A Polyfold proof of Gromov's Non-squeezing Theorem
Franziska Beckschulte, Ipsita Datta, Irene Seifert, Anna-Maria Vocke, Katrin Wehrheim
aa r X i v : . [ m a t h . S G ] O c t A POLYFOLD PROOF OFGROMOV’S NON-SQUEEZING THEOREM
FRANZISKA BECKSCHULTE, IPSITA DATTA, IRENE SEIFERT,ANNA-MARIA VOCKE, AND KATRIN WEHRHEIM
Abstract.
We re-prove Gromov’s non-squeezing theorem by applying Poly-fold Theory to a simple Gromov-Witten moduli space. Thus we demonstratehow to utilize the work of Hofer-Wysocki-Zehnder to give proofs involving mod-uli spaces of pseudoholomorphic curves that are relatively short and broadlyaccessible, while also fully detailed and rigorous. We moreover review thepolyfold description of Gromov-Witten moduli spaces in the relevant case ofspheres with minimal energy and one marked point. Introduction
The non-squeezing theorem, proven by Mikhail Gromov in 1985, essentially ex-cludes nontrivial symplectic embeddings between balls B R and cylinders Z r of ra-dius R, r > B R : = B nR := (cid:8) ( x i , y i ) i =1 ,...,n ∈ R n (cid:12)(cid:12) P ni =1 x i + y i ≤ R (cid:9) , and Z r : = B r × R n − := (cid:8) ( x i , y i ) i =1 ,...,n ∈ R n (cid:12)(cid:12) x + y ≤ r (cid:9) , in any dimension 2 n ≥
4. More precisely, we equip both the closed balls and closedcylinders above with the standard symplectic form ω st = P ni =1 dx i ∧ dy i as subsetsof R n with coordinates x , y , . . . , x n , y n . Then it is easy to see that for any choiceof R and r there are volume preserving embeddings B R ֒ → Z r , due to the infinitelength of the cylinder. If we only consider symplectic embeddings ϕ : B R ֒ → Z r with ϕ ∗ ω st = ω st , there are trivial embeddings for R ≤ r . However, symplecticembeddings cannot exist for R > r , as was shown by Gromov [Gro85] – with variousmore detailed proofs published subsequently, e.g. [Hum97, HZ94, MS04, Wen]. Theorem 1.1.
If there is a symplectic embedding ϕ : B R ֒ → Z r , then R ≤ r . The idea of the proof is to construct an almost complex structure J on thecylinder that pulls back to the standard complex structure ϕ ∗ J = J st on theball, and find a non-constant J -holomorphic curve C ⊂ Z r passing through ϕ (0)with symplectic area at most πr . Then the pullback ϕ − ( C ) ⊂ B R is a J st -holomorphic curve, and thus a minimal surface with respect to the standard metricon R n . As it passes through the center of the ball 0 ∈ B R , comparison with theflat disk of area πR implies R ≤ r via a monotonicity lemma. To find such a Date : October 15, 2020. Theorem 1.1, stated for closed balls and cylinders, implies the analogous statement for openballs and cylinders as follows: Assume there is a symplectic embedding ϕ of the open ball of radius R into the open cylinder of radius r , where R > r . Then there is ε > R − ε > r + ε ,and ϕ restricts to a symplectic embedding of the closed ball of radius R − ε into the closed cylinderof radius r + ε . This is a contradiction to Theorem 1.1. J -holomorphic curve, one observes that the disk cross-section of Z r at the heightof ϕ (0) has the required properties, except that it is holomorphic with respect tothe standard complex structure J . The ideas is then to establish the existenceof J t -holomorphic curves for a path J t of almost complex structures connecting J to J . This requires subtle geometric analysis that is best performed by studyingspheres in the closed symplectic manifold CP × T n − as described in § CP × { pt } ] through a fixed point. Theorem 1.2.
Given any point p and compatible almost complex structure J on CP × T , there exists a J -holomorphic sphere u : S → CP × T with p ∈ u ( S ) and homology class u ∗ [ S ] = [ CP × { pt } ] . Assuming this existence, § Remark 1.3.
We will prove Theorem 1.2 for any compact symplectic manifold(
T, ω T ) with ω T ( π ( T )) = 0, which excludes bubbling in the given homology classof minimal symplectic area. The torus satisfies this assumption since π ( T ) = 0.For more general symplectic manifolds, our line of argument still applies but thepolyfold setup would require the inclusion of bubble trees. Moreover, this onlyproves the result with a possibly nodal J -curve.Classical non-squeezing proofs establish Theorem 1.2 only for “generic” J andrequire a delicate analysis of linearized Cauchy-Riemann operators to show thattheir surjectivity can be achieved alongside the condition ϕ ∗ J = J st . Our proof ofthis more general result demonstrates what proofs of geometric statements look likewhen they can build on abstract polyfold theory [HWZ] and an existing polyfolddescription of the relevant moduli spaces, like for Gromov–Witten spaces given in[HWZ17]. When analytical difficulties are outsourced to polyfold theory, the proofof Theorem 1.2 becomes a transparent geometric argument: • The space M ( J ) of solutions in Theorem 1.2 modulo reparameterization isthe zero set of a Fredholm section σ J : B → E J as described in § • For the standard complex structure J = J we show in Lemma 2.2 that M ( J ) = { [ u ] } consists of a unique solution u ( z ) = ( z, π T ( p )). Theo-rem 3.17 moreover shows that the linearized operator D [ u ] σ J is surjective. • Given any other J = J , § J t ) of compat-ible almost complex structures connecting it to J gives rise to a compactfamily of moduli spaces F t ∈ [0 , M ( J t ). § σ ( t, [ u ]) = σ J t ([ u ]) over [0 , × B . • If we assume M ( J ) to be empty, this implies transversality of σ over theboundary { , } × B . Then § § σ + p ) − (0) between M ( J ) = { [ u ] } and M ( J ) = ∅ . This proves Theorem 1.2by contradiction. Remark 1.4.
The exact meaning of ‘Fredholm section’ in the first step determinesthe class in which the last step provides a contradiction. Minimal work in thefirst step would be to cite [Fil, HWZ17] for a general description in which B isa polyfold (possibly containing nodal curves or curves with nontrivial isotropy).However, this would force us to work with multivalued perturbations p and discuss POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 3 weighted branched orbifolds in the last step. Instead, we show in § B . Then p is single-valued, and the perturbed solution set ( σ + p ) − (0)is a manifold, contradicting the fact that compact 1-manifolds have an even numberof boundary points.The above outline uses polyfold theory entirely as a “black box” with two fea-tures: (a) it describes compactified moduli spaces as zero sets of ‘Fredholm sections’;(b) such ‘Fredholm sections’ can be perturbed to regularize the moduli space. Todemystify feature (a), § B is the space of maps S → CP × T (of Sobolev class W , ) modulo M¨obiustransformations of S that fix a marked point. Moreover, we show that this spacehas trivial isotropy, that is, none of these maps – holomorphic or otherwise – isinvariant under reparameterization with a nontrivial M¨obius transformation. Thisgives the ambient space B the structure of a sc-Hilbert manifold. We describe thisnotion in the following section as part of a brief introduction to polyfold theory.This section also demystifies feature (b) by stating the perturbation theorem in thetrivial isotropy case that is relevant for the non-squeezing proof. Acknowledgements.
This project was initiated by the ‘Women in Symplectic andContact Geometry and Topology’ workshop (WiSCon) at ICERM in 2019. We aredeeply grateful to all the organizers and fellow participants for giving us the spaceand inspiration for this work! Further gratitude to Ben Filippenko and WolfgangSchmaltz for helpful discussions.This work is supported by Deutsche Forschungsgemeinschaft (DFG) under Ger-many’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUC-TURES Excellence Cluster), CRC/TRR 191 and RTG 2229, and by the US Na-tional Science Foundation grants DMS-1708916 and DMS-1807270.1.1.
Polyfold notions and regularization theorems.
Polyfold theory was de-veloped by Hofer, Wysocki, and Zehnder (see [HWZ] and the citations therein)as a general solution to the challenge of regularizing moduli spaces of pseudoholo-morphic curves. The expectation is that any compact moduli space M that isdescribed as zero set of a section can be regularized by appropriate perturbationsof the section. For smooth sections in finite dimensions this is proven in e.g. [GP74,ch.2]. Theorem 1.5 (Finite dimensional regularization) . Let E → B be a smooth finitedimensional vector bundle, and let s : B → E be a smooth section such that s − (0) is compact. Then there exist arbitrarily small, compactly supported, smooth pertur-bation sections p : B → E such that s + p is transverse to the zero section, andhence ( s + p ) − (0) is a manifold of dimension dim( B ) − rank( E ) . The perturbation p is generally a ‘multisection functor’ resulting in f M p := ( σ + p ) − (0) beinga weighted branched orbifold. The contradiction to its boundary (in appropriate orientation) ∂ f M p = M ( J ) being a single point (of trivial isotropy and weight 1) then arises from Stokes’Theorem 0 = R f M p d(1) = R ∂ f M p § BECKSCHULTE, DATTA, SEIFERT, VOCKE, WEHRHEIM
Moreover, the perturbed zero sets ( s + p ′ ) − (0) and ( s + p ) − (0) of any two suchperturbations p, p ′ : B → E are cobordant. A direct generalization of this theorem applies when B has boundary givingrise to perturbed zero sets with boundary ∂ ( s + p ) − (0) = ( s + p ) − (0) ∩ ∂B .Unfortunately, moduli spaces of pseudoholomorphic curves generally do not havenatural descriptions to which this theorem applies. There are several reasons:(1) The space of pseudoholomorphic maps u : S → M , with fixed domain S andtarget M , is the zero set of a Fredholm section of a Banach bundle. But dueto ‘bubbling and breaking’ phenomena, its compactification contains mapsdefined on different domains . To put a topology on the resulting set ofmaps from varying domains, one uses ‘pregluing constructions’ to transfermaps from ‘nodal or broken’ domains to nearby smooth domains. However,these constructions do not provide homeomorphisms to open subsets ofBanach spaces (see e.g. [FFGW16, p.10]), so they do not yield local chartsfor any classical generalization of (topological or smooth) manifolds.(2) The moduli spaces typically arise as quotients of spaces of pseudoholo-morphic maps by groups of reparameterizations of their domains. If thesegroups act with nontrivial isotropy, then we expect an orbifold structureon any ambient space that contains the moduli space.(3) If we wish for a differentiable structure on this ambient space, then we needto ensure that reparameterizations act differentiably. However, this is notthe case for the classical Banach manifold structures on spaces of maps. Asa result, while there are local charts for maps-modulo-reparameterizationwith fixed domain (constructed as local slices to the group action), thetransition maps between different charts are nowhere differentiable.The classical regularization constructions for moduli spaces of pseudoholomor-phic curves, such as [MS04], work around these problems by finding geometric per-turbations that achieve transversality for spaces of maps with fixed domain. Oncethey achieve finite dimensional spaces of perturbed solutions, this requires furthersteps to take quotients and compactify. Whether or not there are sufficiently manygeometric perturbations that are both equivariant and compatible with gluing con-structions depends on the particular geometric setting. The classical proof of thenon-squeezing theorem makes use of the geometric setting of ‘least energy’ to ruleout (1) nodal curves as well as (2) isotropy (due to multiple covers), so that only (3)the differentiability challenge is present. The latter is resolved by finding a regularchoice of J with ϕ ∗ J = J st . This requires showing that holomorphic maps in thedesired homology class all pass injectively through a part of ( CP × T ) \ ϕ ( B R ),where we can freely vary the almost complex structure. While this approach yieldsa rigorous proof, it requires – even in the simplest case – sophisticated analysiscombined with specific geometric properties of the holomorphic curves.Polyfold theory, on the other hand, uses the above challenges as a guide togeneralize the notion of a section in Theorem 1.5 so that the abstract perturbationtheory applies to the desired moduli spaces, and no further case-specific analysis orgeometric properties are needed. The main features are as follows:(1) The ‘pregluing constructions’ generalize open subsets of Banach spaces toimages of retractions as new local models. For example, the neighbourhood Some ‘SFT neck stretching’ moduli spaces also vary the target space M . POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 5 of a nodal sphere is described by an open subset of the model space im ρ = S a ∈ C { a } × π a ( V ) that arises from a family of projections ρ : C × V → C × V, ( a, v ) ( a, π a v ) on a Banach space V , which are centered at π = idand im π a ( V for a = 0. For further details see e.g. [FFGW16, 2.3].(2) The orbifold structure is captured by formulating the notion of an atlas asa groupoid. This provides a nonsingular structure as in (1) on the objectspace (where e.g. perturbations are constructed), with isotropy appearingin the morphisms (e.g. forcing the perturbations to become multivalued).(3) Differentiability of transition maps between different local charts (resp. thestructure maps in the groupoid) is achieved by defining a new notion ofscale-differentiability for maps between Banach spaces equipped with an ad-ditional scale structure. These notions are obtained by formalizing the dif-ferentiability features of reparameterization maps between Sobolev spacesinto a notion that satisfies a chain rule (see [FFGW16, 2.2]).Restricted to finite dimensional Banach spaces, the retractions in (1) are trivial,and (3) coincides with classical differentiability, so (2) reproduces the notion ofan orbifold being represented by a proper ´etale groupoid (see e.g. [Moe02] for anintroduction). In infinite dimensions, these generalizations yield the following newnotions. Here and throughout, we restrict the notions of [HWZ] to metrizabletopologies and sc-Hilbert spaces . In each case, sc-compatibility means that thetransition maps are scale-smooth. • A sc-Hilbert manifold is a metric space equipped with sc-compatiblelocal homeomorphisms to open subsets of sc-Hilbert spaces. • An M-polyfold is a metric space equipped with sc-compatible local home-omorphisms to open subsets of scale-smooth retracts in sc-Hilbert spaces. • A polyfold is a metric space equipped with sc-compatible local homeo-morphisms to finite quotients of open subsets of scale-smooth retracts insc-Hilbert spaces. These domains with group actions and lifts of transi-tion maps form a proper groupoid whose object and morphism spaces areM-polyfolds, and whose structure maps are local sc-diffeomorphisms. • A sc-Hilbert manifold/M-polyfold/polyfold B with boundary is a spacewith compatible charts as before but allowing for open subsets of [0 , ∞ ) × H ,where H is a sc-Hilbert space. Its boundary ∂ B is the union of all preimagesof { } × H . Remark 1.6.
