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The Geometry of Symplectic Energy

arXiv:math/9306216v1 [math.DG] 12 Jun 1993The Geometry of Symplectic EnergyFran¸cois Lalonde ∗UQAM, Montr´ealDusa McDuff†SUNY, Stony BrookNovember 26, 2024IntroductionOne of the most striking early results in symplectic topology is Gromov’s“Non-Squeezing Theorem” which says that it is impossible to embed a largeball symplectically into a thin cylinder of the form R2n × B2, where B2 isa 2-disc. This led to Hofer’s discovery of symplectic capacities, which givea way of measuring the size of subsets in symplectic manifolds.

Recently,Hofer found a way to measure the size (or energy) of symplectic diffeomor-phisms by looking at the total variation of their generating Hamiltonians.This gives rise to a bi-invariant (pseudo-)norm on the group Ham(M) ofcompactly supported Hamiltonian symplectomorphisms of the manifold M.The deep fact is that this pseudo-norm is a norm; in other words, the onlysymplectomorphism on M with zero energy is the identity map. Up to now,this had been proved only for sufficiently nice symplectic manifolds, and byrather complicated analytic arguments.In this paper we consider a more geometric version of this energy, whichwas first considered by Eliashberg and Hofer in connection with their studyof the extent to which the interior of a region in a symplectic manifold deter-mines its boundary.

We prove, by a simple geometric argument, that bothversions of energy give rise to genuine norms on all symplectic manifolds.Roughly speaking, we show that if there were a symplectomorphism of M∗Partially supported by grants NSERC OGP 0092913 and FCAR EQ 3518†Partially supported by NSF grant DMS 91030331

which had “too little” energy, one could embed a large ball into a thin cylin-der M × B2. Thus there is a direct geometric relation between symplecticrigidity and energy.The second half of the paper is devoted to a proof of the Non-Squeezingtheorem for an arbitrary manifold M. We do not need to restrict to manifoldsin which the theory of pseudo-holomorphic curves behaves well.

This is ofinterest since most other deep results in symplectic topology are generalisedfrom Euclidean space to other manifolds by using this theory, and hence arestill not known to be valid for arbitrary symplectic manifolds.1The Main ResultsIn [6], Hofer defined the energy ∥φ∥H of a compactly supported Hamiltoniandiffeomorphism φ : (M, ω) →(M, ω) as follows:∥φ∥H = infH(supx,t H(x, t) −infx,t H(x, t)),where (x, t) ∈M ×[0, 1] and H ranges over the set of all compactly supportedHamiltonian functions H : M × [0, 1] →R whose symplectic gradient vectorfields generate a time 1 map equal to φ. It is easy to check that, for all φ, ψ,•∥φ∥H = ∥φ−1∥H;•∥φ ◦ψ∥H ≤∥φ∥H + ∥ψ∥H; and•∥ψ−1 ◦φ ◦ψ∥H = ∥φ∥H.Thus ∥· ∥H is a symmetric and conjugation-invariant semi-norm on thegroup Ham(M) of all compactly supported Hamiltonian diffeomorphisms ofM, and it follows that the associated function ρH given by:ρH(φ, ψ) = ∥φ−1ψ∥H,is a bi-invariant pseudo-metric.

However, it is harder to show that ∥· ∥H isa norm, or, equivalently, that ρH is a metric. Hofer established this whenM is the standard Euclidean space, using quite complicated analytical argu-ments.

This norm is still rather little understood. A good introduction toits properties may be found in [7, 8].2

In this paper, we will consider the generalized Hofer semi-norm ∥· ∥which is defined as follows.Let M be a symplectic manifold of dimen-sion 2n. If ∂M ̸= ∅, define Ham(M) as the group of all compactly sup-ported Hamiltonian diffeomorphisms which are the identity near the bound-ary.

Consider embeddings Φ of the strip M × [0, 1] in the product manifold(M × [0, 1] × R, ω + dt ∧dz) which are trivial, i.e. equal to (x, t) 7→(x, t, 0),for t near 0 and 1 and for x outside some compact subset of Int M, andare such that all leaves of the characteristic foliation on the hypersurfaceQ = Im Φ beginning on M × {(0, 0)} go through the hypersurface and reachM × {(1, 0)}.

The induced diffeomorphism φ from M = M × {(0, 0)} toM = M × {(1, 0)} is the monodromy of Q. Further, the energy of Q isdefined to be the minimum length of an interval I such that Q is a subset ofthe product M × [0, 1] × I.We define ∥φ∥to be the infimum of the energy of all hypersurfaces Qwith monodromy φ.Since φ−1 is the monodromy of the hypersurface Qwhen read in the opposite direction, ∥· ∥is symmetric.

Further, because thetime 1 map of the isotopy generated by the function H(x, t) is exactly equalto the monodromy of the embedding:(x, t) 7→(x, t, −H(x, t)),we find that∥φ∥≤∥φ∥Hfor all φ ∈Ham(M).This semi-norm was first considered in [1].It isrelevant, for instance, when one is trying to understand the extent to whichthe boundary of a region is determined by its interior, since the boundarycan always be C0-approximated by a sequence of hypersurfaces lying insidethe region: see [2]. Note that the two norms defined here might coincide,since no example is yet known where they differ.As in Hofer’s proof of the non-degeneracy of ∥· ∥H in R2n, we will provethat ∥·∥is a norm on Ham(M) by establishing an energy-capacity inequalitywhich gives a lower bound for the disjunction energy of a subset in terms ofits capacity.

Since all our arguments will rely on properties of embeddedballs, the appropriate capacity to use in the present context is Gromov’sradius c. Thus for any subset A ⊂M, we definec(A) = sup{u : there is a symplectic embedding B2n(u) ֒→Int A}.3

Here, we use the notation B2n(u) to denote the standard ball in standardEuclidean space (R2n, ω0) of capacity u and radiusqu/π. Thus the capacityof a ball of radius r is πr2.

In order to distinguish the standard balls in R2nfrom their images in M, we will reserve the dimensional upperscript to theformer only.The disjunction (or displacement)energy of A ⊂Int M is definedto be:e(A) = inf{∥φ∥: φ ∈Ham(M), φ(A) ∩A = ∅}.We will also need to consider maps φ which not only disjoin A, but also moveA to a new position which is sufficently separated from the old one. This givesus the notion of proper disjunction energy.

