Contact three-manifolds with exactly two simple Reeb orbits
Dan Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings, Hui Liu
aa r X i v : . [ m a t h . S G ] F e b Contact three-manifolds with exactly two simpleReeb orbits
Dan Cristofaro-Gardiner ∗ , Umberto Hryniewicz † ,Michael Hutchings ‡ , Hui Liu § February 10, 2021
Abstract
It is known that every contact form on a closed three-manifold has atleast two simple Reeb orbits, and a generic contact form has infinitely many.We show that if there are exactly two simple Reeb orbits, then the contactform is nondegenerate. Combined with a previous result, this implies that thethree-manifold is diffeomorphic to the three-sphere or a lens space, and thetwo simple Reeb orbits are the core circles of a genus one Heegaard splitting.We also obtain further information about the Reeb dynamics and the contactstructure. For example the contact structure is universally tight; and in thecase of the three-sphere, the contact volume and the periods and rotationnumbers of the simple Reeb orbits satisfy the same relations as for an irrationalellipsoid.
Contents ∗ Partially supported by NSF grant DMS-1711976. † Partially supported by the DFG SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebraand Dynamics’, Projektnummer 281071066-TRR 191. ‡ Partially supported by NSF grant DMS-2005437. § Partially supported by NSFC (Nos. 11771341, 12022111). The ECH index and perturbations 13
Let Y be a closed oriented three-manifold. Recall that a contact form on Y is aone-form λ on Y such that λ ∧ dλ >
0. A contact form λ has an associated Reebvector field R defined by the equations dλ ( R, · ) = 0 , λ ( R ) = 1 . A Reeb orbit is a periodic orbit of R , i.e. a map γ : R /T Z −→ Y, γ ′ ( t ) = R ( γ ( t )) , for some T >
0, modulo reparametrization of the domain by translations. Thenumber T is the period, also called the symplectic action , of γ . We say that theReeb orbit γ is simple if the map γ is an embedding. Every Reeb orbit is the k -foldcover of a simple Reeb orbit for some positive integer k .The three-dimensional case of the Weinstein conjecture, which was proved in fullgenerality by Taubes [39], asserts that a contact form on a closed three-manifoldhas at least one Reeb orbit; see [27] for a survey. It was further shown in [11] thata contact form on a closed three-manifold has at least two simple Reeb orbits. Thislower bound is the best possible without further hypotheses: Example 1.1.
Recall that if Y is a compact hypersurface in R = C which is“star-shaped” (transverse to the radial vector field), then the standard Liouvilleform λ = 12 X i =1 ( x i dy i − y i dx i ) (1.1)restricts to a contact form on Y . If Y is the three-dimensional ellipsoid ∂E ( a, b ) = (cid:26) z ∈ C (cid:12)(cid:12)(cid:12)(cid:12) π | z | a + π | z | b = 1 (cid:27) , a/b is irrational, then there are exactly two simple Reeb orbits, correspondingto the circles in Y where z = 0 and z = 0, with periods a and b respectively.One can also take quotients of the above irrational ellipsoids by finite cyclicgroup actions to obtain contact forms on lens spaces with exactly two simple Reeborbits.It is conjectured that in fact, every contact form on a closed connected three-manifold has either two or infinitely many simple Reeb orbits. This was provedin [8] for contact forms that are nondegenerate (see the definition below), extendinga result of [12]. It was also shown by Irie [34] that for a C ∞ -generic contact form ona closed three-manifold, there are infinitely many simple Reeb orbits, and moreovertheir images are dense in the three-manifold.The goal of this paper is to give detailed information about the “exceptional”case of contact forms on a closed three-manifold with exactly two simple Reeb orbits.To state the first result, let ξ = Ker( λ ) denote the contact structure determinedby λ . This is a rank 2 vector bundle with a linear symplectic form dλ . If γ : R /T Z → Y is a Reeb orbit, then the derivative of the time T flow of R restricts toa symplectic linear map P γ : ( ξ γ (0) , dλ ) −→ ( ξ γ (0) , dλ ) , (1.2)which we call the linearized return map . We say that γ is nondegenerate if 1 is notan eigenvalue of P γ ; this condition is invariant under reparametrization of γ . We saythat the contact form λ is nondegenerate if all Reeb orbits (including non-simpleones) are nondegenerate. The set of nondegenerate contact forms is residual in theset of all contact forms with the C ∞ -topology.The Reeb orbit γ is called hyperbolic if P γ has eigenvalues in R \ {± } . TheReeb orbit γ is called elliptic if the eigenvalues of P γ are of the form e ± πiφ , and irrationally elliptic if moreover φ is irrational. If γ is irrationally elliptic, then γ andall of its covers are nondegenerate, because the linearized return map for the k -foldcover of γ has eigenvalues e ± πikφ . Theorem 1.2.
