Theorem 2]. Theorem 3.1.
Theorem 2 holds when the isotropic embeddings u , u are replaced by sym-plectic embeddings u , u : D ֒ → W such that u ∗ ω = u ∗ ω = ω st . Proof of theorem 2:
Let i : D −→ C × C × C = C , ( x, y ) ( x, y, u : D −→ C × C × C z ( z, , D into C . Let also f k : D (2) → D / k be an area-preserving immersion and u k : D −→ C × C × C ( x, y ) ( x, y, f k ( x + iy )) . Then, u k is a symplectic embedding of D into C with d C ( u k , i ) < k . Let finally consideran isotropic embedding i lk of D into C with d C ( i lk , u k ) < l . Although less explicit thanthe previous embedding in dimension 6, it certainly exists because one can approximate thestandard symplectic embedding u by isotropic ones of the form z ( z, f l ( z ) , W k ( δ ) := { z ∈ C | d ( z, Im u k ) < δ } and W ( ε ) := { z ∈ C | d ( z, Im i ) < ε } .
9t is enough to construct a sequence ϕ , ϕ , . . . of compactly supported in C symplecticdiffeomorphisms, such that for an increasing sequence of indices k = 0 < k < k < . . . we have ϕ i ◦ u k i = u k i +1 , and such that moreover, the sequence Φ i = ϕ i ◦ ϕ i − ◦ · · · ◦ ϕ uniformly converges to a homeomorphism Φ of C . We construct such a sequence ϕ i byinduction. Let C = U ⊃ U ⊃ U ⊃ · · · ⊃ u ( D ) be a decreasing sequence of open setssuch that ∩ U i = u ( D ). In the step 0 of the induction, we let k = 1, and choose ϕ to beany symplectic diffeomorphism with compact support in C such that ϕ ◦ u = u k .Now we describe a step i >
1. From the previous steps we get k < · · · < k i , andsymplectic diffeomorphisms ϕ , . . . , ϕ i − . Denote Φ i − = ϕ i − ◦ · · · ◦ ϕ . By the step i − u k i = Φ i − ◦ u and Φ i − ( U i − ) ⊃ W ( ε i ), where ε i = ki . The choice for ε i impliesthat W ( ε i ) ⊃ u k i ( D ), and moreover by u k i = Φ i − ◦ u we get Φ i − ( U i ) ⊃ u k i ( D ), so weconclude Φ i − ( U i ) ∩ W ( ε i ) ⊃ u k i ( D ). Hence we can choose a sufficiently large l i > k i suchthat Φ i − ( U i ) ∩ W ( ε i ) ⊃ W k i ( δ i ) ⊃ i l i k i ( D ), where δ i = li ε i . Note that d C ( i l i k i , i ) d C ( i l i k i , u k i ) + d C ( u k i , i ) < l i + 12 k i ε i , and moreover i ( D ) , i l i k i ( D ) ⊂ W ( ε i ). Hence by the convexity of W ( ε i ) and by theorem1, there exists a Hamiltonian diffeomorphism ϕ ′ i supported in W ( ε i ) such that i = ϕ ′ i ◦ i l i k i and d C ( ϕ ′ i , Id ) < ε i . Note that in particular, ϕ ′ i ( W k i ( δ i )) ⊃ i ( D ).We claim that there exists a homotopy of a small size between the (symplectic) disc ϕ ′ i ◦ u k i and the (isotropic) disc i , inside ϕ ′ i ( W k i ( δ i )). Indeed, the open set W k i ( δ i ) contains thediscs u k i ( D ) , i l i k i ( D ). Also we have d C ( u k i , i l i k i ) < δ i . Hence the linear homotopy ρ i ( z, t ) :=(1 − t ) u k i ( z ) + ti l i k i ( z ), ( z ∈ D , t ∈ [0 , d C ( u k i ( z ) , ρ i ( z, t )) < δ i for all z ∈ D , t ∈ [0 , W k i ( δ i ), this homotopy ρ i lies inside W k i ( δ i ). We moreover conclude that the size of ρ i is less than δ i , and therefore the homotopy ϕ ′ i ◦ ρ i between ϕ ′ i ◦ u k i and ϕ ′ i ◦ i l i k i = i , lies inside ϕ ′ i ( W k i ( δ i )), and has size less than δ i + 8 ε i ε i (recall that d C ( ϕ ′ i , Id ) < ε i ).We therefore have ϕ ′ i ( W k i ( δ i )) ⊃ i ( D ), and moreover the homotopy ϕ ′ i ◦ ρ i between ϕ ′ i ◦ u k i and i , lies inside ϕ ′ i ( W k i ( δ i )), and is of size less than 9 ε i . Hence by choosing asufficiently large k i +1 > k i and denoting ε i +1 = ki +1 , we get ϕ ′ i ( W k i ( δ i )) ⊃ W ( ε i +1 ) ⊃ u k i +1 ( D ) , and moreover the homotopy between ϕ ′ i ◦ u k i and u k i +1 , given by the concatenation of ϕ ′ i ◦ ρ i and of the linear homotopy between i and u k i +1 , lies in ϕ ′ i ( W k i ( δ i )) and still has sizeless than 9 ε i . Applying the quantitative h -principle for symplectic discs [BO16], we get aHamiltonian diffeomorphism ϕ ′′ i supported in ϕ ′ i ( W k i ( δ i )), such that ϕ ′′ i ◦ ϕ ′ i ◦ u k i = u k i +1 and d C ( ϕ ′′ i , Id ) < ε i .As a result, the composition ϕ i := ϕ ′′ i ◦ ϕ ′ i is supported in W ( ε i ) ⊂ Φ i − ( U i − ), we have ϕ i ◦ u k i = u k i +1 , ϕ i ◦ Φ i − ( U i ) = ϕ ′′ i ◦ ϕ ′ i ◦ Φ i − ( U i ) ⊃ ϕ ′′ i ◦ ϕ ′ i ( W k i ( δ i )) = ϕ ′ i ( W k i ( δ i )) ⊃ W ( ε i +1 ) , d C (Id , ϕ i ) d C (Id , ϕ ′ i ) + d C (Id , ϕ ′′ i ) < ε i . This finishes the step i of the inductive construction.To summarize, we have inductively constructed a sequence of Hamiltonian diffeomor-phisms ϕ , ϕ , . . . with uniformly bounded compact supports in C , such that:(i) ϕ i has support in W ( ε i ) ⊂ Φ i − ( U i − ) where Φ i − = ϕ i − ◦ · · · ◦ ϕ ,(ii) d C (Id , ϕ i ) < ε i = ki ,(iii) u k i +1 = ϕ i ◦ u k i .It follows by (ii) that Φ i is a Cauchy sequence in the C topology, hence uniformlyconverges to some continuous map Φ : C → C . Next, since u k i +1 = ϕ i ◦ u k i for ev-ery i >