This is easiest to define forballs. A disjunction φ of B(c) is said to be proper if (some parametrizationof) B(c) extends to a ball B(2c) such that φ(B(c)) ∩B(2c) = ∅, and theproper disjunction energy ep(B(c)) is the infimum of the energies of allproper disjunctions of B(c).

Similarly, φ is said to be a proper disjunction ofA if each ball B(c) ⊂A may be extended to a ball B(2c) such thatφ(A) ∩(A ∪B(2c)) = ∅,and the proper disjunction energy ep(A) is the minimum energy of such a φ.ΠΠAs usual, if there are no (proper) disjunctions of A in M, we define its(proper) disjunction energy to be infinite.Our main result is:Theorem 1.1 Let (M, ω) be any symplectic manifold, and A any compactsubset of Int M. Then(i)ep(A) ≥c(A),and(ii)e(A) ≥12c(A).Corollary 1.2 For any symplectic manifold M, ∥· ∥is a (non-degenerate)norm on Ham(M). Hence ∥· ∥H is also non-degenerate.Remark 1.3 (i) It is very easy to see that the disjunction energy of a ballin R2n is exactly equal to its capacity.

Indeed, when n = 1, an open ballcan be identified with a square and then disjoined by a translation of energy4

equal to its capacity. In higher dimensions, the result follows from this byconsidering the ball as a subset of a product of squares.✷(ii) It is also easy to check that any disjunction of a ball in R2n is a properdisjunction.

(See the proof of Proposition 2.2.) However, very little is knownabout the space Emb(B2n(u), M) of balls of capacity u in an arbitrary sym-plectic manifold M. For example, if n > 2 it is not even known whetherEmb(B2n(u), B2n(u′)) is path-connected when u < u′.

Therefore, even if onerestricts to balls B of capacity less than c(M)/2, it is not clear what therelation is between e(B) and ep(B).✷(iii) Our arguments actually prove more than what is stated above, becausethey are local: they use only the part of the hypersurface Q swept outby the characteristics emanating from A.We will say that a piece Q ofhypersurface in M × [0, 1] × R disjoins A ⊂M if, for all x ∈A, the point(x, 0, 0) is one end of a characteristic on Q, the other end of which is a pointin (M −A) × {(1, 0)}. Then we can define the energy e(Q) of Q to be theminimal length of an interval I such that Q ⊂M × [0, 1] × I, and our resultreads:e(Q) ≥12c(A)if Q disjoins A, ande(Q) ≥c(B)if Q properly disjoins the ball B.In a similar way, we can define and estimate the energy of a local Hamiltoniandiffeomorphism of M.✷(iv) Polterovich, using geometric arguments which are very similar in spiritto ours, established in [16] that ∥· ∥H is a norm on rational symplecticmanifolds which are tame at infinity.His result is not as sharp as ours,because he considered the disjunction energy of Lagrangian submanifolds,which are more unwieldly than balls.✷Our methods also allow us to prove Gromov’s Non-Squeezing Theoremin full generality.Theorem 1.4 (Non-Squeezing Theorem) Let (M, ω) be any symplecticmanifold, and denote by M × B2(λ) the product of M with the disc B2(λ) ofarea λ, equipped with the product form.

Thenc(M × B2(λ)) ≤λ.5

Remark 1.5 This result was first proved for manifolds such as the stan-dard R2n and T 2n by Gromov in [5]. Its range of validity was extended byimprovements in the understanding of the behavior of pseudo-holomorphiccurves.

However, this method has definite limitations and is not yet knownto apply to all manifolds. (The best result which can be obtained in thisway is described in §3.) We manage to overcome these limitations by usingthe techniques which we developed to prove Theorem 1.1.

As we shall seein Remark 2.3 below, these two theorems are very closely connected, and wewill, in fact, deduce Theorem 1.1 from Theorem 1.4. To the authors’ knowl-edge, these are the first deep results in symplectic topology which have beenestablished for all symplectic manifolds.

Arnold’s conjecture, for example,has still not been proved, even for all compact manifolds.Finally, we observe that the methods developed here permit the con-struction of some new embeddings of ellipsoids into balls. In particular, it ispossible to solve a problem posed by Floer, Hofer and Wysocki in [4].

Thisis discussed further in Remark 2.4.Throughout this paper, all embeddings and isotopies will always be as-sumed to preserve the symplectic forms involved.We are grateful to Yakov Eliashberg and Leonid Polterovich for usefuldiscussions on some basic ideas developed in this paper and to Lisa Traynorfor showing us explicit full embeddings of balls that inspired our Lemma 2.1.The first author thanks Stanford University for a stay during which part ofthis work was undertaken.2The energy-capacity inequality in R2n.This section presents a very simple proof of the energy-capacity inequalityfor subsets of R2n. The basic idea is that if a hypersurface Q of small energydisjoins a large ball in M, one can construct an embedded ball in the productM × B2 whose capacity is larger than the area of the B2 factor.

But theNon-Squeezing Theorem states that when M = R2n this area is an upperbound for the capacity of embedded balls in M × B2.6

We need an auxilliary lemma about decompositions of the ball, which wasinspired by Traynor’s constructions in [17]. Given any set A, we will writeN (A) to denote some small neighborhood of it.Lemma 2.1 Suppose that 0 < c < C, and let Y = P1 ∪L ∪P2 ⊂R2 be theunion of two rectangles, P1 of area C −c and P2 of area c, joined by a linesegment L. Further, letZC,c = B2n(C) × P1 ∪B2n(c) × (L ∪P2).Then, there is a symplectic embedding B2n+2(C) ֒→N ( ZC,c) in any neigh-borhood of ZC,c.Proof:Let π be the projection B2n+2(C) →B2(C) which is induced byprojection onto the last two coordinates.

This represents B2n+2(C) as a kindof fibration over the disc, with fibers which are concentric balls of differentcapacities. Note that the set{ x ∈B2(C) : c(π−1(x)) ≥c}is exactly B2(C −c).It is easy to see that there is an area preservingembedding g : B2(C) ֒→N (Y ) which takes B2(C −c) into a neighborhoodof P1.