Every manifold M (with boundary) is a sc-Hilbert manifold: It islocally homeomorphic to open subsets of R n (or [0 , ∞ ) × R n − ). Here each R k is asc-Hilbert space with trivial scale structure; see [FFGW16, Ex.4.1.8].With this language in place, the application of polyfold theory to a given modulispace M – for example the moduli space in Theorem 1.2 – has two steps: Metrizability of polyfolds is guaranteed by paracompactness and [HWZ, Thm.7.2]. Hilbert spaces equipped with scale structures automatically admit scale-smooth bump func-tions by [HWZ, § For an introduction to the notion of corners in polyfold theory see [FFGW16, § BECKSCHULTE, DATTA, SEIFERT, VOCKE, WEHRHEIM (a) Describe
M ∼ = σ − (0) as the zero set of a section σ : B → E over apolyfold or M-polyfold B with E a ‘strong bundle’ and σ ‘scale-Fredholm’as defined in [FFGW16, HWZ]. Intuitively, B is the same space of possibly-nodal-maps-modulo-reparameterization as M but allows for general mapsin some Sobolev space, with the pseudoholomorphic condition encoded inthe section σ ([ u ]) = [ ∂ J u ] of an appropriate bundle E . This step is bestachieved by combining existing polyfold descriptions such as [HWZ17] withgeneral construction principles such as restrictions [Fil], pullbacks [Sch],quotients [Zho20]. For our example, we describe this in detail in § Theorem 1.7 (M-polyfold regularization) . Let
E → B be a strong M-polyfoldbundle, and let σ : B → E be a scale-smooth Fredholm section such that σ − (0) iscompact. Then there exists a class of perturbation sections p : B → E supported near σ − (0) such that ( σ + p ) − (0) carries the structure of a smooth compact manifoldof dimension index( σ ) with boundary ∂ ( σ + p ) − (0) = ( σ + p ) − (0) ∩ ∂ B .Moreover, for any other such perturbation p ′ : B → E , there exists a smoothcobordism between ( σ + p ′ ) − (0) and ( σ + p ) − (0) . Remark 1.8.
Suppose that the section σ in Theorem 1.7 restricts to a transversesection on the boundary, i.e. σ | ∂ B : ∂ B → E| ∂ B has surjective linearizations atall points in σ − (0) ∩ ∂ B . Then we can choose the perturbation section p to besupported in the interior, i.e. p | ∂ B ≡
0. As a result, ( σ + p ) − (0) has boundary ∂ ( σ + p ) − (0) = ( σ + p ) − (0) ∩ ∂ B = ( σ | ∂ B ) − (0).This can be proven by following the proof of the regularization theorem in [HWZ].It is explicitly stated and proven in the last part of [FW, Thm.A9]. In our case,the map e : X → ∅ and submanifolds C i = ∅ are trivial, and the polyfold X = B has trivial isotropy. So, the ‘multisection’ λ will be represented by a perturbationsection p : B → E . Our transversality assumption on the boundary means that the‘trivial multisection λ δ representing’ p | ∂ B ≡ { x ∈ ∂ B | σ ( x ) = 0 } .The conclusion is the existence of a perturbation section p with p | ∂ B ≡ σ + p is ‘admissible’ and in ‘general position’, as required for the conclusions ofTheorem 1.7.For readers interested in regularization theorems that resolve the challenge (2)of nontrivial isotropy, we recommend the brief overview [FFGW16, Rmk.2.1.7] andthe in-depth discussion of the finite dimensional case [McD06] before diving intothe technicalities of [HWZ] or their summary in [Sch]. Despite a lot of notationaloverhead, the general polyfold regularization theorem [HWZ, Thm.15.4] can beunderstood as a direct combination of the regularization theorems for sections overM-polyfolds and finite dimensional orbifolds. POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 7 Outline of the Proof
Let us consider a symplectic embedding ϕ : B R ֒ → Z r for radii R > r > R ≤ r by showing that R ′ ≤ r + ε for any choice of 0 < R ′ < R and ε >
0. These choices are needed for constructions in the following section.2.1.
Compactifying the target space.
The proof uses the theory of pseudoholo-morphic curves. Since the analytic setup is simpler for closed manifolds, we preferto work with a compact target space. For that purpose we fix an ε >
0. Then wecan understand ϕ as an embedding ϕ : B R ֒ → ˚ Z r + ε into the slightly larger open cylinder ˚ Z r + ε = ˚ B r + ε × R n − . The first factor of thiscylinder compactifies to a CP . The standard symplectic form on ˚ B r + ε descendsto a symplectic (and thus, area) form ω CP such that CP has area Z CP ω CP = π ( r + ε ) . So we may view ϕ as a symplectic embedding ϕ : ( B R , ω st ) ֒ → ( CP × R n − , ω CP ⊕ ω st ) . Now, we want to compactify the second factor of the cylinder as well. Rememberthat B R is the closed ball. So, the projection to R n − of its image under thecontinuous map ϕ is compact. This means that we can choose N > ϕ ( B R ) ⊂ CP × ( − N, N ) n − . Then we can view ϕ ( B R ) asa subset of the (2 n − T := R n − /N Z n − with standardsymplectic form ω T induced from ω st on R n − . This means we get a symlplecticembedding (again denoted by) ϕ , ϕ : ( B R , ω st ) ֒ → ( CP × T, ω := ω CP ⊕ ω T ) . The proof now proceeds by studying pseudoholomorphic curves in CP × T . Here wewish to work with an almost complex structure J on CP × T so that the pullbackof J -holomorphic maps under the embedding ϕ yields pseudoholomorphic mapsto B R with respect to the standard complex structure J st on B R ⊂ R n . This iscrucial for the last step of the non-squeezing proof in § ω st ( · , J st · ) on B R . To do this rigorously, weneed to shrink the ball slightly to interpolate between almost complex structures. Lemma 2.1.
For any < R ′ < R there is an almost complex structure J on CP × T that is compatible with ω = ω CP ⊕ ω T and satisfies ϕ ∗ J | B R ′ = J st .Proof. The basic idea is to define J = ϕ ∗ J st on the image of ϕ and to set J = J outside a neighbourhood of the image. But as a weighted sum of two almost complexstructures will in general not be an almost complex structure, we can not directlyinterpolate between these. Instead, we interpolate between the corresponding Rie-mannian metrics g := ω ( · , J · ) on CP × T and g ϕ ∗ J st := ω ( · , ϕ ∗ J st · ) on ϕ ( B R ).To do this, we choose a partition of unity ψ + ψ = 1 subordinate to the cover CP × T = U ∪ U where U := CP × T \ ϕ ( B R ′ ) and U := ϕ ( ˚ B R ). (These are opensubsets because ϕ , being an embedding, maps open/closed subsets to open/closedsubsets.) Since ψ i is supported in U i and ϕ ( B ′ R ) ∩ U = ∅ , we have ψ | ϕ ( B ′ R ) ≡ BECKSCHULTE, DATTA, SEIFERT, VOCKE, WEHRHEIM thus obtain a metric g with g | ϕ ( B ′ R ) = g ϕ ∗ J st by interpolating with this partitionof unity, g := ψ · g + ψ · g ϕ ∗ J st . Finally, a pair of a Riemannian metric g and a symplectic form ω determine analmost complex structure J compatible with ω and if g was of the form g = ω ( · , J · ),then the determined almost complex structure is in fact the same J , see [MS98,Prop. 2.50 (ii)]. Thus, g and ω together determine an almost complex structure J that has the required properties. (cid:3) There are two more properties of pullbacks C := ϕ − ( C ) ⊂ B R ′ of J -holomor-phic curves C ⊂ CP × T that are required for their to prove the non-squeezingresult R ′ ≤ r + ε . First, we need C to pass through the point p := ϕ (0) ∈ CP × T ,so that C ⊂ B R ′ passes through the center 0 of the ball. Second, we wish tobound the symplectic area R C ω st ≤ R C ω ≤ π ( r + ε ) . The latter is achievedby prescribing the homology class [ C ] = [ CP × { pt } ] since this determines theintegral of the closed symplectic form ω , Z C ω = Z CP ×{ pt } ω CP ⊕ ω T = Z CP ω CP + Z { pt } ω T = π ( r + ε ) + 0 . Ultimately, we will find a not necessarily embedded curve C = u ( S ) by studying J -holomorphic maps u : ( S , i ) → ( CP × T n − , J ) with a point constraint u ( z ) = p in the homology class [ u ] = [ CP × { pt } ]. Their existence is stated, forgeneral J on a product manifold CP × T , in Theorem 1.2. The proof starts withthe existence of a unique J -holomorphic map for a specific J described in § § § The unique J -holomorphic curve. This section begins our study of pseu-doholomorphic curves in CP × T by considering a split almost complex structure J = i ⊕ J T on CP × T . Here ( T, ω T ) can be any compact symplectic manifold,though its topology will be restriced in following sections. For the nonsqueezingproof, there is a standard complex structure J T on the torus T = R n − /N Z n − ;in general, we choose any ω T -compatible almost complex structure J T on T . Thisensures that J is compatible with ω = ω CP ⊕ ω T , meaning g = ω ( · , J · ) is a Rie-mannian metric. In the following, we view S as a Riemann surface by identifyingit with CP and using the standard complex structure i on CP . Then we find a J -holomorphic sphere passing through any given point p = ( z , m ) ∈ CP × T by combining the identification S ∼ = CP with a constant map to T , u : ( S , i ) → ( CP × T, J ) , z ( z, m ) . The symplectic area of this sphere – a quantity that only depends on the homologyclass – is E ( u ) = Z S u ∗ ω = Z CP ω CP = π ( r + ε ) . (1)The next lemma shows that, up to reparameterization, this is the only sphere withthese properties in its homology class. Lemma 2.2.
Assume u : ( S , i ) → ( CP × T, J ) is J -holomorphic, passes through p , and represents the class [ CP × { pt } ] ∈ H ( CP × T ; Z ) . Then, there is abiholomorphism ψ : ( S , i ) → ( S , i ) such that u ◦ ψ = u . POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 9
Proof.
Since u is ( J = i ⊕ J T )-holomorphic, its composition with projection toeach factor yields holomorphic maps f := pr CP ◦ u : S → CP ∼ = S and g :=pr T ◦ u : S → T . Moreover, the homology condition specifies g ∗ [ S ] = [ { pt } ] =0 ∈ H ( T ) and f ∗ [ S ] = [ S ] ∈ H ( S ). The energy identity R g ∗ ω = R | d g | (see[MS04, Lemma 2.2.1]) then implies R | d g | = 0. So, g must be constant. Since u passes through p = ( z , m ), this means g ( z ) = pr T ( u ( z )) = m . Moreover, f ∗ [ S ] = [ S ] ∈ H ( S ) implies that f is neither constant nor a multiple cover ofanother holomorphic map. Thus, ψ := f − exists and is a biholomorphism of S .Then we obtain the claim as ( u ◦ ψ )( z ) = (cid:0) f ( f − ( z )) , g ( ψ ( z ) (cid:1) = ( z, m ) = u forall z ∈ S . (cid:3) Remark 2.3.
It is part of both the classical and our polyfold proof to show thatthe curve u is transversely cut out of the space of all curves in its homology classpassing through p . This statement will be made precise in Theorem 3.17.2.3. Using the monotonicity lemma.
This section finishes the proof of thenonsqueezing Theorem 1.1 assuming that we have found a J -holomorphic map u : S → CP × T with the same properties as the unique J -holomorphic curve inLemma 2.2, except that J is an almost complex structure as in Lemma 2.1 with ϕ ∗ J = J st . The existence of u follows from Theorem 1.2, proven in § u : S → CP × T , we obtain a J st -holomorphic map on e S := u − ( ϕ ( ˚ B R ′ )) ⊂ S , v := ϕ − ◦ u : e S −→ R n . This map passes through the center ϕ − ( p ) = 0 of the ball B R ′ and has area R v ∗ ω st ≤ R u ∗ ω = π ( r + ε ) , so comparison with the minimal surface through thecenter of the ball – the disk of area π ( R ′ ) – will yield R ′ ≤ r + ε . To deduce thisinequality directly from the monotonicity lemma for minimal surfaces, we wouldhave to show that v is an embedding. Instead, we use a monotonicity lemma forholomorphic maps to a complex Hilbert space. It can be found as Lemma A.2 inthe appendix, together with a detailed proof. Proof of Theorem 1.1.
We will apply Lemma A.2 to the map v : e S → R n with( V, J ) := ( R n , J st ) and open balls ˚ B R k := ˚ B R k (0) ⊂ R n of radii R k → R ′ .The preimage of the center is nonempty, v − ( { } ) = u − ( { p } ) = ∅ , since u passes through p = ϕ (0). The domain e S of v is an open subset of S because ϕ ( ˚ B R ′ ) ⊂ CP × T is the image of an open set under an embedding. To applythe lemma we need to restrict v to a compact subdomain S k ⊂ e S with smoothboundary such that k v ( z ) k ≥ R k for all z ∈ ∂S k . For that purpose we consider thesmooth function ρ : e S → R , z
7→ k v ( z ) k . Since its regular values are dense we canfind a sequence 0 < R k < R ′ with limit lim k →∞ R k = R ′ such that R k are regularvalues of ρ . Then S k := { z ∈ e S | k v ( z ) k ≤ R k } is a domain with smooth boundary ∂S k = ρ − ( R k ). It is compact because S k = v − ( B R k ) = u − ( ϕ ( B R k ) ⊂ S is aclosed subset of the compact S . Moreover, v | S k is nonconstant on each connectedcomponent of S k , since u is nonconstant (as it has positive energy) and its criticalpoints in S are a finite set by [MS04, Lemma 2.4.1]. So, we can apply Lemma A.2 to v | S k : S k → V = R n and the open ball˚ B R k = { q ∈ R n | k q k < R k } centered at p = 0 ∈ R n to obtain πR k ≤ Z v − (˚ B Rk ) v ∗ ω st = Z u − ( ϕ (˚ B Rk )) u ∗ ω ≤ Z S u ∗ ω = π ( r + ε ) . As R k → R ′ , this yields π ( R ′ ) ≤ π ( r + ε ) as claimed; and by taking R ′ → R and ε → R ≤ r . (cid:3) A compact moduli space.
In this and the next subsection, we prove The-orem 1.2, while assuming that the M-polyfold construction in § J . Indeed, inLemma 2.2 we compute the number of pseudoholomorphic curves in the particularhomology class intersecting the given point p to be 1 for J = J . So, by showingthat this count is independent of J , we can show the existence of a J -holomorphicmap in Theorem 1.2.For that, we use the fact that the space of ω -compatible almost complex structuresis contractible (see e.g. [MS98, Prop. 4.1]). Thus, we can choose a smooth path( J t ) t ∈ [0 , of ω -compatible almost complex structures from J to J . Moreover, wefix a marked point z ∈ S that we require is mapped to p by all the consideredmaps.Then, for every t ∈ [0 , J t -holomorphic curves M t := ( u : S → CP × T smooth (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( z ) = p , ∂ J t u = 0 , [ u ] = [ CP × { pt } ] ) , ∼ , (2)where u ∼ u ′ iff there is a biholomorphism ψ : S → S such that u ′ = u ◦ ψ . Here ∂ J t is the Cauchy-Riemann operator for J t , that is, ∂ J t u = (d u + J t ◦ d u ◦ i ).Now consider the moduli space for the family ( J t ) t ∈ [0 , M := { ( t, [ u ]) | t ∈ [0 , , [ u ] ∈ M t } . (3)We will prove Theorem 1.2 by contradiction: Assuming M = ∅ , in § M is a compact 1-dimensional cobordism from M to M . However, M = { [ u ] } consists of exactly one element (see Lemma 2.2).Therefore, this cobordism contradicts that M is empty.The first step in completing the proof of Theorem 1.2 is to establish compactnessof the unperturbed moduli space. This is a special case of Gromov’s compactnesstheorem, where bubbling is excluded in the given homology class using the re-striction of the topology of T from Remark 1.3. For general symplectic manifolds( T, ω T ), we would need to compactify M by bubble trees. So, the subsequentproof of M being nonempty would only show the existence of (possibly singular) J -curves and not necessarily smooth spheres in the required homology class. Theorem 2.4.
