Bulky Hamiltonian isotopies of Lagrangian tori with applications
aa r X i v : . [ m a t h . S G ] M a r BULKY HAMILTONIAN ISOTOPIES OF LAGRANGIANTORI WITH APPLICATIONS
GEORGIOS DIMITROGLOU RIZELL
Abstract.
We exhibit monotone Lagrangian tori inside the standardsymplectic four-dimensional unit ball that become Hamiltonian isotopicto the Clifford torus, i.e. the standard product torus, only when consid-ered inside a strictly larger ball (they are not even symplectomorphic toa standard torus inside the unit ball). These tori are then used to con-struct new examples of symplectic embeddings of toric domains into theunit ball which are symplectically knotted in the sense of J. Gutt andM. Usher. We also give a characterisation of the Clifford torus insidethe ball as well as the projective plane in terms of quantitative consid-erations; more specifically, we show that a torus is Hamiltonian isotopicto the Clifford torus whenever one can find a symplectic embedding ofa sufficiently large ball in its complement. Introduction and results
The focus in this paper is on two-dimensional Lagrangian tori inside thestandard four-dimensional Liouville manifold ( C , ω = dλ ) as well as theprojective plane ( C P , ω FS ). We begin by recalling the basic definitions.The Liouville manifold( C , ω = dx ∧ dy + dx ∧ dy )is equipped with the standard Liouville form λ = 12 X i =1 ( x i dy i − y i dx i ) , dλ = ω . Denote by ζ the corresponding Liouville vector field, which generates theflow φ tλ : ( C , dλ ) → ( C , e − t dλ ) ,φ tλ ( z ) = e t/ z . To set up notation, we will denote by B nx ( r ) and D nx ( r ) the open andclosed balls inside C n , respectively, which are centred at the point x ∈ C n Mathematics Subject Classification.
Primary 53D12; Secondary 53D42.
Key words and phrases.
Symplectic embeddings, Lagrangian tori, Polydiscs, Symplec-tically knotted embeddings, Gromov width.The author is supported by the grant KAW 2016.0198 from the Knut and Alice Wal-lenberg Foundation. and of radius r >
0. Further, denote by S n − ( r ) := ∂D n ( r ) ⊂ C n thesphere of radius r > . We also write B n := B n (1), D n := D n (1), and S n − := S n − (1).The projective plane ( C P , ω FS ) equipped with the standard Fubini–Study form can be obtained by collapsing the boundary S of ( D , ω ) to aline ℓ ∞ ∼ = C P via symplectic reduction; in particular, the symplectic areaof the line class [ ℓ ] ∈ H ( C P ) is then equal to R ℓ ω FS = π .A Lagrangian submanifold is a half-dimensional manifold on which thesymplectic form vanishes. In the case of C the last condition is equivalentto the requirement that λ pulls back to a closed form. A Lagrangianisotopy is a smooth isotopy through Lagrangian embeddings; recall thestandard fact that such an isotopy inside C can be generated by a global Hamiltonian isotopy of the ambient symplectic manifold if and only if thepullbacks of λ are constant in cohomology; see e.g. [30] by A. Weinstein. Ingeneral, we will call a smooth isotopy of a subset of a symplectic manifold Hamiltonian if it can be realised by a Hamiltonian isotopy of the ambientsymplectic manifold.The symplectic action class of a Lagrangian inside C is the coho-mology class σ L := [ λ | T L ] ∈ H ( L ; R ) pulled back to it, which by Stokes’theorem is the value of the symplectic area of any two-chain inside C withboundary in that class. A torus is monotone if the symplectic area of atwo-dimensional chain with boundary on it is proportional to the so-calledMaslov class of the same chain; see V. Arnold [1] for the definition of thelatter characteristic class. In particular, it follows that the Lagrangian product tori S ( a ) × S ( b ) ⊂ C are monotone if and only if a = b. Note that the symplectic action class of he standard monotone product tori S ( r ) × S ( r ) ⊂ ( C , ω ) take values that are integer multiples of πr > . These tori are usually called
Clifford tori .R. Vianna [29] has shown that the classes of monotone Lagrangian toriinside ( C P , ω FS ) exhibit a very rich and interesting structure. In particular,there exists infinitely many different Hamiltonian isotopy classes of such tori.The result [12, Theorem C] by the author together with E. Goodman andA. Ivrii implies that all of Vianna’s tori can be placed inside the open unitball ( B , ω ) = ( C P \ ℓ ∞ , ω FS )after a Hamiltonian isotopy. It thus follows a fortiori that his constructionalso gives rise to infinitely many different Hamiltonian isotopy classes ofmonotone Lagrangian tori inside B . In contrast to this, the only knownHamiltonian isotopy classes of Lagrangian tori inside the complete Liouvillemanifold ( C , ω ) ⊃ B are the product tori, together with linear rescalingsof the “exotic” monotone torus [6] constructed by Y. Chekanov; the lattergoes under the name of the Chekanov torus , and we refer to the work [14]by A. Gadbled for the presentation that we will use here.
ULKY HAMILTONIAN ISOTOPIES OF TORI 3
We expect that Vianna’s tori all become Hamiltonian isotopic to standardtori inside a ball which is strictly larger than the unit ball. (This can beconfirmed by hand for e.g. certain particular Hamiltonian isotopies that takeVianna’s first exotic torus constructed in [28] into the unit ball.)
Remark 1.1.
Even though all Lagrangian tori are Lagrangian isotopic in-side the ball by [12], there is still no classification of Lagrangian tori insidethe plane up to
Hamiltonian isotopy. Under additional assumptions con-cerning a certain linking behaviour with a conic; the author established aHamiltonian classification in [11].Our first result is a criterion for when a monotone Lagrangian torus isHamiltonian isotopic to a Clifford torus in terms of the existence of a largesymplectic ball in its complement.
Theorem 1.1. (1) Let L ⊂ ( B , ω ) be a Lagrangian torus inside theunit ball whose whose symplectic action class takes the values Z πr on H ( L ) for some fixed r ≥ / √ . There exists a Hamiltonian iso-topy inside the ball which takes L to the standard monotone producttorus S ( r ) × S ( r ) if and only if it is disjoint from the interior ofsome symplectic embedding of ( D ( p / , ω ) into ( B , ω ) .(2) A monotone Lagrangian torus L ⊂ ( C P , ω FS ) which is disjointfrom the interior of some symplectic embedding of ( D ( p / , ω ) is Hamiltonian isotopic to the standard Clifford torus S ( p / × S ( p / ⊂ B = ( C P \ ℓ ∞ , ω FS ) contained in the affine chart. We then show that Part (1) of the above theorem is sharp in the followingsense: Even under the stronger assumption that L ⊂ B \ B ( p / − ǫ )is a Lagrangian torus that is Hamiltonian isotopic to S ( r ) × S ( r ) insidethe full plane ( C , ω ) , there are cases when any such Hamiltonian isotopymust intersect S = ∂D at some moment in time. (In other words, theHamiltonian isotopy cannot be confined to the unit ball that contains theoriginal Lagrangian.) More precisely, we establish that Theorem 1.2.