In fact, we may choose g so that it sends an open neighbourhood ofB2(C −c) ∪( (−∞, 0) × {0} ∩B2(C) ) ⊂B2(C)into a neighborhood of P1. Clearly, g is covered by the desired embedding ofB2n+2(C) into N (ZC,c).✷Proposition 2.2 For any compact subset A ⊂R2n,e(A) ≥c(A).Proof: Let Q be a hypersurface of energy e which disjoins A, and let B ⊂Abe the image of a standard ball of capacity c. We must show that e ≥c.

Thiswill follow if, for any δ > 0, we can find an embedding of the ball B2n+2(2c)of capacity 2c into the product R2n × B2(e + c + ε), since the Non-SqueezingTheorem then tells us that2c ≤e + c + ε.7

By Lemma 2.1, it suffices to embed Z2c,c symplectically into R2n × X,where X is an annulus of area e + c. By hypothesis, there is a rectangle Rin [0, 1] × R of area e such that Q ⊂R2n × R. Note that, by hypothesis, Qis flat near its ends, that is, that Q coincides with the hypersurface {z = 0}near the boundary t = 0, 1. (Recall that we use the coordinates (t, z) on[0, 1] × R.) Let R′ be another rectangle in R2 of area c with one edge alongt = 1, chosen so that R ∪R′ is a rectangle of area e + c with one edge alongt = 0 and another along t = t1 > 0.

Then form X by identifying these twoedges.Let g : B2n(c) →B ⊂A be a symplectic embedding, and extend g to anembedding, which we also call g, of B2n(2c) in R2n. This is possible becausethe space of embedded symplectic balls of any given radius in R2n is pathconnected.

(The space of embedded balls of variable radius in any manifoldM is always connected, so long as M is connected, and, when M = R2nwe can fix the size of the radius by composing with appropriate homoth-eties. A similar argument shows that the space of symplectic embeddingsof two balls in R2n is path-connected.) This implies that any ball in R2n isisotopic to a standard one and thus can be extended as much as we wish.Further, we may suppose that the ball g(B2n(2c)) is disjoint from φQ(B),where φQ is the monodromy of Q.

To see this, note that because the spaceof symplectic embeddings of two balls in R2n is path-connected, there is asymplectomorphism τ which is the identity on B and which moves φQ(B) faraway. Hence we may alter Q without changing its energy to a hypersurfacewith the conjugate monodromy τ −1 ◦φQ ◦τ.We now define the embedding Z2c,c →R2n × X as follows.• B2n(2c) × P1 goes to R2n × R′ by g × i, where i : U1 →R′.• B2n(c) × L maps to the hypersurface Q ⊂R2n × R by a map whichtakes each line {x} × L to the corresponding flow line of the characteristicflow on Q.• B2n(c) × P2 goes onto φQ(B) × R′ by the map (φ ◦g) × i.It is easy to check that this map preserves the symplectic form.

Hence,the symplectic neighborhood theorem implies that it extends to the requiredsymplectomorphism from N (Z2c,c) to R2n × B2(e + c + ε).✷Remark 2.3 The above argument clearly proves Part (i) of Theorem 1.1 formanifolds for which the Non-Squeezing theorem holds. For, suppose that the8

monodromy φQ of a hypersurface Q is a proper disjunction of A of energy e.Then, for every ball B(c) ⊂A, φQ disjoins B from a ball ˜B(2c) of twice thecapacity and one can construct an embedding Z2c,c ֒→M × B2(e + c + ε) asabove.Similarly, one can prove Part (ii) of Theorem 1.1 in this case, by usinga slightly different embedding. Note that the ball B2n+2(c) embeds into thesetWc,c = B2n(c) × Ywhere now P1 and P2 both have area c/2.

This gives a ball of capacity c inM × B2(e + c/2 + ε). Therefore, if the Non-Squeezing Theorem holds, wemust havec ≤e + c/2 + ε,for all ε > 0 and hence e ≥c/2, as claimed.✷Remark 2.4 Let E(c1, c2) denote the ellipsoidΣiπ(x2i + y2i )/ci ≤1,where c1 ≤c2.Floer, Hofer and Wysocki show in [4] that if c1 ≥1/2,E(c1, c2) embeds symplectically in B4(1) = E(1, 1) if and only if there is asymplectic linear embedding from E(c1, c2) into E(1, 1), i.e.

only if a2 ≤1.They asked whether this is sharp. In other words, if c1 < 1/2, is there c2 > 1such that E(c1, c2) embeds in B4(1)?

Our embedding methods allow us toanswer this question in the affirmative.The idea is as follows. As Traynor points out in [17], the ellipsoid E(c1, c2)may be considered to be fibered over the disc B2(c2), with fibers which aresmaller by a factor of c1/c2 than those of the corresponding ball B4(c2).

Ifc1/c2 < 1/2, we can therefore fit two of these fibers in the corresponding fiberof the ball. From this, Traynor constructs a full filling of the ball B4(1) bytwo open ellipsoids.Now, suppose that we split E(c1, c2) into two, considering it to be con-tained in a neighborhood of a set Z such as ZC,c above.Then, it is nothard to construct the desired embedding E(c1, c2) →B4(1) by folding thetwo parts of Z on top of each other.

Details of this construction will bepublished elsewhere.✷9

3The Non-Squeezing TheoremIn this section we will use the theory of J-holomorphic curves to prove theNon-Squeezing Theorem under certain hypotheses which look rather artifi-cial. We will see in §4 how to construct families of embeddings which satisfythem.A symplectic manifold (M, ω) is often said to be rational if the homo-morphism induced by [ω] from π2(M) to R has discrete image.

In this caseand if this image is not {0}, we will call the positive generator of this imagethe index of rationality of (M, ω), denoted r(M) or simply r. If the imageis {0}, we set r(M) = ∞and if M is not rational, this index is set equal to0 (this does not quite follow the conventional definitions).We will consider M × S2(λ) with a product form Ω= ω ⊕λσ where σis normalised to have total area 1, and will say that a ball in M × S2(λ) isstandard if it is the image of an embedding of the formB2n+2(c) ֒→B2n(c) × B2(c) ֒→M × S2(λ)where the first map is the obvious inclusion and the second is a product.Clearly the capacity of any standard ball in M × S2(λ) is bounded above byλ. The main result of this section says that this remains true for any ballwhich is isotopic to a standard ball through large balls.Proposition 3.1 Let M be closed and have index of rationality r > 0, andsuppose that gt : B2n+2(ct) →M ×S2(λ) is a family of symplectic embeddingssuch that g0 is standard andct ≥sup{r, λ −r},for all t. Suppose also that the image of gt misses one fiber M × pt for all t.Thenct < λfor all t.We will begin by explaining the usual proof of Gromov’s Non-Squeezingtheorem via pseudo-holomorphic curves.