Let ( T, ω T ) be a compact symplectic manifold with ω T ( π ( T )) = 0 .Then the moduli space M defined in Equation (3) is compact with respect to thequotient topology induced by [0 , × C ∞ ( S , M ) .Proof. This proof follows [Hum97, Chapter 4] and [Ack]. Let ( t n , [ u n ]) be a sequencein the moduli space M . In particular, u n is a sequence of J t n -holomorphic mapsin M . We want to show that there exists a subsequence which converges to an POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 11 element ( t ∞ , [ u ∞ ]) ∈ M , meaning that u ∞ is a J t ∞ -holomorphic map. Here, by convergence we mean the following:(1) t n converges to t ∞ in the usual topology of [0 , ⊂ R ,(2) [ u n ] converges to [ u ∞ ] ∈ M t ∞ in the Gromov sense , that is, there existbiholomorphic maps ϕ n : ( S , j ) → ( S , j ) with ϕ n ( z ) = z such that thereparameterized maps u n ◦ ϕ n : S → Q converge in C ∞ to u ∞ .To achieve (1) we can choose a subsequence of t n ∈ [0 ,
1] with t n convergingto a t ∞ ∈ [0 ,
1] since the interval is compact. Then, as { J t } t ∈ [0 , is a continuouspath, we can deduce C ∞ -convergence of the almost complex structures J t n → J t ∞ .To achieve (2), we consider this subsequence, denoting it again by ( t n , [ u n ]). Thekey observation is that the area functional is uniformly bounded on M . Indeed, all J t n -holomorphic curves u n represent the same homology class [ u ] = [ CP × { pt } ],so that E ( u n ) = Z CP u ∗ n ω = ω ([ CP × { pt } ]) = π ( r + ε ) is constant and hence bounded. Moreover, S is a closed surface. Thus, by Gro-mov’s compactness theorem (e.g. [Hum97, Chapter V, Thm. 1.2]) there exists asubsequence of [ u n ] converging in the Gromov sense to a J t ∞ -holomorphic cuspcurve u ∞ of the same energy E ( u ∞ ) = π ( r + ε ) .Actually, this cusp curve consists of a single J t ∞ -holomorphic sphere. Indeed, let[ v ] , . . . , [ v k ] be the non-constant components of u ∞ of energy E ( v n ) = R CP v ∗ n ω > E ( v ) + . . . + E ( v k ) = E ( u ∞ ) = π ( r + ε ) . Since the symplectic form ω = ω CP ⊕ ω T splits, the energies are the sums E ( u n ) = ω CP ( α n )+ ω T ( β n ) of sym-plectic areas of the projections α n := [pr CP ◦ u n ] and β n := [pr T ◦ u n ] to the factors CP and T . Here we have ω T ( β n ) = 0 because of the assumption ω T ( π ( T )) = 0,and ω CP ( α n ) ∈ Z π ( r + ε ) since H ( CP ) is generated by [ CP ] which has sym-plectic area π ( r + ε ) by construction. Thus each nontrivial component has energyat least E ( v n ) ≥ π ( r + ε ) , but since the total energy of the bubble tree is π ( r + ε ) this implies k = 1. This means that the limit cusp curve u ∞ has one non-constantcomponent. Since it has only one marked point (arising from z in the defini-tion of M t ), it cannot have ghost components, and thus u ∞ consists of a single J t ∞ -holomorphic sphere u ∞ : S → CP × T .The meaning of Gromov-convergence [ u n ] → u ∞ = [ u ∞ ] is exactly as statedin (2) above, so we have shown that M is sequentially compact. Finally, com-pactness follows from the fact that the Gromov-topology is metrizable; see [MS04,Theorem 5.6.6.]. (cid:3) Applying the polyfold regularization scheme.
This section proves The-orem 1.2. We will use the notation and facts established in § § §
3, and the polyfold regularization scheme from § Proof of Theorem 1.2.
Assume that M = ∅ . Under this assumption, we will thepolyfold regularization scheme in Theorem 1.7 to perturb M just enough to achievea smooth structure, while not loosing compactness, or changing its boundary at t = 0 ,
1. With that, we obtain a compact cobordism between M = { [ u ] } and M = ∅ . For that purpose we construct the following objects in § : We introduce a candidate space B in (4), then we establish the M-polyfold structure on anopen subset B ′ ⊂ B , that we rename into B here for ease of notation. The exact choice of B ′ as • an ambient M-polyfold [0 , × B ⊃ M modeled on sc-Hilbert spaces (The-orem 3.5), • a tame strong M-polyfold bundle E → [0 , × B (Theorem 3.10), • a sc-Fredholm section σ : [0 , × B → E such that M = σ − (0) ⊂ [0 , × B (Theorem 3.14).The use of sc-Hilbert spaces guarantees the existence of sc-smooth bump functionson B by [HWZ, § σ istransverse to the zero section at { }×B in Theorem 3.17. Moreover, the assumption M = ∅ implies transversality of σ at { } × B (see Remark 3.18). Now we applythe M-polyfold regularization scheme, see Theorem 1.7. This gives a perturbationsection p : [0 , × B → E , such that ( σ + p ) − =: M p is a compact 1-dimensionalmanifold. By Remark 1.8, we can assume p to be supported inside (0 , × B . So,the boundary of M p is ∂ ( M p ) = M p ∩ ∂ ([0 , × B ) = M p ∩ ( { , } × B ) ∼ = M ⊔ M = { [ u ] } . Thus, the boundary ∂ ( M p ) consists of only one point. Such a manifold does notexist. Therefore, the assumption M = ∅ was false and we have proven Theo-rem 1.2. (cid:3) Remark 2.5.
If we dropped the condition u ( z ) = p in the construction of themoduli spaces M t in Equation (2), then we could directly use the polyfold setupfor Gromov-Witten moduli spaces with one marked point from [HWZ17]. Thenthe above arguments would provide a (2n+1)-dimensional cobordism M p betweenthe 2n-dimensional manifolds M and M p , where the latter is obtained from M by the perturbation p | { }×B . One would need to choose the perturbation to besupported away from J -curves intersecting the point p ∈ CP × T (which byassumption do not exist), so that the evaluation map on the perturbed modulispace ev : M p → CP × T does not contain p in its image and hence has degree 0.Moreover, one would need to formulate Lemma 2.2 in a way that the evaluation mapev : M → CP × T is a bijection and thus has degree 1. This difference betweendegrees is then in contradiction to the fact that the evaluation map B → CP × T extends both ev | M p and ev | M to a continuous map ev : M p → CP × T on thecobordism. 3. Polyfold setup
This section provides the polyfold description of the compact moduli space M that is the basis of the proof of Theorem 1.2 in § T, ω T , J T ) denotes the torus with a compatible pair of symplecticform and almost complex structure (e.g. the one constructed in § ω T ( π ( T )) = 0, as explainedin Remark 1.3. Moreover, we fix a point p and compatible almost complex struc-ture J on ( Q, ω ) = ( CP × T, ω CP × ω T ). We choose a smooth path ( J t ) t ∈ [0 , of ω -compatible almost complex structures from J = i × J T to J = J . Then, § M = { ( t, [ u ]) | t ∈ [0 , , [ u ] ∈ M t } of moduli discussed in Remark 3.6 is immaterial, since the moduli space M and all its regular perturbationswill be automatically contained in [0 , × B ′ . POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 13 spaces M t = ( u : S → CP × T smooth (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( z ) = p , ∂ J t u = 0 , [ u ] = [ CP × { pt } ] ) , ∼ , with the equivalence relation u ∼ v : ⇐⇒ ∃ ψ : S → S biholomorphism with ψ ( z ) = z and u = v ◦ ψ. The polyfold setup starts with a choice of an ambient space that contains M (asa compact zero set of a sc-Fredholm section). For the family M = S t ∈ [0 , M t , thenatural choice of ambient space is [0 , × B , where B is an ambient space for eachof the moduli spaces M t , given by B := ( u : S → CP × T of class W , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( z ) = p , [ u ] = [ CP × { pt } ] ) , ∼ . (4)This uses the same equivalence relation ∼ as in the definition of M t , but we equip B with the quotient topology induced by the metrizable topology on the Hilbertmanifold H = W , ( S , CP × T ), unlike the smooth topology in Theorem 2.4.The condition [ u ] = [ CP × { pt } ] specifies some connected component(s) of H , and u ( z ) = p cuts out a further submanifold, so that B is the quotient of a Hilbertmanifold. However, the action by reparameterization with biholomorphisms is notdifferentiable (see e.g. [FFGW16, § B does not inherit the smoothstrucure of a Hilbert manifold. Instead, we will show in Theorem 3.5 that it carriesthe structure of a sc-Hilbert manifold. We sometimes write an equivalence class as α ∈ B instead of [ u ] ∈ B , indicatingthat there is no preferred representative in the class. To build the bundle E → [0 , × B , we consider for each ( t, α ) ∈ [0 , × B , the Hilbert space quotient E ( t,α ) := ( ( u, ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ u ] = αξ ∈ Λ , J t (cid:0) S , u ∗ T( CP × T ) (cid:1) of class W , ) , ∼ , (5)where Λ , J ( S , u ∗ T Q ) denotes the 1-forms on S with values in the pullback bundle u ∗ T Q that are complex antilinear with respect to i on S and J on Q = CP × T .The equivalence relation ∼ is given by( u, ξ ) ∼ ( v, ζ ) : ⇐⇒ ∃ ψ : S → S biholomorphism with ψ ( z ) = z and u = v ◦ ψ and ξ = ζ ◦ d ψ. Now the bundle
E → [0 , × B is given by the total space E := n ( t, [( u, ξ )]) (cid:12)(cid:12)(cid:12) t ∈ [0 , , α ∈ B , [( u, ξ )] ∈ E ( t,α ) o (6)with the projection to [0 , × B . This projection is well-defined since ( u, ξ ) ∼ ( v, ζ )implies u ∼ v .Finally, the section σ : [0 , × B → E , ( t, [ u ]) ( t, [( u, ∂ J t u )])(7) Strictly speaking, our proofs establish the polyfold structures not for B and E → B as stated,but after restriction to a W , -open neighbourhood B ′ ⊂ B of the dense subset of smooth curves B ∞ ⊂ B . Additional estimates could prove B ′ = B , but applications of the polyfold descriptionyield the results for any B ∞ ⊂ B ′ ⊂ B ; see Remark 3.6. cuts out the moduli space σ − (0) = M . To apply the M-polyfold regularizationTheorem 1.7, we need to equip the spaces B and E with sc-smooth structures suchthat σ is sc-smooth, and moreover show that σ is a sc-Fredholm section.3.1. The Gromov-Witten space of stable curves.
The sc-smooth structureon the base space B is obtained in § B ⊂ Z of aspace of stable curves in the manifold Q = CP × T . The space Z := (cid:8) u : S → Q of class W , (cid:12)(cid:12) [ u ] = [ CP × { pt } ] (cid:9) . ∼ (8)does not satisfy the condition u ( z ) = p , but the marked point z is present inthe definition of the equivalence relation ∼ , that equals the one in (4). Thus, thereexists a well defined evaluation map ev : Z → Q, ev([ u ]) := u ( z ).Then, the base space B can be viewed as the preimage of the point p ∈ Q underthe evaluation map ev, that is B = { [ u ] ∈ Z | u ( z ) = p } = ev − ( { p } ) ⊂ Z. (9)The space Z is a subspace of a polyfold that Hofer, Wysocki and Zehnder constructin [HWZ17], which we will denote by Z HWZ . In [HWZ17], Hofer, Wysocki andZehnder also construct a strong polyfold bundle W → Z HWZ and a sc-Fredholmsection ∂ : Z HWZ → W , that cuts out holomorphic curves. More precisely, for anynumbers g, k ∈ N and nontrivial homology class A ∈ H ( Q ), the polyfold Z HWZ has a component Z HWZ g,k,A so that ∂ − (0) ∩ Z HWZ g,k,A = M g,k ( A )is the compactified Gromov-Witten moduli space of (possibly nodal) pseudoholo-morphic curves in class A of genus g , with k marked points. In the following, wewill explain how considering genus g = 0, one marked point, and homology class A := [ CP × { pt } ] ∈ H ( Q ) will provide an identification Z ∼ = Z HWZ0 , ,A ⊂ Z HWZ .In general, the space Z HWZ = { ( S, j, M, D, u ) | . . . } / ∼ is defined in [HWZ17,Definition 1.4,1.5] as a set, and given a topology in [HWZ17, § ( S, j, M, D, u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( S, j, M, D ) connected nodal Riemann surface ,u ∈ W , ,δ ( S, Q ) , R C u ∗ ω ≥ C ⊂ S, R C u ∗ ω > C ⊂ S . Here (
S, j ) is a (not necessarily connected) Riemann surface, M ⊂ S is a finite setof marked points, and D is a finite set of nodal pairs in S . These nodal pairs areidentified in order to obtain a noded Riemann surface – which is required to beconnected. The maps u : S → Q are then required to descend to a continuous mapfrom the noded Riemann surface, and they are required to be of weighted Sobolevclass W , ,δ on the complement S \ D of the nodal points.Two stable maps ( S, . . . , u ) ∼ ( S ′ , . . . , u ′ ) are equivalent if there is a biholomor-phism ψ between the corresponding marked noded Riemann surfaces (i.e., compat-ible with M and D ), such that u ′ = u ◦ ψ . The conditions on the symplectic areain Z HWZ are known to guarantee that pseudoholomorphic curves of this type arestable, meaning that their isotropy groups are finite. We show in Lemma 3.2 thatall maps in Z HWZ have finite isotropy. In fact, Lemma 3.3 proves that all mapsin our particular case Z HWZ0 , ,A have trivial isotropy. To begin with, the following These components are open and closed but not necessarily connected.
POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 15
Lemma simplifies the description of this space to the description given in Equation(8).
Lemma 3.1.