Let Y be a closed three-manifold, and let λ be a contact form on Y with exactly two simple Reeb orbits. Then λ is nondegenerate, and moreover bothsimple Reeb orbits are irrationally elliptic. As a corollary, we obtain the following topological constraint:
Corollary 1.3.
Let Y be a closed three-manifold, and let λ be a contact form on Y with exactly two simple Reeb orbits. Then Y is diffeomorphic to a lens space .Moreover, the two simple Reeb orbits are the core circles of a genus one Heegaardsplitting of Y . Here and below, our convention is that S is a lens space, but S × S is not. roof. This was shown in [32, Thm. 1.3 and § λ is nondegenerate. By Theorem 1.2, this nondegeneracy holds automatically. Remark 1.4.
A special case of Theorem 1.2, where Y is a compact convex hyper-surface in R with the restriction of the standard Liouville form (1.1), was previouslyshown in [42, Thm. 1.4].We also obtain additional dynamical information. To state the result, recall thatthe contact volume of ( Y, λ ) is defined byvol(
Y, λ ) := Z Y λ ∧ dλ. Theorem 1.5.
Let Y be a lens space and let λ be a contact form on Y with exactlytwo simple Reeb orbits, γ and γ . Then:(a) Let p = | π ( Y ) | < ∞ , let T i ∈ R denote the period of γ i , and let φ i ∈ R denotethe “Seifert rotation number” of γ i , see Definition 4.3. Then vol( Y, λ ) = pT T = T /φ = T /φ . (b) λ is dynamically convex, and the contact structure ξ = Ker( λ ) is universallytight . Example 1.6.
For the ellipsoid in Example 1.1, we have T = a , T = b , φ = a/b , φ = b/a , p = 1, and vol = ab . Thus Theorem 1.5(a) implies that if Y = S , thenthe periods T i , the rotation numbers φ i , and the contact volume satisfy the samerelations as for an ellipsoid. For Y = S , under the additional assumptions that λ isnondegenerate and ξ is the standard contact structure, it was previously shown in[6, 18], that “action-index relations” hold, implying that the periods T i and rotationnumbers φ i satisfy the same relations as for an ellipsoid. The equation vol = T T that we prove in this case answers [4, Question 4]. Remark 1.7.
There also exist contact forms on S with exactly two simple Reeborbits which are not strictly contactomorphic to ellipsoids, because the Reeb flowis ergodic with respect to the Liouville measure. This was shown by Albers-Geiges-Zehmisch [1] by suspending pseudorotations from [15, 35]; see Remark 1.11 below. Remark 1.8.
As shown in [22, Prop. 5.1] (see [10, p. 17] for more explanation), eachlens space has either one or two universally tight contact structures up to isotopy,and when there are two they are contactomorphic (and one is obtained from the Recall that a contact form on a three-manifold Y with c ( ξ ) | π ( Y ) = 0 is called dynamicallyconvex if CZ( γ ) ≥ γ , where CZ denotes the Conley-Zehnderindex (see § γ . A contact structure on Y is universally tight if its pullback to the universal cover of Y is tight. Remark 1.9.
We also obtain information about the knot types of the simple Reeborbits γ and γ . It follows from the Heegaard splitting in Corollary 1.3 that theseare p -unknotted. We further show in § − p = 1; similar arguments show that for general p , their rational self-linking number,as defined in [2], equals − /p . Remark 1.10.
The contact forms studied here are analogous to “pseudorotations”,defined in various ways as maps in some class with the minimum number of periodicorbits. For example, in [17] a
Hamiltonian pseudorotation of C P n is defined to bea Hamiltonian symplectomorphism of C P n with n + 1 fixed points and no otherperiodic points (see e.g. [7, 37, 38] for generalizations to other symplectic manifolds).It is shown in [9], see also [16], that for a Hamiltonian pseudorotation of C P , eachfixed point is strongly nondegenerate, meaning that the linearized return map andits higher powers are nondegenerate, and moreover the fixed points are irrationallyelliptic, similarly to Theorem 1.2. It is unknown whether all fixed points of aHamiltonian pseudorotation of C P n must be strongly nondegenerate when n > Remark 1.11.