We assume that the reader is famil-iar with the basics of the theory of pseudo-holomorphic curves as explained10

in [5, 10] for example. The most general argument works when the mani-fold (V, Ω) = (M × S2(λ), ω + λσ) is semi-positive (or semi-monotone): see[12].

This condition says that, for generic tame J there are no J-holomorphicspheres with negative Chern number. It is satisfied by all manifolds of di-mension ≤6 and by manifolds for which there is a constant µ ≥0 suchthatc1(α) = µ[ω](α),for all α ∈π2(M),where c1 is the first Chern class of (M, ω).Lemma 3.2 Let g : B2n+2(c) →M × Int B2(λ) be a symplectic embedding,and suppose that (V, Ω) = (M × S2(λ), ω + λσ) is semi-positive.

Thenc < λ.Proof: Clearly, we may consider g as an embedding into V which misses onefiber. Let JB be an almost complex structure tamed by Ωwhich extends theimage by g of the standard complex structure on R2n+2.

In order to show thatc ≤λ, it is enough, by Gromov’s monotonicity argument, to show that thereexists a JB-rational curve C of symplectic area smaller or equal to λ passingthrough the center g(0) = p0 = (q0, z0) of the ball g(B2n+2(c)). The reason isthat the part of this curve in g(B2n+2(c)) pulls back to a holomorphic curveS through the center of B2n+2(c).

Since S is minimal with respect to theusual metric on B2n+2(c), the monotonicity theorem implies that its area isat least c. Thusc ≤area S =ZS ω0

(For more details of this step see [10, 15]. )Take any almost complex structure J1 tamed by ω on M, integrable ina neighbourhood U of q0, take the usual complex structure J0 on S2, anddenote by Jspl the split structure J1⊕J0.

It is easy to see that for all q ∈U the11

rational curves {q}×S2 in class A0 = [{pt}×S2] are regular for this complexstructure in the Fredholm sense. (This means that, given any holomorphicparametrization f of these curves, the points (f, Jspl) are regular points ofPA0.) Note also that P −1A0 (Jspl) = {q×S2 : q ∈M}, because any holomorphicmap from CP 1 to the split M × S2 in class A0 induces, by projection on thefirst factor, a holomorphic map to M, which is null-homologous and henceconstant.

Obviously all points p ∈V that project to U are regular values ofthe evaluation mapev : P −1A0 (Jspl) ×G CP 1 −→V,where G is the conformal group of CP 1, and these p have pre-image ev−1(p)containing exactly one point. It follows that there exists a structure J′ nearJspl which is generic, that is regular for all projections PA, and is such thatsome point p′ ∈U × S2 is still a regular value for the evaluation map onP −1A0 (J′) ×G CP 1, with exactly one point in its pre-image.Now, let J′′ be a generic almost complex structure in the neighbourhoodof JB and Γ a path in J from Γ(0) = J′ to Γ(1) = J′′, transverse to allprojections PA. Then P −1A0 (Γ) is a smooth manifold, and we consider a shortpath γ in V from γ(0) = p′ to a point γ(1) = p′′ in a neighbourhood of p0,such that the obvious evaluation mapevΓ,A : P −1A (Γ) ×G CP 1 −→V × [0, 1]is transverse to γ×id : [0, 1] →V ×[0, 1] for all A.

(Here evΓ,A maps onto [0, 1]by projection through Γ.) Denote by N the one-dimensional submanifoldev−1Γ,A0(γ × id).

By construction, there is exactly one point of N which mapsto (γ(0), 0). Therefore, if N were compact, it would have at least one otherpoint over (γ(1), 1).

In other words, there would exist a J′′-rational curvein class A0 passing through p′′. And if this were also true for a sequence ofpaths in J whose end points converge to JB and a sequence of paths in Vwhose end points converge to p0, this would give, by Gromov’s compactnesstheorem (see [5]), a sequence of holomorphic spheres (weakly) converging to aJB-cusp-curve.

The component of this cusp-curve passing through p0 wouldhave area smaller or equal to λ and so would be the desired JB-curve.Let us suppose now that one of these manifolds N is not compact. Therewould then exist a sequence of Ji-curves fi : CP 1 →V with {Ji} convergingto some J = Γ(t0) and {fi} diverging.

Since V is compact, the compactness12

theorem implies that some subsequence would converge weakly to a cusp-curve passing through p = γ(t0).This cusp-curve would be a connectedunion of J-curves in classes A1, . .

. , Ak, whereA0 = A1 + .

. .

+ Ak.Therefore, the proof may be finished if we put a hypothesis on M whichensures that a generic path (Γ, γ) does not meet any such cusp-curve. Forexample, since Ω(Aj) > 0 for j = 1, .

. ., k, it is clearly enough to assumethat π2(M) = 0 or more generally that λ = Ω(A0) ≤r.

The real trou-ble comes from the possible presence of multiply-covered curves of negativeChern number, and it is shown in [12, §4] that it suffices to assume that Vis semi-positive.✷Proof of Proposition 3.1There is no semi-positivity hypothesis here: we get around the problemcaused by cusp-curves by considering a very special path Γ. First, let usconsider a path from J0 = Jspl to J1 such that, at each time t, Jt is equal tothe push-forward by the embedding gt of the standard structure on R2n+2.By assumption on the embeddings, we may also suppose that one fiber M×ptis Jt-holomorphic, for each t. Suppose that there were a Jt-holomorphic A-cusp-curve through the center gt(0) of the ball for some t with homologydecompositionA = A1 + .

. .

+ Ak.Let C be the component of this cusp-curve through gt(0). We may supposethat [C] = A1.