The polyfold Z HWZ , ,A for A = [ CP × { pt } ] ∈ H ( Q ) is naturallyidentified with Z . The same holds true if we replace the torus T in Q = CP × T by any compact symplectic manifold with ω T ( π ( T )) = 0 .Proof. First recall that the symplectic area of a map u : S → Q depends onlyon its homology class. Thus, for [ u ] = [ CP × { pt } ] = A ∈ H ( Q ), we obtain R S u ∗ ω = ω ( A ) = π ( r + ε ) >
0, as computed in Equation (1). This is the symplecticarea of all equivalence classes of (possibly nodal) maps in Z HWZ0 , ,A = ( [( S, j, M, D, u )] ∈ Z HWZ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) genus of (
S, j, D ) is g = 0 , M = 1 , [ u ] = A = [ CP × { pt } ] ) . We claim that this can be simplified to the formulation given in (8). For that, wefirst recall that all the nodal surfaces of genus g = 0 are trees of spheres. Next, notethat the assumption ω T ( π ( T )) = 0 guarantees that all components S ≃ C ⊂ S have energy R C u ∗ ω ∈ ω ( π ( Q )) = ω CP ( π ( CP )) = Z π ( r + ε ) . Recall here that wechose the symplectic form on CP in § ω CP ([ CP ]) = R CP ω CP = π ( r + ε ) . As in the proof of Theorem 2.4, the homology condition[ u ] = [ CP × { pt } ] implies that S can only have one component, on which u is non-constant. Indeed, the total symplectic area R S u ∗ ω = ω ([ CP × { pt } ]) = π ( r + ε ) isthe sum of non-negative areas of all components, but each non-constant componenthas energy R C u ∗ ω ≥ π ( r + ε ) . Moreover, S cannot have so-called ghost components C ⊂ S with R C u ∗ ω = 0 because stability of such components would require at leastthree special points, while there is only one marked point from k = 1 and at mostone nodal point connecting C to the unique non-constant component. Thus allnodal surfaces ( S, j, D ), that are needed for the component Z HWZ0 , ,A ⊂ Z HWZ , aresingle spheres, i.e. D = ∅ .The absence of nodes, i.e. D = ∅ , also explains why we do not need to considerweighted Sobolev spaces but can directly work with maps of Sobolev class W , .Moreover, the topology specified in [HWZ17, § W , -maps.Finally, we will use the fact that any compact genus 0 Riemann surface ( S, j )without nodes is biholomorphic to ( S , i ), so that for each point in Z HWZ0 , ,A we canchoose representatives [( S , i, M, ∅ , u )]. The remaining equivalence relation is thenby biholomorphisms of ( S , i ), which we can use to fix the marked point M = { z } and reduce the equivalence relation to reparameterization with biholomorphisms ψ : S → S that fix the marked point ψ ( z ) = z as in Equation (4). Thisidentifies the polyfold given in [HWZ17] with Z as defined in Equation (8) in thefollowing way: Z HWZ0 , ,A = ( S, j, M, ∅ , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ∈ W , ( S, Q ) . . . [ u ] = [ CP × { pt } ] , ( S, j, M, u ) ∼ ( S ′ , ψ ∗ j, ψ − ( M ) , u ◦ ψ ) ∼ = ( ( S , i, { z } , ∅ , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ∈ W , ( S , Q )[ u ] = [ CP × { pt } ] ) , ∼∼ = (cid:8) u ∈ W , ( S , Q ) (cid:12)(cid:12) [ u ] = [ CP × { pt } ] (cid:9) (cid:14) ∼ = Z. (cid:3) Trivial isotropy.
In this section, we show that all (not necessarily pseudo-holomorphic) maps in the space Z defined in (8) have trivial isotropy due to theirspecific homology class. We start by showing that maps of nontrivial finite energyhave finite isotropy groups for any compact domain and target. Lemma 3.2.
Let ( Q, ω ) be any symplectic manifold, (Σ , j ) a compact connectedRiemann surface, and let u : Σ → Q be a C -map with positive symplectic area R Σ u ∗ ω > . Then it has a finite isotropy group G u := { ψ : (Σ , j ) → (Σ , j ) biholomorphic | u ◦ ψ = u } . Proof.
Having positive symplectic area implies that there exists an open ball B ⊂ Σ,so that u is injective on B . Indeed, since R u ∗ ω > p ∈ Σ suchthat ( u ∗ ω ) p does not vanish as a bilinear form on T p Σ, i.e. there are vectors v, w ∈ T p Σ with ( u ∗ ω ) p ( v, w ) >
0. This is equivalent to ω u ( p ) (d u ( p )( v ) , d u ( p )( w )) > ω is skew-symmetric, we know that d u ( p )( v ) , d u ( p )( w ) are linearly indepen-dent. Thus d u ( p ) has maximal rank 2, and we can find a ball B around p such that u | B is injective and R B u ∗ ω > g ( B ) of B under the automorphisms g ∈ G u areall disjoint. Assume this is not the case. Then there exists a g ∈ G u \ { id } such that B ∩ g ( B ) ⊃ U contains a nonempty open set U . For every p ∈ U we have p = g ( q p )for some q p ∈ B . Since u ◦ g = u we have u ( p ) = u ( q p ), so that the injectivityof u | B implies that p = q p . This shows g | U ≡ id, so that unique continuationfor biholomorphisms on the connected surface Σ implies g = id, contradicting theassumption.Therefore, S g ∈ G u g ( B ) is a disjoint union of open sets, and each restriction u | g ( B ) has the same positive energy Z g ( B ) u ∗ ω = Z B g ∗ u ∗ ω = Z B u ∗ ω =: δ > . If u ∗ ω is everywhere non-negative, this implies that G u cannot have more than δ − R Σ u ∗ ω elements. This is the case for u being pseudoholomorphic. To provefiniteness of G u in general, we pick metrics on Σ and Q with respect to which d u and ω are bounded. Then we have R g ( B ) u ∗ ω ≤ C Vol(g(B)) for some constant
C >
0, and hence Vol(g(B)) ≥ δ C . Since the total volume of Σ is finite, and the sets g ( B ) are disjoint, this implies that G u must be finite. (cid:3) For spheres in the specific homology class [ u ] = [ CP × { pt } ] in Q = CP × T wecan extend this argument to show that the isotropy groups are in fact trivial. Lemma 3.3. If [ u ] ∈ Z , then G u = { id } .Proof. The Sobolev embedding W , ( S , Q ) ⊂ C ( S , Q ) and Lemma 3.2 implythat elements of Z have finite isotropy groups. To prove that they are trivial, weconsider u ∈ W , ( S , Q ) with finite but nontrivial isotropy group G u = { id } andwe will show that [ u ] = [ CP × { pt } ], and thus [ u ] / ∈ Z .Note that G u ⊂ Aut( S , i ) = PSL(2 , C ) is a subgroup of the M¨obius group. Since G u is finite, it must consist of elements of finite order. M¨obius transformations areclassified into parabolic, elliptic and hyperbolic/loxodromic ones, corresponding totheir geometric and algebraic properties. The only M¨obius transformations of See for example the lecture notes [Ols] for a detailed geometric description of the M¨obiusgroup, especially [Ols, Cor. 12.1] for the statement about elements of finite order.
POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 17 finite order k > πk aroundtwo different fixed points in the extended complex plane. Since G u is assumed tobe nontrivial, it must contain some such rotation f ∈ Aut( S , i ). We can moreoverchoose a biholomorphism ψ of C ∪ {∞} that maps the fixed points of the rotationto 0 and ∞ . Then the map u ′ := u ◦ ψ − represents the same homology class as u and its isotropy group contains g := ψ ◦ f ◦ ψ − , which is a rotation of order k > ∞ . Thus, g : C ∪ {∞} → C ∪ {∞} is given by g ( z ) = e πi/k z , and wehave u ′ ( re iθ + m · πi/k ) = u ′ ( re iθ ) for all m ∈ Z since g m ∈ G u ′ . This allows us tofactorize u ′ = v ◦ ρ k with v ( re iθ ) := u ′ ( re iθ/k ) and ρ k ( re iθ ) := re kiθ for r ∈ (0 , ∞ ), θ ∈ [0 , π ]. By identifying S ∼ = C ∪ {∞} , this defines continuous maps v : S → Q and ρ k : S → S with v (0) = u (0), v ( ∞ ) = u ( ∞ ), ρ k (0) = 0, and ρ k ( ∞ ) = ∞ .Finally, this implies [ u ] = [ u ′ ] = deg( ρ k ) · [ v ] ∈ H ( Q ), where deg( ρ k ) = k > ρ k is a k -fold cover of S . This contradicts [ u ] = [ CP × { pt } ], since k [ CP ] ∈ H ( CP ) is not representable by a map pr CP ◦ v : S → CP . (cid:3) The base space.
Within this section, we explain how to equip the base space B defined in (4) (and thus also [0 , × B – see Corollary 3.9) with a polyfoldstructure. Since the isotropy is trivial by Lemma 3.3, this means we give B an M-polyfold structure, as discussed in § Remark 3.4.
Before stating this result rigorously, we need to introduce one morepiece of polyfold notation (also see [FFGW16, § Z contains a dense subset Z ∞ ⊂ Z of so-calledsmooth points and a nested sequence of subsets Z ∞ ⊂ . . . ⊂ Z k +1 ⊂ Z k ⊂ . . . Z = Z. Each of these is equipped with its own metrizable topology, so that, in particular,the inclusion maps Z k +1 ֒ → Z k are continuous.In most applications, the smooth points z ∈ Z ∞ are the smooth maps moduloreparameterization, whose domains may be nodal. For the Gromov-Witten polyfold Z in Equation (8), the points [ u ] ∈ Z k are given by maps u : S → Q of class W k, ,and Z k is equipped with the quotient of the W k, -topology. Correspondingly, Z ∞ consists of the equivalence classes of smooth maps. Theorem 3.5.
After replacing the base space B with an open neighborhood B ′ ⊂ B of the smooth points B ∞ = B ∩ Z ∞ , it carries the natural structure of a sc-Hilbertmanifold and thus of an M-polyfold. Remark 3.6.
In practice, we expect B ′ = B by an analogue of the estimates in[HWZ17, Theorems 3.8, 3.10], which guarantee that charts constructed on neigh-bourhoods of smooth points b ∈ B ∞ := B ∩ Z ∞ cover all of B .We need to center charts at smooth points in both proof approaches that we willpresent. Since we avoid all avoidable estimates, this proves a sc-Hilbert structure onan open neighbourhood B ′ ⊂ B of B ∞ . That is, B ′ contains all equivalence classesof smooth maps u : S → Q . (In fact, B ′ contains a W , -neighbourhood of eachsuch smooth point [ u ]). However, we may have [ v ] ∈ B \ B ′ for some v ∈ W , \ C ∞ .While this makes the description of the base space less explicit, it does not affectthe rest of the proof. In fact, we could even allow for a more drastic restriction of the base space B ′ ⊂ B as follows. Proving Theorem 3.5, by applying [Fil, Thm. 5.10 (I)] directly,establishes an M-polyfold structure on a Z -open subset B ′ ⊂ B ∩ Z such that B ∩ Z ∞ ⊂ B ′ . This would remove all maps in W , \ W , from B ′ and guaranteeonly that B ′ contains a W , -neighbourhood of smooth maps.However, the moduli spaces M t and their perturbations automatically lie inthe ∞ -level B ∞ due to the regularizing property of sc-Fredholm sections [HWZ,Def.3.8]. So neither a shift to Z -topology nor restricting to a neighbourhood of B ∞ affects how we can use the M-polyfold regularization scheme (see Theorem 1.7)in the proof of Theorem 1.1.There are several ways to prove Theorem 3.5. We will first explain how thenatural polyfold structure for Z ⊂ Z HWZ constructed in [HWZ17] induces apolyfold structure on its subset B = ev − ( { p } ) ⊂ Z . This proof applies an implicitfunction theorem by Filippenko [Fil] to the evaluation map ev : Z → Q . Proof of Theorem 3.5 by Implicit Function Theorem.
In [HWZ17], the space Z HWZ and thus also its (open and closed) component Z is given a polyfold structure. Fol-lowing the proof in [HWZ17], one sees that for the component Z considered here(i.e. genus 0, homology class [ CP × { pt } ]), the model spaces are sc-Hilbert spaces(i.e. all retractions are identity maps). Their scale structure corresponds to thedense subsets Z m = { [ u ] ∈ Z | u ∈ W m, ( S , Q ) } ⊂ Z . Moreover, by Lemma 3.3we have trivial isotropy. So [HWZ17] actually constructs Z as a sc-Hilbert manifold.Now we will use the description of B = ev − ( { p } ) in Equation (9) as a preimageof p ∈ Q under the evaluation map ev : Z → Q, [ u ] u ( z ). Note that theevaluation map is classically smooth on each level Z m ⊂ Z , so it is sc-smooth by [HWZ, Cor. 1.1]. Furthermore, [Fil, § Q , so in particulartransverse to { p } ⊂ Q . Now we can apply [Fil, Thm. 5.10 (I)] to deduce that B = ev − ( { p } ) is an M-polyfold. Though, strictly speaking, [Fil] constructs an M-polyfold structure on an open neighbourhood of the ∞ -level B ∞ := B ∩ Z ∞ in the1-level B ∩ Z . The proof of nonsqueezing could work with such a neighbourhood(see Remark 3.6), but we will argue that the proof of [Fil, Thm. 5.10(I)] in ourspecific setting does not actually require a shift in the topology – just a restrictionto a neighbourhood B ′ ⊂ B of B ∞ .By [Fil, Rmk.1.4 (ii)], the linear model for B near [ u ] ∈ B is given by the kernelof the differential D ev ([ u ]) : T [ u ] Z → T p Q , which we need to equip with a sc-structure. This yields two reasons for the restrictions in [Fil]: First, the tangentspace T [ u ] Z of an M-polyfold carries a sc-structure only at [ u ] ∈ Z ∞ . This iswhy [Fil] builds charts centered at smooth points only, and so do we. Second, thedifferential of a general sc-smooth map Z → Q is defined only at points in the1-level Z . In our case, the evaluation map ev is classically smooth on Z , so thereis no need to restrict to [ u ] ∈ Z when using the differential D ev ([ u ]).Moreover, the construction of local charts for B uses a local submersion nor-mal form, which [Fil, Lemma 2.1] guarantees only if the map is C (see [Fil, The choices made in the general construction for Gromov-Witten spaces, of a “gluing profile”and sequence 2 π > . . . δ m +1 > δ m > . . . > The manifold Q is finite-dimensional and so carries the constant scale structure where Q m = Q for all m . POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 19
Rmk.1.4 (iii)]). Since [Fil] considers general sc-smooth maps Z → Q , this re-quires a restriction to the C -map Z → Q . In our case there is no need for thisrestriction, since the evaluation map ev : Z → Q is C without a shift.So the charts for B constructed by [Fil] are homeomorphisms between openneighbourhoods of smooth points [ u ] ∈ B ∞ and open subsets of the kernel of thedifferential D ev([ u ]). (No retractions appear here due to the absence of nodes in B .) Since these kernels are sc-Hilbert spaces, this induces a sc-Hilbert manifoldstructure on the subset B ′ ⊂ B that is covered by the charts. (cid:3) Another way to construct the sc-Hilbert manifold structure on B is to directlyincorporate the condition u ( z ) = p in the construction of charts from [HWZ17].Going through this proof should also serve to illuminate the general approach of[HWZ17] in this simplified setting. Proof of Theorem 3.5 by construction of charts.