There exists a Lagrangian torus L ⊂ ( B , ω ) which isHamiltonian isotopic inside ( C , ω ) to the standard product torus S (1 / √ × S (1 / √ , but where every such Hamiltonian isotopy necessarily satisfies φ t H t ( L ) ∩ S = ∅ for at least one value t ∈ [0 , . In addition, we may assume that one ofthe following holds: • L ⊂ B \ B ( p / − ǫ ) , whenever ǫ > is sufficiently small, or • L ⊂ B ( √ − r ) ⊂ B , whenever r ∈ (0 , / . The torus L is constructed in Section 4 by explicit means involving aprobe, which is a tool that was invented in [23] by D. McDuff; see Figures 4 DIMITROGLOU RIZELL and 5 for its depiction. These examples are thus of a more elementary kindthan the tori constructed by Vianna (which are rather cumbersome to de-scribe explicitly inside the ball). In fact, the example that we consider canbe identified with the monotone Chekanov torus inside C P , but where theembedding of B ֒ → C P that contains the torus is obtained by removing aline in the complement of the torus which is different from the “standard lineat infinity”. In order to distinguish L from a product torus inside the unitball it suffices to compactify the ball to C P = B , and then to use the clas-sical result by Y. Chekanov and F. Schlenk [7] that the monotone Chekanovtorus is not Hamiltonian isotopic to a product torus inside C P . See Section1.4 for a discussion about how the holomorphic curves distinguish L fromthe product torus in this case.Additionally, in conjunction with Theorem 1.1, we can conclude that L is exotic also in the following sense which (at least a priori) is stronger: Corollary 1.3.
The Lagrangian torus L in Theorem 1.2 not in the image of S (1 / √ × S (1 / √ under any symplectomorphism ( B , ω ) ∼ = −→ ( B , ω ) . Gromov width of the complement of Lagrangians.
The Gro-mov width is a well-studied symplectic capacity that was introduced byM. Gromov in [15], which for a symplectic manifold ( X , ω ) is equal to thesupremum sup (cid:8) π · r ; ∃ ϕ : ( B ( r ) , ω ) ֒ → ( X, ω ) (cid:9) taken over all symplectic embeddings of open balls.There are previous computations of the Gromov width of the complementof certain Lagrangian submanifolds; we refer to [3] by P. Biran as well as [4,Section 6.2] by P. Biran and O. Cornea, who showed that the Gromov widthof the complement of the monotone Clifford torus inside C P is equal to π /
3. In [9, Theorem 1.6] K. Cieliebak and K. Mohnke give a strong boundfor the Gromov width of the complement of arbitrary Lagrangian tori in C P n ; their bound in particular implies that the complement of an arbitrarymonotone torus L ⊂ C P has Gromov bounded from above by π / π/ ω FS . (Warning: in the latter paper the Fubini–Studyform on C P is rescaled by a multiple of two relative our convention.) Remark 1.2.
Part (2) of Theorem 1.1 gives a partial answer to [20, Conjec-ture 4.2], since it shows that the Clifford torus is the unique monotone torusinside ( C P , ω FS ) that admits an open symplectic ball ( B ( p / , ω ) inits complement, albeit under the additional assumption that the embeddinghas a symplectic extension to the closed ball.The latter condition could be removed, if it is true that any monotoneLagrangian torus contained inside the closure of such an open symplectic ballis Hamiltonian isotopic to a product torus (which is the case whenever thereis a symplectic extension of the embedding to the closed ball D ( p / ULKY HAMILTONIAN ISOTOPIES OF TORI 5
On the negative side, we note that the Gromov width of the complementfails to distinguish at least some of the monotone tori in C P : Proposition 1.4.
The monotone Clifford torus, Chekanov torus, as wellas Vianna’s first torus from [28] inside C P (i.e. T (1 , , , T (1 , , and T (1 , , using Vianna’s notation) all have complements with Gromov width π / .Proof. For the Clifford torus we can take the obvious symplectic ball cen-tered at the origin. For the Chekanov torus we can find symplectic balls ofradius s for any s < p /
3; see Figure 1. For the torus T (1 , ,
25) we leave itto the reader to check that the construction of T (1 , ,
25) from [28], i.e. themutation of T (1 , ,
4) along a suitable embedded Lagrangian disc, can beperformed in the complement of these balls. The key point is that one ofthe two Lagrangian discs considered by Vianna itself lives in the complementof these balls. (cid:3) u /πu /π r + ǫ / /
21 1 / BL Cl ℓ ∞ L Ch Figure 1.
For any ǫ >
0, the version of the Chekanov torus L Ch inside B whose symplectic action assumes the values Z πr with r ∈ [1 / √ ,
1) can be placed over the diagonal in-side the region { u /π < r + ǫ } . (When r = 1 / C P .) For any neigh-bourhood of B = { u /π ≥ r + ǫ } there exists a symplecticembedding of B ( √ − r − ǫ ) which moreover can be takento be disjoint from ℓ ∞ ; see L. Traynor’s work [27].It is interesting that for tori in B , the Gromov can in fact be used todistinguish certain “large” tori which are monotone inside the ball . Proposition 1.5. (1) The Gromov width of B \ S ( r ) × S ( r ) is equalto π r whenever r ∈ [ p / , .(2) If a Lagrangian torus L ⊂ B has symplectic action which assumesthe values Z πr for some fixed r ∈ [1 / √ , and satisfies the propertythat the Gromov width of B \ L is strictly greater than π / , thenit is Hamiltonian isotopic to a product torus. DIMITROGLOU RIZELL
Proof. (1): Considering the round balls centered at the origin we concludethat the Gromov width is bounded from below by π r . On the other hand,it is also bounded from above by the same number by [9, Theorem 1.6].(Alternatively one could argue along the lines of [4, Corollary 6.5].)(2): This is a direct consequence of Part (1) of Theorem 1.1. (cid:3) Example 1.3.
The Chekanov torus inside ( B , ω ) whose symplectic actionclass assumes the values Z πr admits balls in its complement of radius √ s for any s < / − r ; see Figure 1. Its complement thus has Gromov widthequal to some value w ∈ [ π (2 / − r ) , π / Application to knotted symplectic embeddings.
A typical sym-plectic embedding problem concerns the question whether there exists anembedding ( Y n , dλ Y ) ֒ → ( X n , Cdλ X )of a symplectic manifold into e.g. an open Liouville domain ( X, Cλ X ) forsome C > . Here we assume that X is the interior of a compact Liouvilledomain with smooth boundary, while Y is compact subset of a symplecticdomain with a sufficiently well-behaved boundary. Typically one is inter-ested in the case when an obvious, or even canonical, such embedding existsfor all C ≫ C > Y and X . One notable such instance is the seminalwork [24] by D. McDuff, which answers the question when an ellipsoid canbe embedded into a ball.Many of the natural examples of domains of the form Y ⊂ ( C n , ω ) thathave been studied in the literature have the feature that ∂Y is foliated by(possibly degenerate) Lagrangian standard product tori. Most attentionhas been given to domains for which the standard Liouville vector field ζ moreover is transverse to ∂Y. Such domains include closed balls D n ( r ) , closed ellipsoids E ( a, b ) := { π k z k /a + π k z k /b ≤ } , as well as polydiscs D ( a ) × D ( b ) (the latter has a smooth boundary with corner equal to aLagrangian product torus). Domains of this type are typically depicted bytheir image under the standard momentum map µ : C → R , ( z , z ) ( u , u ) = π ( k z k , k z k ) . ULKY HAMILTONIAN ISOTOPIES OF TORI 7
In this manner we obtain a direct connection between symplectic embeddingproblems and embedding problems for families of Lagrangian tori. This di-rection was taken in the work [18] by R. Hind and S. Lisi, [9] by K. Cieliebakand K. Mohnke, [16] by J. Gutt and M. Hutchings, and [19] by R. Hind andE. Opshtein.In the case when there exists a symplectic embedding ( Y n , dλ Y ) ֒ → ( X n , dλ X ) one can further ask the question whether two different suchembeddings have images that can be made to coincide after a symplecto-morphism of the ambient space ( X, dλ X ) . It was shown by J. Gutt andM. Usher [17] that this is not necessarily the case, even if such a symplec-tomorphism exists for the completion of (
X, dλ X ) to a Liouville manifold( X, dλ X ). The same authors calls an embedding symplectically knotted (relative some other embedding) if there exits an ambient symplectomor-phism inside the completed Liouville domain that takes the image of oneembedding to the other, but when no such symplectomorphism exists of theoriginal Liouville domain.We now show that, in view of Corollary 1.3, the embedding of a domaincan be shown to be symplectically knotted by considering Lagrangian toricontained inside its boundary. Theorem 1.6.