The argument at the beginning of Lemma 3.2 shows thatΩ(A1) =RC Ω> ct. Hence,Ω(Aj) < λ −ct ≤r,ifj > 1.Thus the classes Aj, j > 1, do not lie in H2(M). On the other hand, bypositivity of intersections (see [5, 11]), the fact that a fiber is Jt-holomorphicimplies that the intersection number Aj · [M] is ≥0 for all j.

It follows thatone of the Aj has the form A −Bj for some Bj ∈H2(M), and that theothers are all elements of H2(M). Putting all this together, we see that thedecomposition must have the formA = B + (A −B),13

for some B ∈H2(M) such that Ω(B) > ct.Note that neither component of this cusp-curve can be multiply- covered.For if the component in class B were a k-fold covering for some k ≥2, wewould have to have ω(B) > 2ct, which is impossible by assumption on ctsince we must have ω(B) < λ. Further, the component in class A−B cannotbe multiply covered since A −B is not a multiple class.This argument shows that, for all the elements Jt in our special path,the only Jt-holomorphic A-cusp-curves are of type (B, A −B).

By the com-pactness theorem, a similar statement must hold for every J in some neigh-borhood of this path. Thus we may assume that the points on the regularpath Γ considered above have this property.

The arguments in [12] show thatthese cusp-curves are well-behaved, and fill out a subset of V of codimensionat least 2. Therefore, a generic path (Γ, γ) will not meet these cusp-curves,and the argument may be finished as before.✷The next lemma is the key step in extending our results to non-compactmanifolds.Lemma 3.3Let M be a non-compact symplectic manifold of index of ra-tionality r > 0.

For each compact subset K of M, there is a number ζ > 0such that whenever gt : B2n+2(ct) →K × S2(λ) is a family of symplecticembeddings missing one fiber M × pt for all t and beginning with a standardembedding g0, thenct ≥λ −min(r, ζ)for all t⇒ct < λfor all t.Proof: Let K1 be a compact subset of M which contains K in its interior. Inorder to make the previous argument go through, we just have to ensure thatthe A-curves in N do not escape outside K1 × S2.

Suppose that ∂K1 = Σ isa smooth hypersurface and let U be a compact neighbourhood of Σ disjointfrom K.Since all balls lie in K we may assume that the special almostcomplex structures Jt are all equal to the Ω-compatible split structure Jsplon U. Because U is compact, it is easy to see that there exists ζ, s bothsmall enough so that, given any Jt-holomorphic curve C passing throughsome point p ∈Σ, the Ω-area of C ∩Bs(p) is larger than 3ζ.

Choosing now,in the proof of Proposition 3.1, the generic path of almost complex structuresso that each be sufficiently close to Jspl on U, we get a lower bound equal to2ζ on the area of C ∩Bs(p). If C is a curve in the path N, the Ω-area of the14

part of C which lies in the ball Im gt must be at least ct. But ct + 2ζ > λ,by hypothesis.

Therefore, none of the A-curves in N meet Σ.✷4Embedding balls along monodromiesBy Remark 2.3, all theorems will be proved if we show that the Non-SqueezingTheorem holds for all manifolds.Our tool to do this is Proposition 3.1.Thus, given a ball B(c) in a cylinder M × B2(λ) with c > λ, we aim toconstruct another ball B′′(c) of capacity c, which is contained in some cylinderM′′ × B2(λ′′) with c > λ′′ and which is isotopic to a standard ball throughlarge balls. Proposition 3.1 then implies that the ball B′′(c) cannot exist,and it follows that B(c) does not exist either.It is quite a delicate matter to obtain a ball B′′ with the required prop-erties, and as our notation implies we do this by a two-step process, firstconstructing an intermediate ball B′, and then using that to get B′′.

Webegin by explaining the basic procedure which constructs these balls, andthen will give the proof.4.1The N-fold wrapping constructionBecause the original ball B may not extend to a ball of capacity 2c, wemust use a multiple wrapping process to maintain the capacity of our balls.Therefore, instead of using the set Y of Lemma 2.1, we will use the setsYN ⊂R2 described below.It will be often convenient to use rectanglesrather than discs. As always, the label of a set will indicate its capacity (orarea), and as before we will distinguish the standard balls (the domain ofour maps) from their images by reserving the dimensional upperscript to theformer only.Let V 2m be any symplectic manifold, g : B2m(κ) →V a symplecticembedding of a ball of capacity κ, whose image is denoted B(κ), and φs, 0 ≤s < ∞, a diffeotopy of V such that φs is periodic in s with period 1.

Weassume that φs has a 1-periodic generating Hamiltonian H : V ×[0, ∞) →Rwhich satisfies• H vanishes near any integral value of s;• for each s, minV Hs = 0.15

When referring in this section to the energy of such a diffeotopy φs, wewill always mean the maximum over s ∈[0, 1] of the total variation of Hs.Further, we will say that φs strictly disjoins a ball B ⊂V if the ballsB, φ1(B), φ2(B), . .

. , φk(B), .

. .are all disjoint.Now choose any positive integer N, and, for i = 1, .

. ., N define ai, bi ∈Rbyai = (i −1)(1 + 1/N),bi = ai + 1/N.Thus ai+1 = bi + 1.

Let YN = YN(κ) ⊂R2 be the union of N rectanglesPi, 1 ≤i ≤N, of area κ/N with N −1 lines Li, 1 ≤i ≤N −1, of length 1,where:Pi = {(u, v) : ai ≤u ≤bi, 0 ≤v ≤κ},Li = {(u, v) : bi ≤u ≤ai+1, v = 0}.Observe that any embedding of YN(κ) extends to some neighborhoodN (YN(κ)) and hence induces an embedding of the disc B2(κ), since this fitsinside any neighborhood of YN(κ). Therefore an embeddingG : B2m(κ) × YN(κ) ֒→V × R2,induces an embedding G ◦i of the ball B2m+2(κ) byB2m+2(κ)i֒→B2m(κ) × N (YN(κ))G֒→V × R2where i is the obvious inclusion.Now define G : B2m(κ) × YN(κ) ֒→V × R2 byG(p, u, v)=(φi−1(g(p)), u, v),when(u, v) ∈Pi,=(φ(i−1)+u−bi(g(p)), u, H(φ(i−1)+u−bi(g(p)), u −bi)),when(u, v) ∈Li.It is easy to check that G is symplectic and so extends to a symplecticembedding (which we will also call G) of B2m(κ) × N(YN(κ)) into V × R2.The corresponding ball, which is the image of the compositeB2m+2(κ)֒→B2m(κ) × N (YN(κ))G֒→V × R216

will be called the unwrapped ball associated to B(κ) = Im g and φs, andwill be denoted B(B(κ), φ). Note that it exists for any isotopy φs, not onlyfor disjoining isotopies.