To begin, one needs to check that B is a metrizable space. In general, this is proven in [HWZ17, Thm.3.27] usingthe Urysohn criteria (Hausdorff, second countable, and completely regular) thatimply metrizability. These criteria are easily checked in our setting: First notethat the topology on W , ( S , Q ) can be obtained by viewing it as a subset ofthe Hilbert space W , ( S , R N ) via some choice of embedding M ⊂ R N . Now B = b B / Aut( S , i, z ) is the quotient of a subset b B ⊂ W , ( S , Q ) (given by speci-fying the homology class and value u ( z ) = p ), modulo reparameterization by thebiholomorphisms in Aut( S , i, z ) that fix z ∈ S . The relative topology on thesubset b B is automatically metric (thus Hausdorff) and second countable, howevernot all these properties are inherited by the quotient. The metric induces a pseu-dometric on the quotient, which implies that the quotient topology is completelyregular. To show that the quotient is Hausdorff and second countable, it suffices toprove that the quotient map π : b B → B is open. To check this we consider any opensubset b U ⊂ b B and show that π ( b U ) is open by checking that any u ∈ π − ( π ( b U ))has an open neighbourhood contained in π − ( π ( b U )). Note that u ◦ ψ = b u ∈ b U forsome ψ ∈ Aut( S , i, z ) and { w ∈ b B | k w − b u k W , < ε } ⊂ b U for some ε > b U is open. Now let C > ψ up to thirdorder, then we claim that the εC ball around u is contained in π − ( π ( b U )). Indeed, k v − b u k W , < εC implies k v ◦ ψ − b u k W , < ε , thus v ◦ ψ ∈ U and v ∈ π − ( π ( b U )). Thisproves that the quotient map is open and thus finishes the proof of metrizability.Next, the main technical work is the construction of a chart for a neighbourhoodof a given point α ∈ B . For that purpose we pick a representative u : S → Q of α = [ u ]. Since B is a quotient by the reparameterization action, the chartsare constructed as local slices to this action, which involves choices of additionalmarked points and transverse constraints. More precisely, we need to choose gooddata centered at u in the sense of [HWZ17, Def. 3.6]. Such good data exists by[HWZ17, Prop. 3.7]. Recall here that the isotropy group G = G u = { id } is trivialin our case. Then good data in our setting consists of the following objects withthe following properties We essentially use the numbering of [HWZ17, Def. 3.6], but merged (5) into 8.), merged (7)into 2.) and 4.), merged (8) into 7.), and left out (9), (10) which are trivially satisfied in our case.
These exist by [HWZ17, Lemma3.2]. In our case a stabilization of ( S , i, { z } , ∅ ) for the map u consists of twopoints Σ = { z , z } ⊂ S that satisfy the following conditions: • z , z , z ∈ S are pairwise different. • Denote p := u ( z ) , p = u ( z ). Then p , p , p are pairwise different. • For i = 1 ,
2, the map d u ( z i ) is injective, the bilinear form u ∗ ω ( z i ) is non-degenerate, and it determines the correct orientation on T z i S .Now the Riemann surface with the additional marked points ( S , i, { z , z , z } ) isstable; in fact its isotropy group is trivial in our case. [HWZ17] also requires a choiceof good uniformizing family parameterizing variations of the surface and markedpoints, but in our case, since the Deligne-Mumford space of three marked pointson a sphere is trivial, this family is constant. It remains to choose small diskstructures , that is holomorphic embeddings of the closed disk D ≃ D z i ⊂ S with center 0 ≃ z i that are disjoint for i = 1 , Open neighbourhoods U ( p i ) ⊂ Q of p i for i = 1 , ψ i : ( U ( p i ) , p i ) → ( R n ,
0) are chosen so that • U ( p ) and U ( p ) are disjoint; • u | D zi is an embedding for i = 1 ,
2, with image contained in U ( p i ).Here and in the following we denote by U ρ ( p i ) := ψ − i ( { x ∈ R n | | x | < ρ } ) thepreimages of balls of any radius ρ > on Q is chosen such that it agrees on the open sets U ( p i ) with the pullback of the standard metric on R n by ψ i .Moreover, we choose an open neighbourhood of the zero-section e O ⊂ T Q ,which is fiberwise convex and such that for every q ∈ Q , the exponential mapinduced by the chosen metric, exp : e O q := e O ∩ T q Q → Q is an embedding. We choose submanifolds M p i ⊂ U ( p i ) ⊂ Q ofcodimension 2 for i = 1 , p i ∈ M p i and • ψ i ( M p i ) ⊂ R n is a linear subspace; • T p i Q = im d u ( z i ) ⊕ H p i for H p i := T p i M p i ; • { z i } = D z i ∩ u − ( M p i ) is the only point in D z i that u maps to M p i .This in particular implies that u | D zi is transverse to M p i for i = 1 , We choose SD z i ⊂ D z i for i = 1 , D ≃ D z i . U ⊂ W , ( S , u ∗ T Q ) of 0 is chosen such that • every section η ∈ U takes values in u ∗ e O ; • for every η ∈ U and i = 1 ,
2, the map u ′ := exp u ( η ) : S → Q satisfies u ′ ( D z i ) ⊂ U ( p i ), and u ′ | D zi is an embedding transverse to M p i that intersects M p i at a single point p ′ i = u ′ ( z ′ i ), the preimage of some z ′ i ∈ SD z i .From now on we will assume that we chose good data for a smooth map u ∈ C ∞ ,since it suffices to construct charts centered at smooth points α ∈ B ∞ . Now theonly place where our constructions for B = { [ u ′ ] | u ′ ( z ) = p } ⊂ Z differ from the [HWZ17, Thms.3.8, 3.10] imply that for every [ u ′ ] ∈ Z there is a smooth u : S → Q andgood data centered at u such that u ′ = exp u ( η ) for some η ∈ U . This means that the chartscoming from good data centered at the smooth points of Z will cover all of Z . This can beextended to include the conditions u ′ ( z ) = p and η ( z ) = 0, and thus prove the same for B . POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 21 constructions in [HWZ17] for Z is the definition of a linear subspace of the sc-space (cid:0) W k, ( S , u ∗ T Q ) (cid:1) k ∈ N . Our choice E u := { η ∈ W , ( S , u ∗ T Q ) | η ( z ) = 0 , η ( z i ) ∈ H p i for i = 1 , } (10)adds the condition η ( z ) = 0 that linearizes the condition u ′ ( z ) = p = u ( z ). Wethen consider its open subset O := E u ∩ U and claim that the map O → B , η [exp u η ](11)is a homeomorphism onto a neighbourhood of [ u ]. While this is not explicitly statedin [HWZ17, § Continuity: (11) is the composition of two continuous maps: the pointwise ex-ponential map and a quotient projection. The latter is continuous by definition ofthe quotient topology on B . The first map is the composition η E ◦ η with thesmooth exponential map E : u ∗ T Q → Q . Checking that this is continuous between W , -topologies requires local estimates, which hold since 3 · > dim S (see e.g.[Weh04, Lemma B.8]). Injectivity:
Consider η, e η ∈ O with [exp u η ] = [exp u e η ] ∈ B . This means that thereis a biholomorphism ψ : S → S with ψ ( z ) = z and exp u η = exp u e η ◦ ψ . Byproperty 4.), the maps exp u η and exp u e η intersect the submanifolds M p i , i = 1 , p ′ i , e p i ∈ M p i with unique preimages z ′ i , e z i ∈ SD z i . It follows that ψ ( z ′ i ) = e z i for i = 1 ,
2. On the other hand, η ( z i ) , e η ( z i ) ∈ H p i = T p i M i and thefact that M i ⊂ U ( p i ) is totally geodesic imply exp u η ∈ M i and exp u e η ∈ M i , sothe uniqueness of intersection points in property 7.) implies z ′ i = z i and e z i = z i for i = 1 ,
2. Thus we have ψ ( z i ) = z i for i = 0 , ,
2, which implies ψ = id S . From thiswe deduce exp u η = exp u e η and thus η = e η . Continuity of the inverse:
To show that the map in (11) is a homeomorphism,it remains to check that it maps open subsets of E u to open subsets of B . So wefix some η ∈ O and need to show that any α ∈ B sufficiently close to [exp u η ]can be written as α = [exp u η ′ ] for some η ′ ∈ O . First note that v := exp u η satisfies v ( z ) = p along with the slicing conditions v ( z i ) ∈ M p i for i = 1 ,
2. Next, α being close to [ v ] in the quotient topology of B means that α = [ v ′ ] for somerepresentative v ′ that is W , -close to v . By construction of B , this map satisfies v ′ ( z ) = p . Moreover, since v is locally transverse to the slicing conditions (asspecified in 4.), we will have v ′ ( z ′ i ) ∈ M p i for some z ′ i ≈ z i . Now we can compose v ′ with a small M¨obius transformation that fixes z and maps z i to z ′ i to obtaina new representative α = [ w ] that satisfies w ( z ) = p and w ( z i ) ∈ M p i . Thisadjustment in slicing conditions guarantees that η ′ ( z ) := exp − u ( z ) ( w ( z )) defines asection η ′ ∈ E u . Moreover, the construction is done such that w is still W , -closeto v , which guarantees η ′ ≈ η and thus η ′ ∈ O for α sufficiently close to [ v ]. Thisproves openness and thus continuity of the inverse.Thus we have constructed C -charts for B centered at any point [ u ]. It remains toequip the local models with sc-structures and show that the transition maps betweenthese charts are sc-smooth. This is where we (just as [HWZ17, § u ] ∈ B ∞ represented by smooth maps u : S → Q .This regularity is required to give the pullback bundle u ∗ T Q a classically smoothstructure, so that we can define the Sobolev spaces W k, ( S , u ∗ T Q ) of sectionsby closure of the smooth sections. Then we obtain a sc-structure on the Hilbertspace E u in (11) by the subspaces E u ∩ W k, ( S , u ∗ T Q ) for k ∈ N . Finally, compatibility of the charts follows directly from [HWZ17], since our tran-sition maps are the same maps as theirs, just restricted to subsets of their domains.More precisely, transition maps between different charts for Z can be obtained fromits ep-groupoid description by composing local inverses of the source map with thetarget map. Both of these structure maps are local sc-diffeomorphisms by the´etale property in [HWZ, Def.7.3] (for the Gromov-Witten case this is established in[HWZ17, Prop.3.19]). So sc-smoothness of transition maps follows since they arecompositions of local sc-diffeomorphisms. (cid:3) Remark 3.7 (sc-smoothness) . The key point why sc-smoothness appears here isthe following: The choice of stabilization points z , z above depended on u . Fordifferent u ′ we might have other z ′ , z ′ , and so the transition map between thecharts centered at u and u ′ needs to reparameterize all the vector fields η . Butthe reparameterization biholomorphism is not fixed for one transition map, butdepends also on the vector field. So we get a map of the form O ∋ η η ◦ d ψ η ∈ O ′ , which is not classically differentiable w.r.t. any of the usual Sobolev or H¨oldernorms. It is however scale-smooth; see [FFGW16, § Remark 3.8 (Good data centered at u ) . For the holomorphic map u ( z ) = ( z, m )from § u is the identity, and inthe second factor it is the constant map to m ∈ T . For this map, any choice of twodifferent points z , z ∈ S \ { z } yields a stabilization as required. Then M p i := { z i } × T are transverse hypersurfaces since their tangent spaces H p i = { } × T m T satisfy im d u ( z i ) ⊕ H p i = T z i CP ⊕ T m T = T u ( z i ) Q. This choice of charts will be used in § E u := { η ∈ W , ( S , u ∗ T Q ) | η ( z ) = 0 , η ( z i ) ∈ { } × T m T for i = 1 , } . (12) Corollary 3.9.
The product [0 , × B is a sc-Hilbert manifold with boundary ∂ ([0 , × B ) = { , } × B .Proof. As a finite dimensional manifold, the interval [0 ,
1] is trivially a sc-Hilbertmanifold and the notion of boundary is the same as the classical notion (see alsoRemark 1.6). This means that the product [0 , × B also is a sc-Hilbert manifold.In fact, for every pair ( t , [ u ]) ∈ [0 , × B we can choose an open interval I t ⊆ [0 , t and a chart O → B centered at [ u ] as in (11), then a sc-Hilbertmanifold chart for a neighbourhood of ( t , [ u ]) is given by I t × O −→ [0 , × B , ( t, η ) ( t, [exp u η ]) . Sc-smoothness of transition maps between these charts follows directly from sc-smooth compatibility of the charts for B . Finally, O has no boundary, and boundaryof I t arises only from ∂ [0 ,
1] = { , } . Since boundary and corner stratificationsare determined in local charts, this proves the claim. (cid:3) The bundle.
The purpose of this section is to give the projection
E → [0 , ×B defined in (6) the structure of a tame strong polyfold bundle. To achieve this,we first describe the bundle W → Z HWZ from [HWZ17] restricted to Z ⊂ Z HWZ .The fibers of W are defined in [HWZ17, § J on the target manifold. Using the previous simplifications for POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 23 Z ⊂ Z HWZ (i.e. genus 0, S = S , homology class [ CP × { pt } ]) we see that the fiber W α over α ∈ Z is the quotient space W α = ( ( u, η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ u ] = αη ∈ Λ , J (cid:0) S , u ∗ T( CP × T ) (cid:1) of class W , ) , ∼ of complex antilinear 1-forms of class W , with values in the pullback bundle alonga representative u . Here complex antilinearity of η is required with respect to afixed almost complex structure J on CP × T and the standard complex structureon S . The equivalence relation is given by( u, η ) ∼ ( v, µ ) : ⇐⇒ ∃ ψ : S → S holomorphicwith u = v ◦ ψ and η = µ ◦ d ψ. For α ∈ B ⊂ Z we can choose a representative [ u ] = α with u ( z ) = p , and restrict-ing to such representatives reduces the equivalence relation to biholomorphisms ψ with ψ ( z ) = z . So if we choose J = J t for some t ∈ [0 , E ( t,α ) = W α with the fibers of a tame strong polyfold bundle constructed in[HWZ17]. However, we wish to simultaneously extend and restrict the base space:We extend by allowing the almost complex structure to vary, and we restrict tocurves through the fixed point p ∈ Q . Theorem 3.10.
Let B ′ ⊂ B be as in Theorem 3.5 , then E| [0 , ×B ′ → [0 , × B ′ is a tame strong M-polyfold bundle. There are again several ways to prove this. Filippenko [Fil] has a result aboutthe restriction of bundles to sub-polyfolds which we explain in Remark 3.13. Un-fortunately, this would require an existing polyfold description for Gromov-Wittenmoduli spaces with varying J , which we discuss in Remark 3.11. In any case, theproof of transversality of the section σ at t = 0 requires a fairly explicit bundlechart, so our actual proof is an adaptation of [HWZ17] – to our simplified setting,but extending the constructions to varying J . Remark 3.11.
The analysis in [HWZ17] is formulated for a fixed almost complexstructure J on the target manifold Q . This might be deemed as sufficient due tothe following graph trick:Given a smooth family ( J t ) t ∈ R k of almost complex structures, parameterized bya finite dimensional space R k , we can identify the moduli space of J t -holomorphiccurves in Q for some t ∈ R k with a moduli space of certain pseudoholomorphiccurves in the product manifold e Q := C k × Q as follows. We define an almost complexstructure e J on e Q by e J ( t + is, q ) := J C k × J t at the point ( t + is, q ) ∈ e Q = C k × Q .Then e J -holomorphic maps e u : S → e Q = C k × Q in a class e A := [ { pt } ] × A , A ∈ H ( Q ) are constant in C k , so that a constraint e u ( z ) ∈ R k × Q at a markedpoint z ∈ S picks out the maps e u that are of the form z ( t, u ( z )) for some t ∈ R k and a J t -holomorphic map u : S → Q . Thus pairs ( t, u ) of J t -holomorphic mapsin class A for some t ∈ R k can be identified with e J -holomorphic maps in class e A satisfying the point constraint.Our variation of almost complex structures ( J t ) t ∈ [0 , can be formulated in thisway by choosing J t to be constant near t = 0 and t = 1, so that its constantextension to t ∈ R is smooth. However, the polyfold bundle f W → e Z that [HWZ17] Recall we expect B ′ = B by Remark 3.6. constructs for the almost complex manifold ( e Q, e J ) contains already in its base manymore (reparameterization classes of) maps e u : S → C × Q than those of the form e u ( z ) = ( t, u ( z )). The point constraint e u ( z ) ∈ R × Q does not force the C -componentto be constant. Similarly, the fibers of f W , consisting of anti-linear 1-forms withvalues in C × Q , contain an extra factor Λ , ( S, C ) compared with the fibers W α above that are used in the polyfold description of a moduli space for fixed J . Theseinfinite dimensional extra factors would have to be split off near maps of the form(0 , u ( z )) or (1 , u ( z )) in order to relate the moduli spaces for J and J with partsof the moduli space for e J in this setup.Alternatively, extending such a splitting along ( t, u ( z )) for all t ∈ R would yielda smaller polyfold description in which the base space consists only of maps ofthe form ( t, u ( z )). Rather than attempting such a splitting construction, we willdirectly construct the resulting polyfold bundle. Proof of Theorem 3.10.