Let Y ⊂ {k z k + 2 k z k ≤ } ⊂ ( D , ω ) be any closed symplectic domain that satisfies Y ∩ {k z k + 2 k z k = 1 } = S (1 / √ × S (1 / √ . There exists a symplectic embedding φ : ( Y, ω ) ֒ → ( B , ω ) for which the monotone Lagrangian torus S (1 / √ × S (1 / √ ∈ ∂Y ismapped to a torus L as in Theorem 1.2 and such that, for any ǫ > , onecan find a symplectomorphism Φ : ( B (1 + ǫ ) , ω ) → ( B (1 + ǫ ) , ω ) thatsatisfies Φ( φ ( Y )) = Y . The construction of the symplectic embedding is a slight extension ofthe construction of the Lagrangian L in the proof of Theorem 1.2, and itis performed in Section 4.1. In view of Corollary 1.3 combined with [10,Theorem 1.2] we can now conclude that: Corollary 1.7.
Assume that Y ⊂ D ( p /
3) = {k z k + k z k ≤ / } is satisfied in addition to the above. Then the embedding φ ( Y ) ⊂ ( B , ω ) issymplectically knotted relative the canonical inclusion Y ⊂ ( B , ω ) . Proof.
Any symplectomorphism Φ that takes φ ( Y ) to Y takes the Lagrangiantorus φ ( S (1 / √ × S (1 / √ L to a Lagrangian torus that is containedinside D ( p /
3) and which is extremal in the sense of [9] relative the latter
DIMITROGLOU RIZELL ball. By [10, Theorem 1.2] it then follows that L must be contained entirelyinside the boundary ∂D ( p /
3) = S ( p / Y ∩ S ( p /
3) = S (1 / √ × S (1 / √ . Assuming the existence of such a symplectomorphism Φ : B ∼ = −→ B wecan thus deduce the existence of a symplectic embedding of a closed ballΦ − D ( p / ⊂ B which intersects the exotic torus L only along itsboundary. Using Part (1) of Theorem 1.1 we obtain a Hamiltonian iso-topy from L to a standard product torus inside B . This yields the soughtcontradiction with Theorem 1.2 (cid:3) Example 1.4.
The above method in particularly yields a symplecticallyknotted embedding of the polydisc Y = D (1 / √ × D (1 / √ ⊂ ( B , ω )into the unit ball. This domain was not considered in [17], and seems tobe of a rather different nature than the examples therein. It is unclearto the author if this embedding remains symplectically knotted also inside B (1) × B (1); if this is the case, then it would answer Question 1.9 in theaforementioned paper.1.3. Proposed notion: Bulky Hamiltonian isotopy.
In view of theprevious theorem, we find the following proposed definition to be natural.Consider two subsets A , A ⊂ ( X n , dλ ) of a compact Liouville cobordismwith smooth boundary ∂X = ∂ + X ⊔ ∂X − (the latter being the decomposi-tion into its convex and concave components), and denote by ( X n , dλ ) thecompletion of ( X, dλ ) to a Liouville cobordism with noncompact cylindricalends. In particular,( X \ ( X \ ∂X ) , dλ ) ∼ = ([0 , + ∞ ) × ∂ + X ⊔ ( −∞ , × ∂ − X, d ( e s λ | T ∂X ))where the latter exact symplectic manifold is the symplectisation of theboundary of (
X, dλ ) . Definition 1.1.
A Hamiltonian isotopy from A to A inside the completionof ( X, dλ ) , i.e. a Hamiltonian isotopy φ tH t : ( X, dλ ) → ( X, dλ )which satisfies φ H t = Id , and φ H t ( A ) = A , is said to be a bulky Hamiltonian isotopy from A to A relative X if thereexists no smooth one-parameter family φ t,s of Hamiltonian isotopies of thesame kind that satisfies φ t, = φ tH t , while φ t, ( A ) ⊂ X holds for all t ∈ [0 , . ULKY HAMILTONIAN ISOTOPIES OF TORI 9
In other words, we can rephrase the Theorem 1.2 as the statement that“the Hamiltonian isotopies in C that take L to a standard torus are allbulky relative the unit ball.”We end with an additional example, again of rather elementary nature. Example 1.5.
Again consider the monotone product torus L Cl = S (1 / √ × S (1 / √ . There exists a Hamiltonian isotopy which takes L to itself, through toricontained entirely inside S ( p / U (2)which start with the identity and end with ( z , z ) ( z , z ). However,there exists no such Hamiltonian isotopy which is contained entirely inside B ( p / × B ( p / L Cl ⊂ ( C P × C P , (2 / ω FS ⊕ (2 / ω FS )corresponding to the Maslov-two discs that pass through the divisor {∞} × C P ∪ C P × {∞} . More precisely, we need to consider a version of the superpotential whichalso keeps track of the relative homology classes if the discs. We again referto [7] for the computation of the superpotential, and note that the relativehomology classes of these discs are invariant for Hamiltonian isotopies of thetorus that stay away from the divisor. Rephrased using our newly definednotion: “the Hamiltonian isotopies inside C that interchange the two fac-tors of L Cl are bulky with respect to any Liouville domain contained inside B ( p / × B ( p / Holomorphic curve invariants.