For the sake of clarity, we will sometimes write Gφfor the corresponding embedding G.If e is the energy of φs, then H takes values in [0, e] and the image Im Gof B2m(κ) × YN by G lies in the setV ×aPi ∪aQi= V × S ⊂V × R2,where Qi, i = 1, . .

. , N −1 are the rectanglesQi = {(u, v) : bi ≤u ≤ai+1, 0 ≤v ≤e}.Denote by τS the translation of V × S by 1 + 1/N in the u-direction ofR2.

(We will make no distinction between the translation of S ⊂R2 and itslift to the product V × S.) This sends Pi to Pi+1 and Qi to Qi+1, and it iseasy to see that, if φs strictly disjoins B(κ), then τS strictly disjoins Im G,that is, all the ballsIm G, τS(Im G), (τS)2(Im G), . .

.are disjoint. Thus, if X is the annulus obtained by quotienting S by thetranslation τS, X has area A = κ/N + e and, as in Proposition 2.2, we getan embedded ball in V × B2(A + ε) by the composite:B2m+2(κ) ֒→B2m(κ) × N (YN(κ))G֒→V × S →V × X ֒→V × B2(A + ε).This ball wraps N times round the annulus X, and will be called the wrappedball BW(B(κ), φ) generated by B(κ) and φs.Remark 4.1Note that if e < κ, we may, by choosing N large enough,arrange that A be arbitrarily close to e. Thus, for sufficently small ε > 0, weget a ball BW = BW(B, φ) of capacity κ inside a cylinder in V × R2 of areaA + ε < e + 2ε < κ.The next result is obvious.Lemma 4.2 If Bt = Im (ft : B2m(κt) →V ) and {φt} vary smoothly withrespect to a parameter t, the corresponding wrapped ball BW(Bt, φt) variessmoothly.17

Lemma 4.3 The translation τS of V × S which disjoins the unwrapped ballB(B, φ) may be extended to a 1-periodic diffeotopy {σs}0≤s<∞of V × R2 insuch a manner that• σ1 = τS;• the diffeotopy σs strictly disjoins B(B, φ); and• the energy of σs is ≤A + ε.Proof:Suppose first that the rectangles Pi, Qi in T all have the same v-height. Then S has the form R × I, and one can extend τS to have the form(x, u, v) 7→(x, u + β(v), v) on V × R × I, for some suitable bump functionβ which equals 1 + 1/N on the interval I.This map has the generatingHamiltonian H(x, u, v) =R v β which has energy ≤R β.

The general resultfollows because there is an area-preserving map which commutes with τS andtakes S into a set S0 of the form R × I with area S0/τ < A + ε.✷Of course, the wrapped ball BW(B, φ) may also be strictly disjoined bya diffeotopy (a translation) of energy < A + ε < κ. But for the first step ofour argument we consider instead B(B, φ) with disjoining isotopy σs sincethe latter is more flexible.4.2Regularity and the plan of the proofWe can now make more precise the plan briefly outlined in the introductionof the section.

We fix a small constant ε > 0 of size to be determined later.We start with the ballB2n+2(c)g֒→B(c) ⊂M × B(λ) ⊂M × R2 = M′(with λ < c) given in the statement of the Non-Squeezing Theorem. Thisis strictly disjoined by a diffeotopy φs of energy < λ + ε which translatespoints in the R2 direction.

We first construct the unwrapped ball B′(c) =(B(c), φ) ⊂M′×R2, together with the disjoining diffeotopy σs. By Lemma 4.3above, we may choose N so large that σs has energy <λ + 2ε.Thusthe wrapped ball B′′(c) = BW(B′(c), σs) of capacity c lies in a cylinderC′′ = M′ × R2 × B2(λ + 3ε).Proposition 4.4 The ball B′′(c) in C′′ = M′ × R2 × B2(λ + 3ε) is isotopicthrough balls of capacity ≥λ to a standard ball.18

Corollary 4.5 The Non-Squeezing Theorem holds for any symplectic man-ifold.Proof of the corollary: If (M, ω) is compact and rational with indexof rationality r, choose ε < r/3. The Non-Squeezing Theorem then followsimmediately from Propositions 4.4 and 3.1: simply apply Proposition 3.1 tothe closed manifold M×T 2(Λ)×T 2(Λ) which has the same index of rationalityas M, where Λ is chosen large enough so that M ×T 2(Λ)×T 2(Λ)×B2(λ+3ε)contains the ball B′′(c).

If (M, ω) is compact but not rational, one can slightlyperturb both the form ω on M and the ball B(c) in M × B2(λ) to get a ball˜B sitting inside ˜M × B2(λ), where ( ˜M, ˜ω) is rational. Although ˜B may havecapacity ˜c a little less than c, we may clearly arrange that if c > λ then ˜c > λ.Thus the Non-Squeezing Theorem holds true for all compact manifolds.Suppose now that M is non-compact.

Note that the initial ball B(c) sitsin a compact region of M × B2(λ), so that we may assume both that M is acompact manifold with boundary and that it has positive index of rationalityr. Then B(c) ⊂K ×B2(λ), where K is a compact subset of M. Hence B′′(c)lies in K ×T 2(Λ)×T 2(Λ)×S2(λ+3ε) where Λ increases as ε decreases (andas N increases), because the N-fold wrapping construction does not movethe ball in the M-direction.

This ball is isotopic to a standard ball throughballs in K × T 2(Λ) × T 2(Λ) × S2(λ + 3ε) of capacity equal to λ + 3ε up to asmall quantity 3ε. But note that r(M) = r(M ×T 2(Λ)×T 2(Λ)) and that theconstraining number ζ of Lemma 3.3 depends only on K: it is independantof the size Λ of T 2(Λ) as soon as Λ is large enough.