0, we have i = Φ ◦ u . Finally, we claim that Φ is an injective map, hence ahomeomorphism. To see this, consider two points x = y ∈ U = C . If x, y ∈ u ( D ),then by (iii), Φ( x ) = i ◦ u − ( x ) = i ◦ u − ( y ) = Φ( y ). If x, y / ∈ u ( D ), then x, y ∈ c U i for i large enough, so by (i), Φ i ( x ) = Φ i +1 ( x ) = Φ i +2 ( x ) = ... = Φ( x ), and similarlyΦ i ( y ) = Φ( y ) (because for each j > i , the support of ϕ j lies in Φ j − ( U j − ) ⊂ Φ j − ( U i )), soΦ( x ) = Φ i ( x ) = Φ i ( y ) = Φ( y ). Finally, if x ∈ u ( D ) and y / ∈ u ( D ), then y ∈ c U i for i largeenough, and so Φ( y ) = Φ i ( y ) ∈ Φ i ( c U i ) ⊂ c W ( ε i +1 ) by (i). Since Φ( x ) ∈ Im i ⊂ W ( ε i +1 ),we conclude that also in this case we have Φ( x ) = Φ( y ). (cid:3) C -rigidity Here we address the following question which appeared in our earlier work [BO16]:
Question 1.
Assume that a symplectic homeomorphism h sends a smooth submanifold N to a submanifold N ′ , and that h | N is smooth. Under which conditions h ∗ ω | N ′ = ω | N ? Of course, that question is non-trivial only when dim N is at least 2, which we assumehenceforth. The question is particularly interesting in the setting of pre-symplectic sub-manifolds. Recall that a submanifold N ⊂ ( M, ω ) is called pre-symplectic if ω has constantrank on M . The symplectic dimension dim ω N of a pre-symplectic submanifold N is theminimal dimension of a symplectic submanifold that contains N . One checks immediatelythat dim ω N = dim N + Corank ω | N .In [BO16], we answered question 1 in various cases of the pre-symplectic setting. Theo-rem 2 allows to address almost all the remaining cases. Our next result incorporates theseremaining cases, together with those verified in [BO16]: Theorem 3.
Let N ⊂ ( M n , ω ) be a pre-symplectic disc. Then the answer to question 1 is • Negative if dim ω N n − , or if dim ω N = 2 n − and Corank ω | N > . Positive if dim ω N = 2 n , or if dim ω N = 2 n − and Corank ω | N = 0 . The only case that remains open is when dim ω N = 2 n − ω | N = 1 (i.e.dim N = 2 n −
3, Corank ω | N = 1). Proof of theorem 3:
When dim ω N n − N is not isotropic, the answer is negativebecause we can find a symplectic homeomorphism that fixes N and contracts the symplecticform (by [BO16]). When dim ω N n − r := corank ω | N >
2, there is a localsymplectomorphism that takes N to [0 , r × D k × { } ⊂ C r ( z ) × C k ( z ′ ) × C m ( w ) , where m ≥ r ≥
2. By theorem 2, we can find a symplectic homeomorphism f ( z , z , w ) of C × C which takes [0 , × { } to a symplectic disc. The induced map˜ f : C z ,z ) × C ( w ) × C r − × C k × C m − −→ C n ( z , z , w , z , . . . , z r , z ′ , . . . , z ′ k , w , . . . , w m ) f ( z , z , w ) × Idis obviously a symplectic homeomorphism which takes N to a submanifold on which theco-rank of the symplectic form is reduced by 2. Note that this argument also works whendim ω N n − N is isotropic. The second item of the theorem was proved in [BO16]. (cid:3) References [BHS18] Lev Buhovsky, Vincent Humili`ere, and Sobhan Seyfaddini. A C counterexample to the Arnoldconjecture. Invent. Math. , 213(2):759–809, 2018.[BO16] Lev Buhovsky and Emmanuel Opshtein. Some quantitative results in C symplectic geometry. Invent. Math. , 205(1):1–56, 2016.[EM02] Y. Eliashberg and N. Mishachev.
Introduction to the h -principle , volume 48 of Graduate Studiesin Mathematics . American Mathematical Society, Providence, RI, 2002.[Gro86] Mikhael Gromov.
Partial differential relations , volume 9 of
Ergebnisse der Mathematik und ihrerGrenzgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, 1986.
Lev BuhovskiSchool of Mathematical Sciences, Tel Aviv University e-mail : [email protected] OpshteinInstitut de Recherche Math´ematique Avanc´eeUMR 7501, Universit´e de Strasbourg et CNRS e-maile-mail