As previously mentioned, we rely onthe work [7] in order to show the nonexistence of a Hamiltonian isotopy in-side B which takes the product torus to L in Theorem 1.2. It is worthwhileto elaborate a bit on the precise mechanism which distinguishes betweenthese tori. The tool used in [7] for distinguishing different Hamiltonian iso-topy classes of Lagrangian tori is the theory of pseudoholomorphic curves,more precisely by comparing the superpotentials of the different tori. Thesuperpotential is an invariant which counts the number of families of pseu-doholomorphic Maslov-two discs with boundary on the torus; see e.g. thework [2] by D. Auroux.The lesson that we learn from the examples in Theorem 1.2 is that, eventhough we are interested in obstructing Hamiltonian isotopy inside the ball,it is completely crucial that we consider the superpotential which countsholomorphic discs in all of C P = B . If instead the superpotential insidethe ball was to be considered, it would give the same answer for L and theproduct torus; the reason is that, for a monotone Lagrangian torus insidethe ball, the superpotential is invariant under Hamiltonian isotopy as wellas linear rescalings. In conclusion, for the torus L considered here, it is the terms in the su-perpotential that count the discs in C P passing through the line at infinity ℓ ∞ = C P \ B that distinguish it from the product torus. In general this isnot a well-defined count if the torus is merely assumed to be monotone insidethe ball (being monotone inside C P is a stronger condition). However, inour case the count of Maslov-two discs that pass through the line at infinityis well-defined for the following reason. For an almost complex structureon C P = B which makes the line at infinity holomorphic, the class ofpseudoholomorphic discs of Maslov index two that pass through the line atinfinity are a priori of minimal symplectic area, given the symplectic actionproperties of the tori L under consideration. Hence, the count of these discsis invariant under deformations of the almost complex structure that keepsthe line at infinity holomorphic. However, for a ball which is larger thanthe unit ball, the corresponding discs are no longer of minimal symplecticarea. Since we then cannot exclude bubbling from occurring while varyingthe almost complex structure, we no longer have any reasons to expect thatthe number of such disc is an invariant that can be used to obstruct theexistence of a Hamiltonian isotopy. (Indeed, Theorem 1.1 can provide aHamiltonian isotopy in this case.)2. The proof of Part (1) of Theorem 1.1
Denote by ϕ : ( D ( p / , ω ) ֒ → ( B , ω ) ,L ⊂ B \ ϕ ( B ( p / , the symplectic embedding whose existence is assumed.After the application of the positive Liouville flow φ ǫλ to both L and ϕ ( B ( p / ǫ > φ ǫλ is the conformal sym-plectomorphism given by scalar multiplication with e ǫ ) we may in additionassume that the new Lagrangian torus has the symplectic actions Z πe ǫ r while, of course, it now is disjoint from the rescaled image e ǫ ϕ ( B ( p / e ǫ p /
3. (Here we have madeuse of the assumption in Theorem 1.1 that the closure of the image of ϕ iscontained inside the open unit ball.) If we manage to construct the soughtHamiltonian isotopy after the above rescaling, the general case will then alsofollow immediately. Indeed, it suffices to rescale the produced Hamiltonianisotopy by the negative-time Liouville flow φ − ǫλ . In view of the above, we will now restrict attention to the case when r > / √ ϕ : ( B ( e ǫ p / , ω ) ֒ → ( B \ L, ω ) satisfies ϕ ( B ( e ǫ p / , ω ) ⊂ B \ L, i.e. we can find a closed ball of radius e ǫ p / L . ULKY HAMILTONIAN ISOTOPIES OF TORI 11
A neck-stretching sequence.
Recall that symplectic reduction ap-plied to the boundary ∂D = S → C P produces a compactification B = C P where the latter is equipped with the Fubini–Study symplectic form ω FS for which a line has symplectic are equal to R ℓ ω FS = π . In particular,using ℓ ∞ to denote the line at infinity, we have ( B , ω ) = ( C P \ ℓ ∞ , ω FS ).The main technical ingredient that we will need is neck-stretching arounda hypersurface of contact type that can be identified with a small unit normalbundle around L. Neck-stretching first appeared in work [13] by Y. Eliash-berg, A. Givental, and H. Hofer, and was later made precise in the SFT com-pactness theorem [5] by F. Bourgeois, Y. Eliashberg, H. Hofer, C. Wysocki,and E. Zehnder and independently [8] by K. Cieliebak and K. Mohnke.Roughly speaking, neck-stretching is a conformal limit in which the sym-plectic manifold splits into several pieces, along with the pseudoholomorphiccurves that it contains. We are interested in considering the neck-stretchinglimits of the foliation of pseudoholomorphic lines of C P , which persistsfor arbitrary compatible almost complex structures by Gromov’s classicalresult [15]. We work in the same setting of [12, Section 3] and direct thereader to that article for the technical details.A Weinstein neighbourhood of L becomes a concave cylindrical end(( −∞ , log (4 δ )] × U T ∗ L, d ( e t ( α ))) ֒ → (cid:16) B \ ϕ ( B ( e ǫ p / , ω (cid:17) after removing the Lagrangian torus, i.e. when considered in the symplecticmanifold B \ L. Here α = pdq | T ( UT ∗ L ) is chosen to be the contact form onthe unit cotangent bundle for a choice of flat metric on T , and δ > { log (3 δ ) } × U T ∗ L ֒ → B \ ϕ ( B ( e ǫ p / L . Stretch-ing the neck amounts to choosing a particular sequence J τ , τ ≥ , of com-patible almost complex structures on C P determined as follows. • All J τ are fixed outside of the neighbourhood[log (2 δ ) , log (4 δ )] × U T ∗ L of the above spherical normal bundle of L ; • All J τ are equal to the standard integrable complex structure i nearthe divisor ℓ ∞ , while inside the subset(( −∞ , log (2 δ )] × U T ∗ L, d ( e t ( α )))of the concave end they are equal to the almost complex structure J std defined in [12, Section 4]; and • Near the spherical normal bundle of L the almost complex structure J τ is the pull-back of the cylindrical almost complex structure J cyl2 DIMITROGLOU RIZELL determined by the flat metric under a (non-symplectic!) diffeomor-phism [log (2 δ ) , log (4 δ )] ∼ = −→ [log (2 δ ) , log (4 δ ) + τ ]extended to the U T ∗ L -factor of the symplectisation by the identity.In particular, one obtains a limit compatible almost complex structure on C P \ L that we denote by J ∞ and which is cylindrical on the entire concaveend near L . We refer to [12, Sections 3 and 4] for more details.Now comes the point when we will use the existence of the embeddingof the symplectic ball as stipulated by the assumptions of the theorem; wechoose the neck-stretching sequence so that( † ) the almost complex structures J τ all coincide with the push-forwardof the standard almost complex structure i under the symplectomor-phism ϕ in the subset ϕ ( B ( e ǫ p / ⊂ B \ L holds in addition to the bullet points listed above.For the analysis that we conduct it is crucial that the cylindrical almostcomplex structure is chosen with respect to the contact form on U T ∗ L in-duced by the flat metric on L . The reason is that, for instance, the non-existence of contractible geodesics makes the breaking analysis of pseudo-holomorphic curves significantly simpler. Recall that the SFT compactnesstheorem implies that a sequence of finite energy J τ -holomorphic curves hasa subsequence that converges to a pseudoholomorphic building which con-sists of several levels of punctured finite-energy pseudoholomorphic curves[5], [8]. These finite energy curves are asymptotic to Reeb chords on U T ∗ L, i.e. lifted geodesics for the flat metric in the case under consideration.We will only be interested in the case of a sequence of J τ -holomorphicdegree one curves in C P , which are usually called (pseudoholomorphic)lines . Recall that for any given tame almost complex structure there existsa unique pseudoholomorphic line through any two given points, or throughone fixed point with a given complex tangency, by Gromov’s classical result[15]. In addition, any pseudoholomorphic line is embedded. In the case ofan SFT-limit of lines the corresponding building consists of: • a non-empty top level consisting of punctured J ∞ -holomorphicspheres in C P \ L ; • a (possibly zero) number of middle levels consisting of puncturedpseudoholomorphic spheres in R × U T ∗ L for the cylindrical almostcomplex structure J cyl ; and • a (possibly empty) bottom level consisting of punctured pseudo-holomorphic spheres in T ∗ L for the almost complex structure J std defined in [12, Section 4],where the punctured spheres moreover can be glued along the punctures toyield a continuous map of degree one from a single sphere into C P . SeeFigure 2 for examples. Of course, it is also possible that the limit curveconsists of a single component contained entirely in the top level; this must ULKY HAMILTONIAN ISOTOPIES OF TORI 13 then be a sphere without any punctures, and we call it unbroken (it isa compact J ∞ -holomorphic sphere of degree one in the usual sense). Bypositivity of intersection, established in [21] by D. McDuff, one can deducethat any component arising in the limit is a (trivial or nontrivial) branchedcover of an embedded punctured sphere. Note that the almost complexstructures J std and J cyl used here have the feature that the canonical T -action by isometires on the flat torus L lift to an action by biholomorphisms;see [12, Section 4].In the following we will use the (Fredholm) index ind( u ) of a puncturedsphere u to denote the expected dimension of the component of the modulispace which contains it, where the moduli space consists of curves up toreparametrisations considered without asymptotic constraints at the Bottmanifolds of Reeb orbits. For a k -punctured sphere u inside C P \ L thisindex can be expressed by ind( u ) = k − µ ( u )where u is its compactification of u to a chain in C P with boundary on L and where µ denotes the Maslov class. See [12, Section 3] for more details.Here we recall some crucial results from [12]. Lemma 2.1 (Section 3 [12]) . (1) For any punctured sphere u in C P \ L , the Fredholm index of a branched cover ˜ u satisfies ind(˜ u ) ≥ d · ind( u ) where d ≥ is the degree of the cover;(2) For a generic almost complex structure J ∞ on C P \ L it is the casethat ind( u ) ≥ for any punctured sphere in C P \ L , where thisindex moreover coincides with the dimension of the moduli spacewhich contains u whenever the curve is simply covered. In addition ind( u ) ≥ is odd if u is a plane; and(3) If v , . . . , v N ⊂ C P \ L and w , . . . , w N are punctured spheres whichconstitute the components of a broken line, where w i reside either inmiddle levels R × U T ∗ T or the bottom level T ∗ T , then the equality N X i =1 ind( v i ) − N X i =1 χ ( w i ) = 4 is satisfied, where χ ( w i ) ≤ .Proof. The inequality in (1) was shown in the proof of [12, Lemma 3.3]. Part(2) is simply [12, Lemma 3.3]. The equality in (3) was shown in the end ofthe proof of [12, Proposition 3.5] (the right-hand side here is 4 instead of 2,as in the reference, since the latter considered indices after a generic pointconstraint). (cid:3)
Straight-forward topological considerations of the possibilities of buildingsin the class of a line in conjunction with the previous lemma now give usthe following useful result:
Corollary 2.2.
Assume that J ∞ is a generic almost complex structure on C P \ L . Let v , . . . , v N ⊂ C P \ L and w , . . . , w N be punctured sphereswhich constitute the components of a broken line, where w i reside either inmiddle levels R × U T ∗ T or the bottom level T ∗ T . Then • There are at least two planes among the top-level components { v i } ,and any component v i in the top level satisfies ind( v i ) ∈ { , , , } ;and • If some v i satisfies ind( v i ) ≥ , then – there are precisely two planes among the components { v i } in thetop level, – the remaining components v i , i = i , in the top level satisfy ind( v i ) ≤ , and – all components { w i } in the middle and bottom levels are cylin-ders, while the top components { v i } are either cylinders or planes. Extracting an SFT-limit of lines.
Choose a generic pointpt ∈ ϕ ( B (( e ǫ − p / J τ -holomorphic lines that pass through pt as well as somesecond fixed point on L . (By Gromov’s result [15] there is always a uniquesuch line.) A sequence of such lines for which τ → + ∞ has a convergent sub-sequence by the SFT compactness theorem [5]. Due to the point constrainton L the limit is a pseudoholomorphic building in the class of a “broken”line that passes through both pt ∈ B \ L as well as some point on the torus L. The monotonicity property for the symplectic area of pseudoholomorphiccurves (see [26] by J.-C. Sikorav) applied to the ball ϕ ( B ( p / ⊂ ϕ ( B ( e ǫ p / † ) satisfied by the almost complex structure impliesthat Z A pt ω FS ≥ π / A pt ⊂ C P \ L in the top level of the limit buildingthat passes through the point pt . In particular, since the total area of thesecomponents is R ℓ ∞ ω FS = π , there is a unique such component. From thiswe are able to conclude that: Lemma 2.3.
For a generic point pt ∈ ϕ ( B (( e ǫ − p / and a genericperturbation of the almost complex structure J ∞ in a unit normal bundle of L ∪ ℓ ∞ , we can assume that the component A pt is • disjoint from ℓ ∞ , • of symplectic area πr (where thus r < / √ ) and • of index three (i.e. Maslov index four) and embedded (thus in partic-ular it is not a branched cover). ULKY HAMILTONIAN ISOTOPIES OF TORI 15
Proof.
Every broken pseudoholomorphic line must contain a plane that isdisjoint from ℓ ∞ by the flatness of the metric on L used in the constructionof the neck-stretching sequence; see [12, Section 3]. Since the puncturedspheres inside C P \ ( ℓ ∞ ∪ L ) are of symplectic area equal to kπr > kπ/ k = 1 , , , . . . , by our assumptions on L , and since A pt is of symplecticarea at least π / A pt is disjoint from ℓ ∞ . In fact, we also conclude that its symplectic area isprecisely equal to 2 πr (i.e. k = 2), where thus r < / √ ℓ ∞ except A pt , which implies that A pt is a plane. (Forthese arguments we use the property that a sphere of degree one is of totalsymplectic area equal to π and that every component in the top level ofthe building contributes positively to the symplectic area, while the othercurves contribute zero to the symplectic area.)The Fredholm index of A pt is determined as follows. First we observethat ind( A pt ) is odd and at most equal to three for a generic almost complexstructure by Lemma 2.1 and Corollary 2.2 (to achieve genericity it sufficesto perturb near L ). Finally, since the point pt was chosen to be generic, wecan assume that A pt is of index at least two, and moreover not a branchedcover of a plane of index one, as follows by a simple dimension count. (cid:3) Recall that the plane A pt must be embedded by positivity of intersection[21], since it is not a branched cover. Lemma 2.4.
The J ∞ -holomorphic plane A pt ⊂ C P \ ( ℓ ∞ ∪ L ) of Fredholmindex three (Maslov index four) produced by the above lemma has a simplycovered asymptotic Reeb orbit.Proof. Consider a sequence of J τ -holomorphic lines which satisfy a generictangency condition at a generic point pt ′ ∈ A pt as τ → + ∞ . Using the SFTcompactness theorem, we can extract a limit holomorphic building from aconvergent subsequence.We first claim that the limit component is smooth at the point where thetangency is taken. Indeed, positivity of intersection implies that in someneighbourhood of the point pt ′ , the underlying simply covered curve mustbe smooth; see [21]. There is still the possibility that the building containsa multiple cover of A pt branched at pt ′ (such a curve satisfies any prescribedtangency condition). This is however not possible, since the symplectic areaof a line is equal to π which is strictly less than k R A pt ω for any k > unbroken J ∞ -holomorphic line ℓ ⊂ C P \ L (i.e. a pseudoholomorphic curve without punc-tures) that satisfies the tangency. (Any top component of a broken linecomes in a moduli space of dimension at most three.) Since the connectinghomomorphism H ( B , L ) δ −→ H ( L ) is an isomorphism, we see that A pt • ℓ is divisible by the multiplicity of theorbit. Positivity of intersection [21] allows us to conclude that0 < A pt • ℓ ≤ [ ℓ ∞ ] • [ ℓ ∞ ] = 1 , which shows that this multiplicity is precisely equal to one. (cid:3) C P \ LT ∗ L pt A pt C BDB ∞ A ∞ ℓ ∞ ℓ ∞ γ γ ′ η − η η ′ − η ′ Figure 2.