We construct bundle charts as in [HWZ17, § . Fix apair ( t , [ u ]) ∈ [0 , × B ∞ , a representative u of [ u ], an open interval I t ⊂ [0 , t (whose ‘sufficiently small’ choice will be specified below), and good datacentered at u . These choices determine a chart for the base space as in § I t × O → [0 , × B , ( t, η ) ( t, [exp u ( η )]) . (See § O = E u ∩ U ⊂ W , ( S , u ∗ T Q ).) To build chartsfor the bundle, recall that we abbreviate Q := CP × T . Then, using the chosenrepresentative u , the bundle fiber E t , [ u ] can be identified with the Hilbert space F := Λ , J t ( S , u ∗ T Q ) ∩ W , ( S ) . (13)Here Λ , J ( . . . ) denotes continuous complex-antilinear 1-forms, and to pick out thoseof class W , we view them as functions on S with values in a bundle whose fiberover z ∈ S are the linear maps T z S → T u ( z ) Q . This Hilbert space can beequipped with a scale structure whose k -th level consists of forms of regularity W k, . So we can define a trivial strong bundle I t × O ⊳ F → I t × O (14)which we will use as local model for E → [0 , × B near the pair ( t , [ u ]).To trivialize the bundle, we need to map J t -antilinear one-forms with values in u ∗ T Q to J t -antilinear one-forms with values in v ∗ T Q for pairs ( t, v ) near ( t , u ).We will do this in two steps, first changing the almost complex structure and thenthe pullback bundle. For every t ∈ I t we can define a linear map from the space ofantilinear 1-forms with respect to J to the space of antilinear 1-forms with respectto J t (both with values in u ∗ T Q ), K t : F = Λ , J t (cid:0) S , u ∗ T Q (cid:1) −→ Λ , J t (cid:0) S , u ∗ T Q (cid:1) (15) ξ ( ξ + J t ( u ) ◦ ξ ◦ i ) . Notation here is the same as in [HWZ17], with lots of simplifications because we work witha constant Riemann surface and trivial isotropy. On the other hand, we allow the almost complexstructure to vary in a 1-parameter family, whereas [HWZ17] fixes it. As a set, O ⊳ F is the Cartesian product O × F . The symbol ⊳ means that we considerit as a strong bundle, which is a condition on the scale structure: For η ∈ O on level m , i.e. ofSobolev class W m, , it makes sense to talk about sections along u ′ := exp u η of Sobolev classup to W m, = W k, , i.e. up to level k = m + 1. POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 25
Note that K t is the identity map. Moreover, the explicit form of K t allows us tocheck that this family is smooth with respect to the W k, -topology on Λ , ( . . . )for any k ∈ N . So by choosing I t as sufficiently small neighbourhood of t wecan guarantee that K t is a linear sc-isomorphism for all t ∈ I t .The rest of the bundle chart construction – i.e. the step from the pullback bundle u ∗ T Q to the pullback bundle v ∗ T Q – proceeds as in [HWZ17]; we just need to ensuresc-smooth dependence on the extra parameter. We construct a family of connections e ∇ t on T Q for t ∈ [0 ,
1] as follows: Let ∇ t denote the Levi–Civita connection ofthe metric g t := ω ( · , J t · ) on Q and define a new connection by e ∇ tX Y := ( ∇ tX Y − J t ∇ tX ( J t Y )). Then the family is smooth in t , and e ∇ t is a complex connection on thealmost complex vector bundle (T Q, J t ), that is, it satisfies e ∇ tX ( J t Y ) = J t ( e ∇ tX Y ).Moreover, recall that the good data used to construct the above chart for B includedthe choice of an open neighbourhood e O of the zero section of T Q , such that e O isfiberwise convex and the exponential map (for a fixed metric on Q that does notvary with t ) restricts to an embedding on each fiber e O q , q ∈ Q . Then for a tangentvector η q ∈ e O q , consider the geodesic path [0 , → Q, s exp q ( sη q ) from q to p := exp q ( η q ). Parallel transport with respect to e ∇ t along this path defines a J t -complex linear map Γ t ( η q ) : (T q Q, J t ( q )) −→ (T p Q, J t ( p )) . (16)This in fact is an isomorphism for each t ∈ [0 ,
1] and η q ∈ T q Q , and these isomor-phisms vary smoothly with t ∈ [0 , q ∈ Q and η ∈ T Q .The resulting bundle chart covering the chart (11) of the base space given by I t × O → [0 , × B , ( t, η ) ( t, [exp u η ]) is I t × O ⊳ F −→ E ( t, η, ξ ) (cid:16) t, h(cid:0) exp u η , Ξ( η, ξ ) := Γ t ( η ) ◦ K t ( ξ ) (cid:1)i(cid:17) . (17)Here the complex antilinear 1-form Ξ( η, ξ ) ∈ Λ , J t ( S , v ∗ T Q ) with values in thepullback bundle by v := exp u η is given at each z ∈ S by the complex antilinearmapΓ t ( η ( z )) ◦ K t ( ξ )( z ) : (cid:0) T z S , i (cid:1) → (cid:0) T u ( z ) Q, J t ( u ( z )) (cid:1) → (cid:0) T v ( z ) Q, J t ( v ( z )) (cid:1) . It coincides with the construction in [HWZ17, (3.9)] in case K t = id due to J beingfixed. We can now follow the arguments of [HWZ17, § E as tamestrong M-polyfold bundle. For that purpose first note that each map (17) covers anM-polyfold chart for [0 , × B ′ ⊂ [0 , × B , so the images of these maps cover only E| [0 , ×B ′ . Second, these are bundle charts in the sense that they are linear bijectionson each fiber. They are local homeomorphisms because the topology on the totalspace E| [0 , ×B ′ is defined in this manner as in [HWZ17, Thm.1.9]. Strong sc-smoothcompatibility of these bundle charts is proven for fixed J in [HWZ17, Prop.3.39], For fixed k , the continuity of K t requires u ∈ C k , which is why we center these charts atsmooth points [ u ] ∈ B ∞ . Any smooth family of complex connections suffices for the present construction. In § e ∇ to split along the factors of Q = CP × T , see Remark 3.12. Note here that parallel transport is defined along any path, so there is no issue with the factthat the path is induced by an exponential map for a different metric than the family of metricsused in the construction of the connection. using the language of [HWZ, Prop.3.39]. The strong bundle isomorphism µ thatis considered here, and proven to be a local sc-diffeomorphism, in fact encodes alltransition maps between different bundle charts. The proof of [HWZ17, Prop.3.39]directly extends to the case of varying J thanks to its explicit nature (15) of K t as family of linear 0-th order operators. Here we have to again require u ∈ C ∞ toensure that K t is a bounded operator on each of the W k, -scales. In fact, theseoperators vary smoothly with t ∈ [0 ,
1] on each scale. Thus the charts (17) equip E| [0 , ×B ′ with the structure of a strong M-polyfold bundle. Finally, tameness is acondition on the underlying M-polyfold that is automatically satisfied in our casesince all retractions are trivial; see [HWZ, Def.2.17]. (cid:3) Remark 3.12.
For t = 0, remember that J = i ⊕ J T splits along the factors of Q = CP × T . This means that also the metric splits and so does its Levi-Civitaconnection ∇ . Then also the connection e ∇ splits, and thus the parallel transportmap Γ in (16) preserves the factors of T ( z,p ) Q = T z CP × T p T . Remark 3.13.
An alternative proof of Theorem 3.10 is to construct the bundle
E → [0 , × B in (6) from an implicit function theorem in [Fil].This proof requires as starting point a polyfold description of the Gromov-Wittenmoduli space M for a family ( J t ) t ∈ [0 , of almost complex structures. Such a de-scription is obtained by performing the constructions of Theorem 3.10 over [0 , × Z to obtain a tame strong M-polyfold bundle e p : f W → [0 , × Z which restricts onevery slice { t } × Z to the bundle W → Z from [HWZ17] for the almost complexstructure J t .Now the projection E → [0 , × B in (6) is obtained from f W → [0 , × Z byrestriction to B = ev − ( { p } ) ⊂ Z as in the first proof of Theorem 3.5. Here themap [0 , × Z → Q, ( t, [ u ]) ev[ u ] = u ( z ) is transverse to { p } ∈ Q since theevaluation map – without the [0 , § e B ⊂ [0 , × ( B ∩ Z ) of[0 , × B ∞ such that e p − ( e B ) = E| e B → e B inherits the structure of a tame strongM-polyfold bundle. In our special case, the shift to Z -topology is actually notneeded for the reasons already stated in Theorem 3.5, and since the retracts are alltrivial. Furthermore, e B can be replaced by [0 , × B ′′ for an open neighbourhood B ′′ ⊂ B of B ∞ that may just be somewhat smaller than B ′ from Theorem 3.10.Indeed, e B is given by a union of charts for [0 , × B centered at smooth points,which lift to strong bundle charts. Each of these charts is constructed in productform, and since [0 ,
1] is compact we can find for every b ∈ B ∞ finitely many suchproduct charts that cover [0 , × { b } . This implies [0 , × U b ⊂ e B for some openneighbourhood U b ⊂ B of b . This proves the claim with B ′ = S b ∈B ∞ U b .3.5. The section.
This section finalizes the polyfold description of the modulispace M = σ − (0) by establishing the relevant properties of the section σ : [0 , ×B → E introduced in (7). Up to quotienting by reparameterization in the base, itsprincipal part (given by its values in the fibers) is ( t, u ) ∂ J t u . Theorem 3.14.
Let B ′ ⊂ B be as in Theorem 3.5 , then σ : [0 , × B ′ → E is asc-Fredholm section of Fredholm index . Recall we expect B ′ = B by Remark 3.6. POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 27
As for the bundle structure in Remark 3.13, the Fredholm property of the sectioncould be proven by the restriction results of Filippenko [Fil] – if the Fredholmproperty of the Cauchy-Riemann operator with varying J was firmly established.We explain this approach in Remark 3.15, after giving a direct proof of the Fredholmproperty based on the explicit bundle charts in Theorem 3.10. Proof of Theorem 3.14.
We work in local coordinates centered at a pair ( t , [ u ]) ∈ [0 , × B ∞ that were defined in (17). The principal part of the section σ is givenin these coordinates by f : I t × O −→ F, ( t, η ) K − t (cid:0) Γ t ( η ) − ( ∂ J t exp u η ) (cid:1) , with the family of sc-isomorphisms K t : F → Λ , J t (cid:0) S , u ∗ T Q (cid:1) given in (15), andparallel transport Γ t ( η ) as in (16). Both of these are linear and explicitly given interms of point-wise operations. Thus routine computations show that f is not justsc-smooth, but for any k ∈ N restricts to a classically smooth map with respectto the W k +3 , -norm on O and the W k, -norm on F . (For the present proof, itsuffices to check continuous differentiability.) We can moreover see that the section σ has classical Fredholm linearizations at any zero ( t , [ u ]) ∈ σ − (0). Indeed, in thecoordinates centered at a point with ∂ J t u = 0 we have d f ( t , T, ζ ) : R × E u → F given by(18) d f ( t , T, ζ ) = − T · J t ( ∂ t J t ) | t = t ∂ J t u + (D u ∂ J t ) ζ, with E u given in (10). The first part, T
7→ − T · J t ( ∂ t J t ) | t = t ∂ J t u , is a boundedlinear operator R → F with respect to any W ℓ, -norm on F , hence it is compactwith respect to the W , -norm on F . The second part is a restriction of the classicalCauchy-Riemann operator D u ∂ J t : X := W , ( S , u ∗ T Q ) → F . This classicaloperator is known to be Fredholm, see e.g. [MS04, Thm.C.1.10], and restriction tothe finite codimension subspace E u ⊂ X preserves the Fredholm property. Thisshows that d f ( t ,
0) is classically Fredholm with index given by the index of therestriction D u ∂ J t | E u plus the dimension of the domain R of the compact factor.The index of D u ∂ J t is 2 n + 2 c ([ CP × { pt } ]) by the Riemann-Roch Theorem. TheChern number can be computed with respect to any compatible almost complexstructure on Q , and for J = J CP ⊕ J T we have c ([ CP × { pt } ]) = Z CP ×{ pt } c (T Q, J )= Z CP c (T CP , J CP ) + Z { pt } c (T T, J T ) = 2 + 0 . Finally, restricting an operator to a subspace reduces the Fredholm index (via a mixof its effects on kernel and image) by the codimension of the subspace. In this case, E u ⊂ W , ( S , u ∗ T Q ) has codimension 2 n + 4 since it is given by the codimension2 n condition η ( z ) = 0 and the two codimension 2 conditions η ( z i ) ∈ H p i . Thuswe obtain the claimed Fredholm indexindex(d f ( t , n + 2 c ([ CP × { pt } ]) − n − . This also establishes the classical Fredholm property of the section σ in local coor-dinates. The nonlinear sc-Fredholm property of polyfold theory, however, demands more than just the linearizations being sc-Fredholm. The sc-Fredholm propertyof a section as defined in [HWZ, Def. 3.8] requires three conditions. The first,sc-smoothness, follows from the classical smoothness in local coordinates. The sec-ond condition, σ being regularizing, means that if ( t, v ) ∈ [0 , × W m, with ∂ J t v ∈ W m +1 , , then ( t, v ) ∈ [0 , × W m +1 , . This follows from the corre-sponding property of ∂ J t for fixed t – which is known from elliptic regularity. Thethird condition seems less transparent but is very important for the implicit func-tion theorem in scale calculus: At every smooth point ( t , [ u ]) ∈ [0 , × B ∞ , thesection needs to have the sc-Fredholm germ property [HWZ, Def. 3.7], that is, aftersubtraction of a local sc + -section, (a filled version of) the germ of σ needs to beconjugate to a basic germ as defined in [HWZ, Def. 3.6]. The latter means thatafter splitting off a finite dimensional factor from the domain and projecting to thecomplement of a finite dimensional factor in the image, the germ is the identity plusa contraction mapping. It is this third property that implies that linearizations of asc-Fredholm section are sc-Fredholm operators, see [HWZ, Prop. 3.10]. The indexis then defined as the Fredholm index of the linearization at the lowest level of thescale structure, which we computed above to be 1.To establish the equivalence to a contraction germ normal form, we proceedsimilar to [HWZ17, Prop. 4.26], using the fact that the section is classically differ-entiable in all but finitely many directions. In fact, the local representative f aboveis continuously differentiable in all directions, and thus satisfies the conditions ofbeing sc-Fredholm with respect to the trivial splitting E u ∼ = { } × E u , as defined in[Weh, Def. 4.1]. Indeed, we already established the regularizing property (i). Thedifferentiablity conditions (ii) in the trivial splitting follow from classical continuousdifferentiability. Besides, in this setting the linearized sc-Fredholm property (iii)is only required of D f – though for any (not necessarily holomorphic) base point( t , [ u ]). We can make up for the latter complication by subtracting from f thesc + -section f : I t × O −→ F, ( t, η ) K − t (cid:0) ∂ J t u (cid:1) , which takes the same value at ( t ,
0) as f . Thus the linearization of f − f at ( t , f − f )( t , T, ζ ) = T · ∂ t (Γ t ( η ) − )( ∂ J t u ) + D Γ − t ( ζ )( ∂ J t u ) + D u ∂ J t ζ. To check that this is a linear sc-Fredholm operator we use the conditions of [Weh,Def. 3.1]. (i) It is bounded on each level of the scale structure. (ii) It is regularizingby elliptic regularity for the linearized Cauchy-Riemann operator. Lastly, we haveto check in (iii) the classical Fredholm property on the lowest level of the scale struc-ture. Note that the first two summands actually are bounded with respect to the W , -norm on F , and thus induce compact operators to F with the W , -topology.Thus the Fredholm property again follows from the corresponding property of thelinearized Cauchy-Riemann operator. All in all, we have shown that the section The linear sc-Fredholm property [HWZ, Def. 1.8] is a direct analogue of the classical linearFredholm property – kernel and image need to have complements that respect the sc-structure,these complements need to be sc-isomorphic, and kernel and cokernel should be finite dimensional. In our setting, there is no need for a filling since all retractions are trivial.
POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 29 f − f in local coordinates satisfies all conditions of [Weh, Thm. 4.5], which impliesits contraction germ normal form. This finishes the proof. (cid:3)
Remark 3.15.
An alternative proof of Theorem 3.14 is to combine an implicit func-tion theorem from [Fil] with a polyfold description of the Gromov-Witten modulispace M for a family ( J t ) t ∈ [0 , of almost complex structures. Such a descriptionis obtained as follows. First, one follows Remark 3.13 to construct an M-polyfoldbundle e p : f W → [0 , × Z which restricts on every slice { t } × Z to the bundle W → Z from [HWZ17] for the almost complex structure J t . Then, the section ∂ : [0 , × Z → f W , ( t, [ u ]) (cid:0) t, (cid:2) ( u, ∂ J t u ) (cid:3)(cid:1) is shown to be sc-Fredholm by fol-lowing the arguments of [HWZ17, Thm.4.6] or our proof of Theorem 3.14. Givensuch a description, we claim that the sc-Fredholm property is preserved when werestrict from [0 , × Z to the preimage [0 , × B = e ev − ( { p } ) of the submanifold { p } ⊂ Q under the evaluation map e ev : [0 , × Z → Q, ( t, [ u ]) ev([ u ]). Forthat purpose we can again quote the results by Filippenko: [Fil, § { p } ⊂ Q satisfy all compatibility conditions of [Fil, Theorem 5.10 (III)]. A direct applicationof that result asserts that σ | e B : e B → W | e B is sc-Fredholm for some open subset e B ⊂ [0 , × B containing [0 , × B ∞ . However, the shift in topology is again notneeded since the evaluation map is classically smooth on all levels. Furthermore,as in Remark 3.13, the open subset e B ⊂ [0 , × B can be replaced by [0 , × B ′′′ for an open neighbourhood B ′′′ ⊂ B of B ∞ that may just be smaller than B ′ fromTheorem 3.10.3.6. Linearization.
The goal of § σ at t = 0. For that purpose we will need to consider its linearizationat the unique solution [ u ] ∈ B for t = 0 (see § σ := σ (0 , · ) : B → E| { }×B at [ u ] ∈ B , which is whatwe will compute now. We do the computation in a local chart for the restrictedbundle E| { }×B centered at [ u ], given by O × F → E| { }×B , ( η, ξ ) (cid:0) , [exp u η, Γ t ( η ) ◦ ξ ] (cid:1) . This chart is obtained from the chart (17) for the bundle
E → [0 , × B centered at(0 , [ u ]) as in § O × F ∼ = { } × O ⊳ F ⊂ [0 , × O ⊳ F . Here O ⊂ E u is an open subset of the vector space E u given in (12), F is defined in(13), the exponential map is induced by a fixed metric chosen as part of the gooddata in the second proof of Theorem 3.5, and Γ t is defined in (16). The map K t from (15) does not show up here because we have t = t = 0 and so K t is theidentity map. In this chart, the restricted section σ from (7) is given by σ : O → O ×
F, η (cid:0) η , Γ ( η ) − ◦ ∂ J (cid:0) exp u η (cid:1) (cid:1) . Since u is the center of the chart, it corresponds to η = 0. So the linearizedoperator D u σ in the coordinates of this chart is represented by d σ (0) : E u → F , Strictly speaking, the statement of this theorem shifts the scale structure. We can avoid thisby observing that all the differentiability and linearized Fredholm properties of f : R × E u → F persist w.r.t. the W , -topology on E u and the W , -topology on F . To generalize the Fredholm analysis in [HWZ17] to allow for a finite dimensional family ofalmost complex structures, note that this introduces an extra factor in the domain – in our case[0 ,
1] – along which the section is classically smooth. Since it is finite dimensional, it can also besplit off when constructing the contraction germ normal form. which we can compute for ˆ η ∈ E u as follows:d σ (0)(ˆ η ) = ddθ (cid:12)(cid:12)(cid:12) θ =0 ξ ( θ ˆ η ) = ddθ (cid:12)(cid:12)(cid:12) θ =0 Γ (cid:0) θ ˆ η (cid:1) − | {z } =: A θ ◦ ∂ J (cid:0) exp u θ ˆ η (cid:1)| {z } =: µ θ = A |{z} =id ◦ ddθ (cid:12)(cid:12)(cid:12) θ =0 µ θ + ddθ (cid:12)(cid:12)(cid:12) θ =0 A θ ◦ µ |{z} =0 = ( ∇ ˆ η + J ( u ) ◦ ∇ ˆ η ◦ i ) . Here ∇ denotes the Levi-Civita connection on Q corresponding to the metric g that is compatible with J . It is A = Γ t (0) − = id u ∗ T Q , as for every z ∈ S it is (the inverse of) parallel transport from u ( z ) to itself via the constant path,and µ = ∂ J (cid:0) exp u (cid:1) = ∂ J u = 0 since u is J -holomorphic. In the last stepwe used a formula from [MS04, Prop. 3.1.1] and again the fact that u is J -holomorphic. Thus we have shown that the polyfold-theoretic linearization D u σ in an appropriate bundle chart is given by a restriction of the classical Cauchy-Riemann operator of the complex bundle ( u ∗ T Q, J ( u )),d σ (0) : E u → F, ˆ η ( ∇ ˆ η + J ( u ) ◦ ∇ ˆ η ◦ i ) = D J ( u ) ˆ η. Indeed, this operator differs from the classical Cauchy-Riemann operator D J ( u ) : X → F as in [MS04, Rmk.C.1.2] only by the domain E u being a subspace of X := W , ( S , u ∗ T Q ).3.7. Transversality at the boundary.
The last missing ingredient for the proofof Gromov’s nonsqueezing Theorem 1.2 in § J -holomorphic curve is cut out transversely. Remark 3.16.
The following proof is a first instance of the general principle“classical transversality implies polyfold transversality”. The core difference be-tween the two notions is that, classically, one usually proves surjectivity of a lin-earized Cauchy-Riemann operator D : X → F on a tangent space X = T u X to a space X of all maps (in a given homology class, etc.). In our case (ignoringthe homology class and point constraint u ( z ) = p ), this total space would be X = W , ( S , Q )). This space still carries the action of a group of reparameteriza-tions. In our case a group Aut of automorphisms of S acts on X , and preserves theCauchy-Riemann operator, so that the tangent space of its orbit lies in the kernel,T u { u ◦ ϕ | ϕ ∈ Aut } ⊂ ker D.In contrast, the polyfold setup works with the quotient space B = X / Aut whosetangent space T [ u ] B is represented by a subspace E u ⊂ X = W , ( S , u ∗ T Q ).So, the main challenge in deducing surjectivity of the linearized polyfold sectionD [ u ] σ = D | E u from surjectivity of the classical Cauchy-Riemann operator D isin showing that this quotient construction results in a splitting of the total space X = E u + A with a complement A ⊂ ker D that represents the infinitesimal actionof reparameterizations. Theorem 3.17. σ is transverse to the zero section at t = 0 .Proof. Since u is the only solution for t = 0 (see Lemma 2.2), for checkingtransversality of σ at the boundary t = 0 it suffices to consider the lineariza-tion of σ at (0 , u ) and show that it is surjective. For this it is sufficient toshow that the linearization of σ := σ (0 , · ) is surjective. In § u σ in a local chart centered at u ∈ B to be the restric-tion d σ (0) = D J ( u ) | E u : E u → F of the classical Cauchy-Riemann operator POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 31 D J ( u ) : X → F to the subspace E u ⊂ X = W , ( S , u ∗ T Q ) given by (12). Itscodomain F is defined in (13). We will first show that this classical operator issurjective; see also [Wen, Lemma 5.5].Recall from § u : S → Q = CP × T, z ( z, m ) is the productof the identifying map S ∼ = CP and a constant map to the torus. Thus thecomplex structure along u splits J ( u ) = J CP ⊕ J T ( m ) into the standard complexstructure J CP on T CP ∼ = id ∗ T CP and the constant complex structure J T ( m ) = J st on T m T = R n − . Thus we have a natural splitting of complex vector bundles u ∗ (T Q, J ) = (T CP , J CP ) ⊕ E ( n − , where E ( n − is the trivial complex bundle of rank n − S with fibers( R n − , J st ). The domain of the Cauchy-Riemann operator thus splits into X = W , (cid:0) S , u ∗ T (cid:0) CP × T (cid:1)(cid:1) = W , ( S , T CP ) | {z } =: X CP × W , ( S , E ( n − ) | {z } =: X T , and, analogously, the codomain splits F = F CP × F T into the W , -closures ofsmooth complex antilinear 1-forms on S with values in T CP resp. E ( n − . Thisshows that the classical Cauchy-Riemann operator splitsD J ( u ) = D J CP ⊕ D J st : X = X CP × X T → F = F CP × F T into the classical Cauchy-Riemann operator D J CP : X CP → F CP of the complexline bundle (T CP , J CP ) over CP ∼ = S and the Cauchy-Riemann operator D J st : X T → F T of the trivial bundle E ( n − over S . The latter splits further D J st =D i ⊕ . . . ⊕ D i into n − i of the trivialcomplex line bundle E (1)0 = C × S → S . These complex line bundles satisfy c (T CP ) + 2 χ ( S ) = 2 + 2 · > c ( E (1)0 ) + 2 χ ( S ) = 0 + 2 · > J ( u ) of these surjective operators arising from complex line bundles.Towards proving surjectivity of the restriction d σ (0) = D J ( u ) | E u : E u → F ,note the above surjectivity means D J ( u ) ( X ) = F . If we can now show that X = E u + V can be written as the sum of E u and a subspace of the kernel V ⊂ ker D J ( u ) , then it follows that D J ( u ) ( E u ) = D J ( u ) ( E u + V ) = F , so therestriction d σ (0) = D J ( u ) | E u is surjective as well. To find this subspace V notethat our choice of transverse hypersurfaces in Remark 3.8 was made such that thevector space E u defined in (12) splits as E u = E CP × E T , where E CP = { η ∈ X CP | η ( z i ) = 0 for i = 0 , , } , E T = { η ∈ X T | η ( z ) = 0 } . So we can construct V ⊂ X as product V := V CP × V T of subspaces satisfying(a) X CP = E CP + V CP and D J CP ( V CP ) = 0,(b) X T = E T + V T and D J st ( V T ) = 0.To meet these requirements, we choose the subspaces of holomorphic sections V CP := T id Aut( S ) and V T := { η T : S → T m T constant } . With this choice it is easy to verify (b): Every η ∈ X T is the sum of the constantvector field taking the value η ( z ) and the vector field η ( · ) − η ( z ) ∈ E T . Moreover,D J st ( V T ) = 0 holds since all the elements of V T are constant.To verify (a), first recall that Aut( S ) is the group of biholomorphisms S → CP , i.e. solutions ψ : S → CP of the nonlinear Cauchy-Riemann equation ∂ J CP ψ = 0.So the tangent space T id Aut( S ) to this finite dimensional family of holomorphicmaps at the identity map is a subspace of the kernel of the linearized Cauchy-Riemann operator ker D J CP ⊂ X CP . In particular, V CP = T id Aut( S ) is a sub-space of the space of smooth sections C ∞ ( S , T CP ) ⊂ X CP = W , ( S , T CP )since these tangent vectors are obtained as derivatives of paths of maps in Aut( S ),T id Aut( S ) = (cid:8) dd t (cid:12)(cid:12) t =0 γ ( t ) (cid:12)(cid:12) γ : ( − ε, ε ) → Aut( S ) with γ (0) = id (cid:9) . These derivatives take values dd t (cid:12)(cid:12) t =0 γ ( t )( z ) ∈ T γ (0)( z ) CP = T z CP at z ∈ S .So, strictly speaking, they are sections of the pullback bundle id ∗ T CP under theidentification id : S ∼ = → CP . More concretely, the automorphisms of S = C ∪ {∞} are of the form ∞ 6 = z az + bcz + d and ∞ 7→ ac with a, b, c, d ∈ C such that ad − bc = 0.Thus, each γ : ( − ε, ε ) → Aut( S ) is given by differentiable functions a, b, c, d :( − ε, ε ) → C that satisfy a (0) = 1 = d (0), b (0) = 0 = c (0) and a ( t ) d ( t ) − b ( t ) c ( t ) = 0for all t . We can compute their derivative at any z ∈ S \ {∞} ,dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 γ ( t )( z ) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 a ( t ) z + b ( t ) c ( t ) z + d ( t ) = ˙ a (0) z + ˙ b (0)0 z + 1 − (1 z + 0)( ˙ c (0) z + ˙ d (0))(0 z + 1) = z ˙ a (0) + ˙ b (0) − z ˙ c (0) − z ˙ d (0) . Conversely, any choice of numbers
A, B, C, D ∈ C induces a tuple of functions( − ε, ε ) → C as above, given by a ( t ) = 1 + tA , b ( t ) = tB , c ( t ) = tC , d ( t ) = 1 + tD .These define a section η V = dd t (cid:12)(cid:12) t =0 γ ( t ) : S → T S in X CP , which on S \ {∞} ≃ C is of the form (19) η V ( z ) = z ( A − D ) + B − z C ∈ C ∼ = T z CP . Now, to prove (a), given any η ∈ X CP , we need to find η E ∈ E CP and η V ∈ V CP with η = η E + η V . Since η E needs to satisfy η E ( z i ) = 0 for i = 0 , ,
2, we need η V to agree with η on all marked points. Without loss of generality, we can assumethat z = 0, and by Remark 3.8 we are free to choose the points z , z ∈ S as welike. We choose z := 1 and z := −
1. Then, given η ∈ X CP the requirements η V ( z i ) = η ( z i ) translate by (19) into B = η V (0) = η (0) ,A + B − C − D = η V (1) = η (1) , − A + B − C + D = η V ( −
1) = η ( − . This system of 3 equations for 4 variables can be solved by choosing D = 0, A = η (1) − η ( − , B = η (0) , C = η (0) − η (1)+ η ( − , D = 0 . As explained above, this choice defines a vector field η V ∈ T id Aut( S ) = V CP on S , and using (19) we ensured that it has the desired values η V ( z i ) = η ( z i ). Now η E := η − η V satisfies η E ( z i ) = 0 for i = 0 , ,
2, and hence we have η = η E + η V with η E ∈ E CP . This finishes the proof of (a) and thus proves this theorem. (cid:3) Since a ( t ) d ( t ) − b ( t ) c ( t ) = 1 is an open condition and satisfied at t = 0, by continuity it issatisfied for all small enough t . This computation uses the chart CP \ {∞} ∼ = C . To compute the value of η V at ∞ ∈ CP ,we would have to consider a chart near ∞ . This is not needed here if we just avoid putting amarked point at ∞ . POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 33
Remark 3.18. If M = ∅ , then σ is transverse to the zero section at t = 1. Thestatement that the linearization is surjective for all u ∈ M is then vacuously true. Appendix A. The Monotonicity Lemma for pseudoholomorphic maps
The purpose of this appendix is to give a detailed proof of the monotonicitylemma for J st -holomorphic maps to R n that was used in § C , and with ( R n , J st ) generalized to a complex Hilbert space as follows. Definition A.1. A complex Hilbert space ( V, J ) consists of a Hilbert space V with inner product h· , ·i and a compatible complex structure J , i.e. an endomorphism J : V → V with J = − id V that preserves the inner product. The associatedsymplectic structure ω : V × V → R is ω ( v , v ) = h Jv , v i .A pseudoholomorphic map v : ( S, j ) → ( V, J ) consists of a compact Riemannsurface ( S, j ) with (possibly empty) boundary ∂S and a C -map v : S → V satis-fying the Cauchy-Riemann equation d v ◦ j = J ◦ d v . Lemma A.2.