Therefore, for ε < min(r,ζ)3,the argument of Lemma 3.3 applies.✷In order to explain our strategy for proving Proposition 4.4 it will beconvenient to introduce the following definition.Given δ > 0, we will say that an isotopy φs in some manifold is δ-regularon a ball B(κ) if, for each s ∈[δ, 1], the ballsB(κ), φs(B(sκ)), φ2s(B(sκ)), . .

. , φks(B(sκ)), .

. .are all disjoint (note, in particular, that φs strictly disjoins B(κ)).HereB(sκ) denotes a ball of capacity sκ which is concentric with B(κ).

Noticethat it is not so much the isotopy itself which is important but its relationto the concentric balls B(κ). Further, all the balls φks(B(sκ)) are assumedto be disjoint from the whole initial ball B(κ).19

Regularity is really a 1-dimensional notion: any translation of R at con-stant speed which disjoins a given interval also disjoins subintervals withina time proportional to their lengths. The basic 2-dimensional example ofa regular diffeotopy is a translation which disjoins a rectangle in R2, or itsconjugate which disjoins a disc.

(We will often call the latter a translationtoo. )To be more precise, let φs, 0 ≤s < ∞, be the translation of the strip S =R×[0, h] in the u-direction at speed ν > 1.

(As above, we use the coordinates(u, v) on S.) Note that this isotopy is generated by a Hamiltonian which isa linear function of v on the strip, and vanishes outside some slightly largerstrip. As in Lemma 4.3 above, its total variation may be taken arbitrarilyclose to hν.

Then for any small δ > 0 and any κ < h, consider an embeddingof the standard disk g : B2(κ) ֒→S such that B(sκ) = g(B2(sκ)) lies inside[−s, 0]×[0, h] for all s ∈[δ, 1]. (Such an embedding exists because our choiceof constants h > κ, δ > 0 leaves a little extra room.) It is easy to check thatφs is regular on Im g.With this simple basic example, we can construct higher dimensionalexamples of regular disjoining diffeotopies since regularity is stable underproducts.Lemma 4.6Given a δ-regular pair (f, φ) in R2 and g any symplectic em-bedding of a ball in any manifold, form the product h = g × f : B2n(κ) ×B2(κ) →V = M × R2.

Then the pull-back π∗(φs) of φ by the projectionπ : M × R2 →R2 is also δ-regular on the composite h ◦i of h with thestandard embedding i : B2n+2(κ) ֒→B2n(κ) × B2(κ).Proof:This is clear because i sends each concentric subball of capacityκ′ onto a ball in B2n(κ) × B2(κ) whose projection on the second factor is aconcentric subdisc of same capacity κ′.✷The following lemmas form the heart of our argument. They are provedin the following section.Lemma 4.7 For every δ, ε > 0, there is a 1-parameter family B′t(κt), σts,0 ≤t ≤1, of balls and strict disjoining isotopies in M′ × R2 such that:•B′0(κ0) = B′(c) and σ0s = σs;•the final isotopy σ1s is δ-regular over the ball B′1(κ1);20

•the isotopy σts has energy < λ + 2ε for all t; and•the balls B′t have capacity κt ≥λ for all t, and κ1 = λ.Lemma 4.8 Suppose given a ball B of capacity λ in V with δ-regular strictlydisjoining isotopy σs of energy < λ + 2ε, and let BW be the correspondingwrapped ball in the cylinder C = V × B2(λ + 3ε). Then, if δ is sufficientlysmall, BW is isotopic through balls in C of capacity λ to a standard ball.Proof of Proposition 4.4By Lemmas 4.2 and 4.7, the wrapped ballB′′(c) is isotopic through balls of capacity κt ≥λ which embed in C′′ tothe wrapped ball B′′1 = BW(B′1(κ1), σ1).Since the isotopy σ1s is regular,Lemma 4.8 shows that the wrapped ball B′′1 is isotopic in C′′ to a standardball.✷4.3The Construction of IsotopiesOur first lemma constructs an isotopy of an unwrapped ball, and the secondone for a wrapped ball.

We begin with the proof of the second lemma.Proof of Lemma 4.8The wrapped ball BW is the image of a compositeB2m+2(λ) ֒→B2m(λ) × N (YN(λ))G֒→V × S →V × B2(λ + 3ε),where the last map is essentially the quotient by the translation τS. Theeasiest way to describe the isotopy from BW to a standard embedding is toconstruct an isotopy of the mapB2m+2(λ) ֒→B2m(λ) × N (YN(λ))G֒→V × Swhose image is disjoined by all iterates of τS.The crucial observation is that, just as in Lemma 2.1, the ball B2m+2(λ)fits inside a neighborhood of the subset W of B2m(λ)×YN(λ) which is definedas follows:W =B2m(λ) × P1∪B2m(λ(1 −1/N)) × (L1 ∪P2)∪.

. .∪B2m(λ(1 −i/N)) × (Li ∪Pi+1)∪.

. .∪B2m(λ/N) × PN.21

Thus the interval L1 for example is longer than we need: G maps it into ahypersurface which disjoins the whole ball B2m(λ) while we only need thehypersurface to disjoin the subball B2m(λ(1 −1/N)) from B2m(λ). Becausethe hypersurface comes from the regular isotopy σs, we only need the partof it corresponding to the isotopy σs, 0 ≤s ≤1 −1/N, which sits over aninterval of length 1 −1/N.Here are the details.Let us change the proportions in the set YN bymoving the points bi, keeping the ai fixed.

Thus, for 0 ≤s ≤1, put bsi = bi+s,and letY sN = P s1 ∪Ls1 ∪. .

. ∪P sN,where the rectangles P si have area (1/N +s)λ and the intervals Li have length1 −s.

Then, for each s, B2m+2(λ) embeds in a neighborhood ofW s =B2m(λ) × P s1∪B2m(λ(1 −1/N −s)) × (Ls1 ∪P s2)∪. .

.∪B2m(λ(1 −i/N −is)) × (Lsi ∪P si+1)∪. .

.∪B2m(λ(1/N −(N −1)s)) × P sN.Here, the convention is that B2m(κ) denotes a point when κ = 0 and is emptyfor κ ≤0. Note that when s = s0 = 1 −1/N, W s0 is just B2m(λ) × P s01embedded by a standard embedding and the isotopy may be ended.