The numbers indicate the Fredholm indices ofthe components (i.e. dimension of the moduli space of therespective component without any asymptotic constraint inthe Bott manifolds of periodic Reeb orbits). The asymptoticorbits are lifts of the geodesics on L in the homology classes ± η ∈ H ( L ) and ± η ′ ∈ H ( L ). The bottom componentare the complexifications of the unoriented closed geodesics γ ⊂ L and γ ′ ⊂ L in the two different homology classes.2.3. A condition for Hamiltonian unknottedness.
The monotonicitycombined with Lemma 2.3 now implies that the broken line produced inthe previous subsection consists of precisely two components in its top level:the embedded plane A pt together with an embedded plane A ∞ that passesthrough ℓ ∞ ; both are simply covered and have simply covered asymptotics.Further, the classification of pseudoholomorphic cylinders in [12, Section4] implies that the component in the bottom level is a standard cylinder;roughly speaking, these are the complexifications of the geodesic in the class ± η ∈ H ( L ) to which the planes are asymptotic. Even if the original brokenline does not pass through L, we can replace the cylinder in the bottomlevel with a cylinder that intersects L cleanly precisely in the correspondinggeodesic; such a configuration is shown on the left in Figure 2.Since the involved asymptotic orbits are simply covered by Lemma 2.4,the smoothing technique from [12, Section 5] can then be used to produce asmoothing of the above building to an embedded symplectic sphere. More-over, the resulting symplectic sphere intersects L cleanly along the simplycovered closed geodesic in class ± η ∈ H ( L ) to which the planes A pt and A ∞ are asymptotic. In other words, the assumptions of the below classificationtheorem is met, from which the existence of the sought Hamiltonian isotopythen follows. ULKY HAMILTONIAN ISOTOPIES OF TORI 17
Theorem 2.5.
Consider a Lagrangian torus L ⊂ ( B , ω ) = ( C P \ ℓ ∞ , ω FS ) for which there exists a tame almost complex structure J on C P that isstandard near ℓ ∞ , and for which some J -holomorphic line ℓ has the propertythat ℓ ∩ L is a simple closed curve γ ⊂ L of Maslov index four (computedusing the trivialisation of T B ). Then L is Hamiltonian isotopic inside ( B , ω ) to a product torus. The proof of Theorem 2.5.
By the “refined” version of the nearbyLagrangian conjecture for the cotangent bundle ( T ∗ T , d ( p dθ + p dθ )) of atorus established in [11, Theorem B] it suffices to find a Hamiltonian isotopyof L supported inside B that places the torus inside the subset( C P \ ( ℓ ∞ ∪ { z z = 0 } ) , ω FS ) ∼ = ( T × U, d ( p dθ + p dθ ))in a position which, moreover, makes it homologically essential inside thesame neighbourhood. Here z i denote the standard affine coordinates on C P \ ℓ ∞ ∼ = C , and U = { p + p < π, p , p > } ⊂ R is an open convex subset.To produce a Hamiltonian isotopy of L to the sought position we will relyon the techniques from the proof of [11, Lemma 5.7], by which it sufficesto find two J -holomorphic lines ℓ i ⊂ C P , i = 1 , , that intersect ℓ ∞ intwo distinct points, and for which L ⊂ C P \ ( ℓ ∞ ∪ ℓ ∪ ℓ ) is homologicallyessential. Namely, after a deformation near the nodes of ℓ ∞ ∪ ℓ ∪ ℓ that canbe performed by hand, one can show that there exists a Hamiltonian isotopythat fixes ℓ ∞ setwise while it takes the union ℓ ∞ ∪ ℓ ∪ ℓ of J -holomorphiclines to the three standard lines ℓ ∞ ∪ { z z = 0 } .In order to construct the J -holomorphic lines ℓ i , i = 1 ,
2, we need to againconsider a neck-stretching sequence J τ induced by a flat metric on L. It willfurthermore be crucial that: • J τ = i near ℓ ∞ , and • the line ℓ , whose existence we are assuming, remains J τ -holomorphicfor all τ ≥ ℓ to converge to a building as shown on the left inFigure 2 when taking the limit τ → + ∞ . We use ± η ∈ H ( L ) to denote thehomology class of the unoriented simple closed curve γ = L ∩ ℓ on L . Thetwo planes in the top level of the limit building will be asymptotic to thetwo lifts of a geodesic which coincides with this closed curve for a suitablechoice of flat metric on the torus.To ensure that J τ can be made to satisfy the second bullet point above,we need the following intermediate result. Lemma 2.6.
After a Hamiltonian isotopy supported in a small Weinsteinneighbourhood of L , the line ℓ can be made to coincide with a “complex-ified geodesic” (i.e. a J std -holomorphic cylinder explicitly described in [12, Section 4] ) for the flat metric on L inside some even smaller Weinsteinneighbourhood.Proof. Recall the standard fact that any smooth isotopy of L can be ex-tended to an ambient Hamiltonian isotopy of its Weinstein neighbourhood D ≤ δ T ∗ L. In this manner we can deform ℓ in order to make it intersect L ina closed geodesic in the class [ ℓ ∩ L ] ∈ H ( L ) for any choice of flat metric.The subspace of linear symplectic two-planes inside ( C , ω ) that intersectsome fixed symplectic two-plane non-transversely is a contractible space.This fact can be readily used to deform ℓ in order to, first, make it tangentto the complexification of ℓ ∩ L and, second, to make it even coincide withthe complexification in some tiny neighbourhood. (cid:3) By the above we can assume that ℓ remains J τ -holomorphic for all τ ≥ τ → + ∞ . We then combine this result with theexistence of broken lines from [12]. Lemma 2.7.