Consider a nonconstant ( j, J ) -holomorphic map v : S → V and anopen ball ˚ B R ( p ) := { q ∈ V | k q − p k < R } centered at a point p ∈ v ( S ) in the image,of radius R > such that k v ( z ) − p k ≥ R for all z ∈ ∂S . Then the symplectic areaof v within the ball is at least the area of the flat disk of radius R , that is Z v − (˚ B R ( p )) v ∗ ω ≥ πR . Proof.
We begin by rewriting the 2-form v ∗ ω on S in local holomorphic coordinates s + it ∈ C for ( S, j ) ≃ ( C , i ) as v ∗ ω = ω ( ∂ s v, ∂ t v ) d s ∧ d t = (cid:0) k ∂ s v k + k ∂ t v k (cid:1) d s ∧ d t (20) = (cid:10)(cid:0) ∂ s v d s + ∂ t v d t (cid:1) ∧ (cid:0) ∂ s v d t − ∂ t v d s (cid:1)(cid:11) = h d v ∧ ∗ d v i . Here the Cauchy-Riemann equation ∂ s v + J∂ t v = 0 in local coordinates togetherwith compatibility of ω and J with the inner product implies ω ( ∂ s v, ∂ t v ) = h J∂ s v, ∂ t v i = h− J ∂ t v, ∂ t v i = k ∂ t v k = k ∂ s v k . The final result h d v ∧ ∗ d v i is a well defined global expression (i.e. independent ofcoordinates) that uses the Hodge operator ∗ : T S → T S induced by the complexstructure j . In local holomorphic coordinates the Hodge operator is given by ∗ d s =d t and ∗ d t = − d s . Here and below the notation h α ∧ β i for differential forms withvalues in V denotes the wedge product ∧ on the level of differential forms, with twovalues in V being multiplied via the inner product h· , ·i .The second expression for v ∗ ω in (20) shows that the area A ( r ) := R v − (˚ B r ( p )) v ∗ ω is non-negative and grows monotone with r , since it integrates a non-negative mul-tiple of the area form on S over domains v − ( ˚ B r ( p )) that grow with r . Classical If V is finite dimensional, then v is automatically smooth by elliptic regularity. More precisely, we assume that v is not constant on any connected component of the domain.This excludes the pathological case of v ≡ p on one connected component and on the othercomponents covering just a small symplectic area in V \ B R ( p ). monotonicity proofs now argue that the ratio a ( r ) := r − A ( r ) as function of r ∈ (0 , R ] satisfies dd r a ≥ r → a ( r ) ≥ π , which implies the claim a ( R ) ≥ π .We will follow the same line of argument but avoid differentiability concerns byestablishing a uniform difference estimate(21) A ( r + ε ) − A ( r ) ≥ εr + ε A ( r ) for all 0 < r < R, < ε < R − r. Proof of difference estimate:
To prove (21), we simplify notation by assumingwithout loss of generality that p = 0. This can be achieved by applying a globalshift which does not affect the area. We can estimate the area of v in ˚ B r (0) by(22) A ( r ) = R v − (˚ B r (0)) v ∗ ω ≤ R S f ε ( k v k ) h d v ∧ ∗ d v i , where f ε : [0 , ∞ ) → [0 ,
1] is the continuous, piecewise linear cutoff function with f ε | [0 ,r ] ≡ f ε | [ r + ε, ∞ ] ≡
0, and f ′ ε | ( r,r − ε ) = − ε − . On the other hand, integrationby parts yields R S f ε ( k v k ) h d v ∧ ∗ d v i = R ∂S f ε ( k v k ) h v, ∗ d v i − R S f ′ ε ( k v k ) d k v k ∧ h v, ∗ d v i≤ − R S f ′ ε ( k v k ) k v kh d v ∧ ∗ d v i≤ r + εε R v − (˚ B r + ε (0)) \ v − (˚ B r (0)) 12 h d v ∧ ∗ d v i = r + εε (cid:0) A ( r + ε ) − A ( r ) (cid:1) . (23)Here the first step uses a weak version of the Laplace equation d ∗ d v = 0 whichfollows from the Cauchy-Riemann equation ∂ J v = 0. If v is twice differentiablethen this can be checked in local coordinates, ∗ d ∗ d v = − ∂ s v − ∂ t v = ( − ∂ s + J∂ t )( ∂ s v + J∂ t v ) = ( ∂ J ) ∗ ∂ J v = 0 . Otherwise we first consider smooth functions w : S → V with w | ∂S = 0 andcalculate using integration by parts Z S h d w ∧ ∗ d v i = Z S h d ∗ d w, v i = Z S h ( ∂ J ) ∗ ∂ J w, v i dvol = Z S h ∂ J w, ∂ J v i dvol = 0 . Then we note that this identity extends by continuity to C functions such as w = f ε ( k v k ) v , which vanishes on ∂S since k v ( ∂S ) k takes values in [ R, ∞ ), where f ε vanishes since r + ε < R . This yields the first step in (23), with the integral over ∂S vanishing.The second step in (23) can be checked in local holomorphic coordinates andusing (20) again,d k v k ∧ h v, ∗ d v i = k v k (cid:0) h v, ∂ s v i d s + h v, ∂ t v i d t (cid:1) ∧ (cid:0) h v, ∂ s v i d t − h v, ∂ t v i d s (cid:1) = k v k (cid:0) h v, ∂ s v i + h v, ∂ t v i (cid:1) d s ∧ d t ≥ k v k (cid:0) k v k k ∂ s v k + k v k k ∂ t v k (cid:1) d s ∧ d t = k v k h d v ∧ ∗ d v i . The third step in (23) follows from f ′ ε ( k v k ) ≡ r + ε ≥ k v k ≥ r , and f ′ ε = ε − where it doesn’t vanish. Now combining (22) and (23) proves (21), r + εε (cid:0) A ( r + ε ) − A ( r ) (cid:1) ≥ R S f ε ( k v k ) h d v ∧ ∗ d v i ≥ A ( r ) . Monotonicity of A ( r ) implies that this function is differentiable almost everywhere and hasat most countably many jump discontinuities. Then the same holds for the ratio function a ( r )since r r − is smooth on the domain (0 , R ]. POLYFOLD PROOF OF GROMOV’S NON-SQUEEZING THEOREM 35
Monotone growth of ratio function:
In terms of the ratio function a ( r ) = r − A ( r ), the difference estimate (21) implies( r + ε ) (cid:0) a ( r + ε ) − a ( r ) (cid:1) = A ( r + ε ) − ( r + ε ) r A ( r ) ≥ A ( r ) + εr + ε A ( r ) − ( r + ε ) r A ( r )= r + εr +2 εr − r − εr − ε r − ε ( r + ε ) r A ( r ) = − ε r + εr + ε a ( r ) . Now for fixed 0 < r < r < R we have a ( r + ε ) − a ( r ) ≥ − Cε for all r ∈ [ r , r ]and 0 < ε < R − r , with a non-negative constant C := max r ∈ [ r ,r ] ,ε ∈ (0 ,R − r ) r + ε ( r + ε ) a ( r ) ≤ r + ( R − r ) r Z S u ∗ ω. Then summation with ε = r − r N (with N sufficiently large for ε < R − r ) yields a ( r ) − a ( r ) = P N − n =0 a ( r + nε + ε ) − a ( r + nε ) ≥ − N C (cid:0) r − r N (cid:1) . Taking N → ∞ this implies a ( r ) ≥ a ( r ) for any 0 < r < r < R . Next, takingthe limit r → R for fixed r > A ( R ) ≥ lim r → R A ( r ) ≥ lim r → R r a ( r ) ≥ R a ( r ) . So to prove the claim A ( R ) ≥ πR it remains to establish lim r → a ( r ) ≥ π . Centering the ball at a regular point:
Before studying the r → A ( R ) ≥ πR after replacing p = v ( z )with a sequence p n = v ( z n ) → p of images of regular points S \ ∂S ∋ z n → z ,regular just meaning that they are not critical points of v . Indeed, given such asequence and assuming the area bound holds on all balls ˚ B R ( p n ), we have A ( R ) = Z v − (˚ B R ( p )) v ∗ ω ≥ Z v − (˚ B R −k pn − p k ( p n )) v ∗ ω ≥ π ( R − k p n − p k ) since ˚ B R −k p n − p k ( p n ) ⊂ ˚ B R ( p ). This proves A ( R ) ≥ πR in the limit k p n − p k → v are isolated in S . For dim V < ∞ this is proven in [MS04, Lemma 2.4.1]. Ifdim V = ∞ first note that by the Cauchy-Riemann equation in local coordinates, J∂ s v = ∂ t v , any z ∈ S is either regular (i.e. d z v is injective) or critical (i.e. d z v =0). Next, choose a complex splitting ( V, J ) ≃ ( C , i ) ⊕ ( V ′ , J ′ ) in which the firstcomponent pr C ◦ v is nonconstant (e.g. by splitting off im d z v ∼ = C at a regular point).Then classical complex analysis asserts that the critical points of the nonconstantholomorphic map pr C ◦ v : S → C are isolated, so in particular the critical pointsof v are isolated. Lower bound for ratio as r → Once p = v ( z ) is the image of a regular point z ∈ S \ ∂S of v , we choose a neighbourhood U ⊂ S of z with local holomorphiccoordinates U ∼ = D δ := { ( s, t ) ∈ R | s + t < δ } for some δ >
0, so that z ∼ = (0 ,
0) and v | U is given by v ( s, t ) = p + sX + tJX + h ( s, t ) with a vector X = ∂ s v (0 , ∈ V of length k X k = 1 and an error term h with h (0 ,
0) = 0 andd h (0 ,
0) = 0. Then we can bound the area A ( r ) ≥ R D δ χ (cid:0) k v − p k r (cid:1) v ∗ ω by an integralof the characteristic function χ with χ | [0 , ≡ χ | (1 , ∞ ) ≡
0. Moreover, the limitlim r → a ( r ) exists, since we already proved monotonicity of a and it is boundedbelow by 0. So to prove lim r → a ( r ) ≥ π it suffices to find a sequence r n → with lim n →∞ r − n R D rn χ (cid:0) k v − p k r n (cid:1) v ∗ ω = π . To construct this sequence of radii weuse the continuous differentiability of h to choose 0 < r n ≤ δ for each n ∈ N sothat k d h ( s, t ) k ≤ n and k h ( s, t ) k ≤ n | ( s, t ) | for | ( s, t ) | ≤ r n . Then on D r n we canestimate (cid:12)(cid:12) ω ( ∂ s v, ∂ t v ) − (cid:12)(cid:12) = (cid:12)(cid:12) ω (cid:0) X + ∂ s h , JX + ∂ t h (cid:1) − ω ( X , JX ) (cid:12)(cid:12) = (cid:12)(cid:12) ω ( X , ∂ t h ) + ω ( ∂ s h, X ) + ω ( ∂ s h, ∂ t h ) (cid:12)(cid:12) ≤ n + n + ( n ) ≤ n . To control the distance k v ( s, t ) − p k we use 2 xy ≤ n x + ny to obtain (cid:12)(cid:12) k v ( s, t ) − p k − | ( s, t ) | (cid:12)(cid:12) = (cid:13)(cid:13) sX + tJX + h ( s, t ) (cid:13)(cid:13) − ( s + t ) ≤ | s ||h X , h ( s, t ) i| + 2 | t ||h JX , h ( s, t ) i| + k h ( s, t ) k ≤ n ( s + t ) + n (cid:0) |h X , h ( s, t ) i| + |h JX , h ( s, t ) i| (cid:1) + k h ( s, t ) k ≤ n | ( s, t ) | . This holds for ( s, t ) ∈ D r n and implies χ (cid:0) k v ( s,t ) − p k r n (cid:1) = 1 for | ( s, t ) | ≤ r n √ n − =: ρ n . Now writing π = r − n R D rn d s d t , from the above we obtain (cid:12)(cid:12)(cid:12) r − n R D rn χ (cid:0) k v − p k r n (cid:1) v ∗ ω − π (cid:12)(cid:12)(cid:12) = r − n (cid:12)(cid:12)(cid:12)R D rn (cid:0) χ (cid:0) k v − p k r n (cid:1) ω ( ∂ s v, ∂ t v ) − (cid:1) d s d t (cid:12)(cid:12)(cid:12) ≤ r − n (cid:16)R D ρn (cid:12)(cid:12) ω ( ∂ s v, ∂ t v ) − (cid:12)(cid:12) + R D rn \ D ρn (cid:12)(cid:12) χ (cid:0) k v − p k r n (cid:1) ω ( ∂ s v, ∂ t v ) − (cid:12)(cid:12)(cid:17) ≤ r − n (cid:16)R D ρn n d s d t + R D rn \ D ρn s d t (cid:17) = r − n (cid:0) n πρ n + πr n − πρ n (cid:1) = πn − − n − + π − π − n − −→ n →∞ . Together with (24) this finishes the proof, A ( R ) ≥ R lim n →∞ a ( r n ) ≥ R r − n R D rn χ (cid:0) k v − p k r n (cid:1) v ∗ ω −→ n →∞ R π. (cid:3) References [Ack] Benjamin Ackermann. Pseudo-holomorphic curves and Gromov’s non-squeezing the-orem. (unpublished lecture notes from 2015).[FFGW16] Oliver Fabert, Joel W. Fish, Roman Golovko, and Katrin Wehrheim. Polyfolds: Afirst and second look.
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