Thelength of Ls0iis then1N > δ: the lengths of Lsi are thus larger than δ duringthe whole isotopy.We now map W s to V × S by the obvious map Gs, which, for each0 ≤s ≤1 −1/N is constructed from the isotopy σu, 0 ≤u ≤1 −s. It is nothard to check that the δ-regularity of σs implies that Im Gs is disjoined bythe translation τS along S. Since we may choose everything to vary smoothlywith s, the result follows.✷Proof of Lemma 4.7We construct a family of balls B′t and isotopiesσts from the unwrapped ball B′(c) = B(B, φ) to a ball and isotopy whichare the lifts of a ball and strictly disjoining isotopy in R2.

The first part ofthe isotopy is simple: we just decrease the capacity of the unwrapped ballB′ from c to λ by shrinking the initial ball B(c) while keeping the initialdisjoining diffeotopy φs fixed. Note that τs does not change through thisisotopy.Consider now the ball B′(λ) obtained at the end of this isotopy.

It is clearby the definition of the unwrapped construction that the following diagram22

commutes:B2n+4(λ)i֒→B2n+2(λ) × N (YN(λ))˜Gg֒→M′ × R2↓(π ◦g) × id↓(π × id)B2(λ) × N (YN(λ))Gj֒→R2 × R2where g is the initial embedding B2n(λ) ֒→B′(λ), j is the inclusion B2(λ) ֒→R2, π is the projection onto R2, and we write ˜Gg, Gj for the correspondingmaps which give the unwrapped balls. It follows that:(i)any isotopy Gt, 0 ≤t ≤t0 beginning with G0 = Gj lifts to an isotopy˜Gt beginning with ˜G0 = ˜Gg.

(ii)If, for each t, ρts is a 1-periodic diffeotopy of R4 which strictly disjoinsIm (Gt), then its pull-back ˜ρts to M × R4 also strictly disjoins Im ( ˜Gt) andhas the same energy. (iii)Finally, if at time t0, Gt0 is a productf1 × f2 : B2(λ) × N(YN(λ)) ֒→R2 × R2and ρt0s is (the pull-back of) a translation ψs in the second R2-factor such thatthe pair (f2, ψs) is δ-regular, then ˜Gt0 is again a product whose second factoris δ-regular under ψs.

It follows that the pair ( ˜Gt0 ◦i, ˜ρt0s ) is also δ-regularby Lemma 4.6.This reduces the proof to the construction of the 4-dimensional isotopy Gt.This may be easily defined by using the fact that the translation φs whichstrictly disjoins B2(λ) in R2 is generated by an autonomous HamiltonianH (of total variation λ + ε): for the moment, let us forget the (technical)requirement that our isotopies should be constant when the time s is near aninteger value, and let us suppose that the isotopy σs is a 1-parameter group.Then, for each 0 ≤t ≤1, setσts = σ(1−t)s,and consider the corresponding unwrapped ball B(B2(λ), σt) = Im Gσt. Be-cause the energy e(σt) of σt is (1−t)e(σ) = (1−t)(λ+ε), the ball B(B2(λ), σt)23

sits over a strip St = `i Pi ∪`i Qti where the rectanges Qti have area (1 −t)(λ + ε). Therefore the translation τ t which moves St through the distance1 + 1/N may be extended as in Lemma 4.3 to an isotopy τ ts of energyλN + e(σt) + ε/2 = λN + (1 −t)(λ + ε) + ε/2.Note that τ t does not disjoin B(B2(λ), σt) from itself, since σt does not disjoinB2(λ) at time s = 1.

However, if we follow τ t (which is a movement in thesecond R2 factor) with the translation σt+11= σ−t in the first R2 factor, wedo get a disjoining isotopy. Thus, the isotopyρts = σ−ts ◦τ tsdisjoins B(B2(λ), σt) from itself at time s = 1, and may be extended to anisotopy of R4 with total energyλN + e(σt) + e(σt+1) + ε=λN + (1 −t)(λ + ε) + t(λ + ε) + ε=λN + λ + 2ε.Taking Gt = Gσt, we can therefore satisfy (i) and (ii) above.

Observe thatwhen t = 1, G1 is simply the inclusion of B2(λ) × N(YN(λ)) and ρ1s is(isotopic to) the disjoining translation (σs)−1 in the first R2-direction. It iseasy to see that there is a family of symplectomorphisms βt, 1 ≤t ≤2 of R4which begins with the identity such that, for each t ∈[1, 2], the conjugateisotopies β−1t◦ρ1s ◦βt strictly disjoin Im (G1) and such that the final isotopyψs = β−12◦ρ1s ◦β2 has the form of a translation in the u-direction(x, y, u, v) 7→(x, y, u, v + αs(u))that can be chosen δ-regular on the second factor of G1.

The pair (G1, ψs)then satisfies the conditions in (iii) above.It remains to take into account the requirement that our isotopies shouldbe constant when the time s is near an integral value. But it is sufficient tochoose the initial translation φs, that disjoins B2(λ) in R2, generated by anHamiltonian of the form Hs = hH where H is autonomous and h : R →R isa bump function equal to 1 everywhere except on a µ-neighborhood of eachinteger: choosing µ small enough with respect to δ, we may clearly arrangethat the above argument holds.✷24

References[1] Y. Eliashberg and H. Hofer: An energy-capacity inequality for thesymplectic holonomy of hypersurfaces flat at infinity, Preprint 1992. [2] Y. Eliashberg and H. Hofer: Towards the definition of symplecticboundary, Preprint 1993.

[3] Y. Eliashberg and L. Polterovich: Biinvariant metrics on the groupof Hamiltonian diffeomorphisms, preprint 1991[4] A. Floer, H. Hofer and K. Wysocki: Applications of Symplectic Ho-mology I, preprint 1992. [5] Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds,Invent.

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68 (1993), 48-72. [8] H. Hofer: Symplectic Capacities, Durham Conference, ed: Donaldsonand Thomas, London Math Soc, 1992.

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[14] D. McDuffand L. Polterovich, Symplectic packings and AlgebraicGeometry, preprint 1992[15] D. McDuffand D. Salamon, Notes on J-holomorphic curves , StonyBrook preprint 1993. [16] L. Polterovich: Symplectic displacement energy for Lagrangian sub-manifolds, preprint 1991.

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