Under the assumption of the existence of the line ℓ as above,there is a stretched almost complex structure J ∞ on C P \ L which admitsthe following broken lines.First, there is a one-parameter family of buildings parametrised by t ∈ ( − ǫ, ǫ ) which consist of: • Top level:
Two J ∞ -holomorphic planes A t pt , A t ∞ ⊂ C P \ L , where A t pt is of index index 3, passes through pt ∈ B , and is disjoint from ℓ ∞ , while A t ∞ is of index 1 and intersects ℓ ∞ transversely in a singlepoint; and • Bottom level:
A single J std -holomorphic cylinder C t which is thecomplexification of the geodesic γ t .In addition, the closed geodesics γ t are simply covered and provide a foliationthe open annulus S γ t ⊂ L .Second, there is a building which consists of: • Top level:
Two J ∞ -holomorphic planes B, B ∞ ⊂ C P \ L , where B is of index index 1 and is disjoint from ℓ ∞ , while B ∞ is of index3 and intersects ℓ ∞ transversely in a single point; and • Bottom level:
A single J std -holomorphic cylinder D which is thecomplexification of the geodesic γ ′ .In addition, the closed geodesic γ ′ intersects each geodesic γ t transversely ina single point.These two buildings are depicted in Figure 2.Proof. The building of the first kind for t = 0 can be taken to be the SFT-limit of the J τ -holomorphic lines ℓ for τ → + ∞ ; here we rely on the previousLemma 2.6. The existence of the remaining buildings of the same typecan be deduced by applying the automatic transversality theorem, which ULKY HAMILTONIAN ISOTOPIES OF TORI 19 is due to C. Wendl in the SFT setting [31, Theorem 1], to the two J ∞ -holomorphic planes in the top level of the previously constructed building.To use automatic transversality it crucial that the two components in thetop level are immersed planes of positive index. The family of cylinders inthe bottom level of the building can then be constructed explicitly; see [12,Section 4].The existence of the building in the second bullet point was establishedin [12, Proposition 5.7] by considering a limit of lines with two point con-straints: one point on L and one point near ℓ ∞ .The intersection property of the geodesics γ t and γ ′ is clear by positivityof intersection: they must intersect since their homology classes are notcollinear, while [ ℓ ∞ ] • [ ℓ ∞ ] = 1 implies that they cannot intersect in morethan one point. (cid:3) In the remainder of the proof we produce the sought pseudoholomorphiclines by following an idea due to K. Mohnke [25]: extract the sought linesby taking point constraints at suitable pseudoholomorphic planes, in orderto achieve the required linking behaviour with L . Here it is crucial thatboth classes of buildings supplied by Lemma 2.7 exist; in general we areonly guaranteed the second type of broken line described there.To obtain the lines we will consider the limits of lines that that satisfy asuitable point constraint after stretching the neck. The key technical stepis the following lemma. Lemma 2.8.
Consider the SFT-limit of the unique J τ -holomorphic linesthat pass through two generic points pt ∈ A t pt and pt ∈ B ∪ B ∞ as τ → + ∞ . Then this limit is an unbroken J ∞ -holomorphic line in C P \ L whichpasses through the same two points.Proof. The limit building is a (possibly unbroken) line which intersects theplanes A t ′ pt in a discrete nonempty subset for any t ′ which is sufficiently closeto t . Indeed, either the limit intersects A t pt in a discrete nonempty subset,in which case the statement follows from positivity of intersection, or thelimit building contains a (possibly trivial) multiple cover of the plane A t pt itself as a component in its top level. In the second case we need to use thefact from Lemma 2.7 that A t pt ∩ A t ′ pt = { pt } .If the point pt ∈ A t pt is chosen generically, then we can further useCorollary 2.2 to deduce that the top level component of the limit buildingthat passes through pt must have an underlying simply covered puncturedsphere whose index is at least 2. (A generic such point constraint is of codi-mension two, and if the component is equal to A t pt itself then the statementis automatically true.)If the limit is a broken line, then it consists of precisely two planes ofpositive index together with a number of cylinders in its top, middle, andbottom levels by Corollary 2.2. The classification of J std -holomorphic cylin-ders from [12, Section 4] and the computations of their intersection numbers (see Corollary 4.3 therein) implies that middle and bottom levels of the cylin-der must have asymptotics in the classes ± kη ∈ H ( L ) for some k >
0. (Ifnot, this broken line would intersect the broken lines of the first kind fromLemma 2.7 in more than two points: at least once at pt ∈ C P \ L in the toplevel, and at least once arising from intersections of cylinders in the middleand bottom levels.)If the limit line is broken, then by examining the asymptotic orbits ofthe punctured spheres in the middle and bottom level of the building (thesewere shown to all be geodesics in the homology classes ± kη ), we concludethat the building does not contain any of the two planes B and B ∞ , norany branched cover of these. In particular, the limit building intersects theunion B ∪ B ∞ of planes in a discrete nonempty subset.However, if the building in fact is broken, then we again arrive at a con-tradiction by using positivity of intersection. Indeed, the limit buildingcontains a cylinder in its middle and bottom level with asymptotics in ho-mology classes of the form ± kη by the above and, hence, it must intersectthe cylinder D in the second building of Lemma 2.7 by [12, Corollary 4.3].On the other hand, it also intersects B ∪ B ∞ in a discrete nonempty subset,so these two buildings intersect with a total algebraic intersection numberat least 2, which contradicts [ ℓ ∞ ] • [ ℓ ∞ ] = 1. (cid:3) The above lemma now provides us with the needed pseudoholomorphiclines ℓ and ℓ . More precisely, the line ℓ can be taken to be an unbroken J ∞ -holomorphic line which passes through the two planes A t pt and B , while ℓ can be taken to be the line which passes through the two planes A t pt and B ∞ . The sought linking properties of ℓ i and L now readily follow frompositivity of intersection; these unbroken lines cannot pass through the twoplanes B and B ∞ simultaneously. The obtained configuration of lines isshown schematically in Figure 3.For suitable point constraints, the two lines ℓ i produced above moreoverintersect in a point disjoint from ℓ ∞ . We have thus managed to producethe lines in the sought position, where the previously established linkingproperties imply that L ⊂ C P \ ( ℓ ∪ ℓ ∪ ℓ ∞ ) is homologically essentialas needed. (The latter complement of three lines is diffeomorphic to T × R .) (cid:3) Proof of Part (2) of Theorem 1.1
It may be possible that one can prove Part (2) by reusing the analysisdone in Part (1); however, we here choose to perform a slightly differentneck-stretching argument.First we recall the classical result [22, Corollary 1.5] by D. McDuff whichin particular says that any two symplectic embeddings of closed balls ofthe same radius into C P are Hamiltonian isotopic. From this it suffices ULKY HAMILTONIAN ISOTOPIES OF TORI 21 C P \ LT ∗ L pt A t pt C DB ∞ A t ∞ ℓ ∞ ℓ ℓ ∞ ℓ Bη − η η ′ − η ′ Figure 3.
Two unbroken pseudoholomorphic lines ℓ and ℓ which pass through the different planes of the two build-ings supplied by Lemma 2.7. For the particular configurationshown it follows that L is homologically essential in the com-plement of ℓ ∪ ℓ ∪ ℓ ∞ .to establish the following proposition, and then use the classification of La-grangian tori inside a round sphere (see e.g. [10, Section 3]) which followsfrom elementary techniques. Proposition 3.1.
Any monotone torus L ⊂ C P which is disjoint from B ( p / ⊂ C P is contained entirely inside S ( p / .Proof. The proof is based upon the same idea as the proof of the mainresult in [10] by the author, but where the null-homology of L is given by aneck-stretching argument in a slightly different geometric setting.Assume the contrary, i.e. that L is not contained entirely inside S ( p / S ∼ = S ⊂ C P \ ( D ( p / ∪ L )contained inside a thin Weinstein neighbourhood of L , which thus has thefeature that it links L ⊂ C P nontrivially (recall that L is nullhomologous).From now on the choice of S will remain fixed. In addition, for any classof almost complex structures that are fixed in some given neighbourhood of S , we can now infer that there exists a constant ~ > A ⊂ C P \ L that passesthrough S satisfies the bound Z A \ D ( √ / ω FS ≥ ~ > We stretch the neck around L while keeping the almost complex structurestandard inside D ( p / − δ ) as well as in the above neighbourhood of S .Neck stretching works as described in Section 2.1, with the only differencethat we cannot assume that ℓ ∞ is holomorphic. The crucial feature of oursetup is that we can stretch the neck while keeping the constant ~ > δ > ∈ D ( p / − δ ) closeto 0 must be of area strictly greater than π/
3, and hence consist of a plane A pt of index 3 which passes through pt, an additional plane in the top levelof index 1, together with a single cylinder in the bottom component. Herethe monotonicity of L significantly simplifies the analysis of the possiblebreakings. (It does not make sense to make any claims about whether A pt is disjoint from ℓ ∞ or not, since the latter sphere is not necessarily holomor-phic. This is not a problem, since we do not need to show that A pt has asimply covered asymptotic.)The remainder of the proof is the same as in [10, Section 4.3]. We con-struct a null-homology of L ⊂ C P by evaluation from the one-dimensionalcomponent of the moduli space of planes of index 3 which pass throughthe point pt ∈ D ( p / − δ ) and which contains the plane A pt . (After ablow-up we can consider the strict transform of the moduli problem, whichinstead concerns planes of index 1 that intersect the exceptional divisor.)Here it is important that this component of the moduli space is compact,which is a consequence of the monotonicity of L together with the genericityof the choice of point pt ∈ D ( p / − δ ).Note that the aforementioned planes all have symplectic area π / L , while the area of the same planes concentrated inside theball D ( p / − δ ) is equal to at least Z A pt ∩ D ( √ / − δ ) ≥ π (2 / − δ )by the monotonicity property for symplectic area [26], while using the as-sumption that the almost complex structure is standard inside D ( p / − δ ).Since L and S are nontrivially linked, some of these planes that constitutethe null-homology of L are forced to pass through the subset S , and in viewof (M) their symplectic areas when intersected with C P \ B ( p / − δ )must hence be bounded from below by ~ >
0. Since δ > < δ < ~ / (2 π ). (cid:3) Proof of Theorem 1.2
Probes are a useful tool for constructing Hamiltonian isotopies that wasinvented by McDuff in [23]. For any integer m = 1 , , , . . . we consider the ULKY HAMILTONIAN ISOTOPIES OF TORI 23 u /πu /π u /π u /π − r / / / − r / r / − r / − rP L Cl L Figure 4.
The monotone Clifford torus L Cl = S (1 / √ × S (1 / √
3) can be isotoped to L inside the probe P ⊂ D . We can moreover place L inside B ( √ − r ) for any r ∈ (0 , / . γ γ q q √ r x iy q Figure 5.
Inside the probe P , consider a Hamiltonian iso-topy of the form ψ ( { γ t } × S ) , where ψ ( { γ } × S ) is themonotone product torus.probe P m := µ − { u = π − mu , u > } ⊂ C for the standard momentum map µ : C → ( R ≥ ) ,µ ( z , z ) := ( π k z k , π k z k ) , on ( C , ω ) . We will consider the foliation ψ m : B (1 / √ m ) × S ∼ = −→ P m ⊂ C ∗ × C , (( r, θ ) , ϕ ) (( p − mr , ϕ ) , ( r, θ + mϕ )) , by symplectic discs B (1 / √ m ) ×{ ϕ } in local angular coordinates. Note thatthis map indeed extends smoothly over { } × S . Since the symplectic form ω is pulled back to the standard symplectic form ω on B (1 / √ m ) underthe map ψ m , the characteristic distribution on P m can be seen to be givenby ker( ω | T P m ) = R ∂ ϕ ⊂ T P m . In particular, integrating it, we obtain a trivial symplectic monodromy mapon the symplectic disc leaves. This means that
Lemma 4.1.
For any simple closed curve γ ⊂ B (1 / √ m ) , the image ψ m ( γ × S ) ⊂ ( C , ω ) is an embedded Lagrangian torus. For example, the monotone Clifford torus of symplectic action π/ L := ψ ( S (1 / √ × S ) ⊂ S ( p / . Considering a suitable smooth family of simple closed curves that all boundthe area π/ B (1 / √
3) we obtain a Hamiltonian isotopy L t := ψ ( γ t × S ) ⊂ P ⊂ ( C , ω )of Lagrangian tori. We will take L to be the Clifford torus while L thetorus obtained from the curve γ ⊂ B (1 / √ \ { } shown in Figures 4 and5. Note that this isotopy L t of tori intersects the subset S × { } ⊂ D for some t ∈ (0 , γ t can be disjoint from the origin in B (1 / √
2) for obvious topological reasons
Lemma 4.2 (Gadbled [14]) . The torus L ⊂ B is Hamiltonian isotopic tothe Chekanov torus when considered inside the completion ( C P , ω FS ) ⊃ ( C P \ ℓ ∞ , ω FS ) ∼ = ( B , ω ) ⊃ L of the ball.Proof. The representative of the Chekanov torus described in [14] differsfrom L simple by the linear change of coordinates[ Z : Z : Z ] [ Z , Z , Z ] . In particular, the torus is clearly Hamiltonian isotopic to the standard pre-sentation of the Chekanov torus. (cid:3)
The claim that L is not Hamiltonian isotopic to the standard producttorus inside B then finally follows from the fact that the monotone Cliffordand Chekanov tori inside the compactification ( C P , ω FS ) are not Hamil-tonian isotopic as shown by Chekanov and Schlenk [7]. We can thus take L := L . (cid:3) ULKY HAMILTONIAN ISOTOPIES OF TORI 25
Proof of Theorem 1.6.
The key point is that the Hamiltonian isotopyfrom S (1 / √ × S (1 / √
3) to L inside C that was constructed above canbe taken to fix the hypersurface {k z k + 2 k z k = 1 } ∩ { z = 0 } ⊂ D setwise; this is the hypersurface that contains the “probe” P as well as thetwo Lagrangian tori. To see this, it is convenient to extend the embedding ψ of the probe constructed in the same section to a symplectic embeddingΨ : B ( p (1 − δ ) / × ( − δ, δ ) ֒ → S → C ∗ × C , (( r, θ ) , ( s, ϕ )) ( s + p − r , ϕ ) , ( r, θ + 2 ϕ )) , defined using polar coordinate for some small δ > . Note that ω pulls backto the product symplectic form ω + sds ∧ dϕ on B ( p (1 − δ ) / × (( − δ, δ ) × S ) , while the restriction Ψ | { s =0 } = ψ is the original embedding of the probe P constructed above.One can then realise the Hamiltonian isotopy of the torus in the probeby a suitable lift of a Hamiltonian isotopy of ( B ( p (1 − δ ) / , ω ) that isgenerated by a compactly supported Hamiltonian, to yield a Hamiltonianisotopy of the product( B ( p (1 − δ ) / × (( − δ, δ ) × S ) , ω + sds ∧ dϕ )of symplectic manifolds. The sought Hamiltonian is finally produced bymultiplication with a suitable smooth bump function. (cid:3) References [1] V. I. Arnol ′ d. On a characteristic class entering into conditions of quantization. Funkcional. Anal. i Priloˇzen. , 1:1–14, 1967.[2] D. Auroux. Special Lagrangian fibrations, wall-crossing, and mirror symmetry. In
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