More classically, a pseudorotation of the closed disk is defined tobe an area-preserving homeomorphism of the closed disk with one fixed point andno other periodic points; see e.g. [5, 15]. There is a direct connection between thecontact forms considered in this paper and pseudorotations of the closed disk. Aswe explain in §
5, for a contact form on S with exactly two simple Reeb orbits, theReeb flow has a disk-like global surface of section, for which the Poincar´e returnmap is an area-preserving diffeomorphism of an open disk with one fixed point andno other periodic points. In fact this map can be “completed” to a pseudorotationof the closed disk. Conversely, some smooth pseudorotations of the closed disk canbe “suspended” to contact forms on S with exactly two simple Reeb orbits [1]. We now briefly describe the proofs of Theorems 1.2 and 1.5.A key ingredient in these proofs, as well as in the related papers [11, 12], is the“Volume Property” in embedded contact homology, which was proved in [13]. Theembedded contact homology (ECH) of (
Y, λ ) is the homology of a chain complexwhich is built out of Reeb orbits, and whose differential counts (mostly) embeddedpseudoholomorphic curves in R × Y ; see the lecture notes [29] and the review in §
2. The version of the Volume Property that we will use here asserts that if Y is a5losed connected 3-manifold with a contact form λ , thenlim k →∞ c σ k ( Y, λ ) k = 2 vol( Y, λ ) . Here { σ k } is a “ U -sequence” in ECH, and c σ k is a “spectral invariant” associatedto σ k , which is the total symplectic action of a certain finite set of Reeb orbitsdetermined by σ k ; these notions are reviewed in § γ and γ denote thetwo simple Reeb orbits, and let T and T denote their periods. Simple applicationsof the Volume Property from [11, 12] (just using the k / growth rate of the spectralinvariants and not the exact relation with contact volume) show that the homologyclasses [ γ i ] ∈ H ( Y ) are torsion, and the ratio T /T is irrational. A more preciseuse of the Volume Property then gives the relations φ i = T i vol( Y, λ ) ℓ ( γ , γ ) = T T vol( Y, λ ) (1.3)where φ i ∈ R is the Seifert rotation number that appears in Theorem 1.5(a), while ℓ ( γ , γ ) ∈ Q is the linking number of γ and γ , see Definition 4.2. The proof of (1.3)also depends on a new estimate for the behavior of the ECH index (the grading onthe ECH chain complex) under perturbations of possibly degenerate contact forms,which is proved in § φ = ℓ ( γ , γ ) T T , φ = ℓ ( γ , γ ) T T . (1.4)Since ℓ ( γ , γ ) is rational and T /T is irrational, it follows that φ and φ areirrational. The latter fact implies that γ and γ are irrationally elliptic; see § ℓ ( γ , γ ) = 1 /p , andcombined with (1.4) this proves Theorem 1.5(a). The proof of Theorem 1.5(b) usesadditional calculations in § φ i . Acknowledgments.
Key discussions for this project took place when DCG andHL were visiting Peking University for the “Workshop on gauge theory and Floerhomology” in December 2019. We thank Peking University for their hospitalityduring this visit. This research was completed while DCG was on a von Neumannfellowship at the Insititute for Advanced Study. DCG thanks the Institute for theirwonderful support. We thank Viktor Ginzburg for helpful comments.6
Preliminaries
In this section we review the material about embedded contact homology that isneeded for the proofs of Theorems 1.2 and 1.5. We include a new, slight extensionof the definition of the ECH index to degenerate contact forms.Throughout this section fix a closed oriented three-manifold Y and a contactform λ on Y , and let ξ = Ker( λ ) denote the associated contact structure. We now recall some topological notions we will need, following the treatment in [26].These were originally introduced in a slightly different context in [25].
Definition 2.1. An orbit set is a finite set of pairs α = { ( α i , m i ) } where the α i are distinct simple Reeb orbits, and the m i are positive integers. We define thehomology class of the orbit set α by[ α ] = X i m i [ α i ] ∈ H ( Y ) . Definition 2.2. If α = { ( α i , m i ) } and β = { ( β j , n j ) } are orbit sets with [ α ] = [ β ],define H ( Y, α, β ) to be the set of 2-chains Z in Y with ∂Z = P i m i α i − P j n j β j ,modulo boundaries of 3-chains. The set H ( Y, α, β ) is an affine space over H ( Y ).Given orbit sets α and β as above, let Z ∈ H ( Y, α, β ), and let τ be a homotopyclass of symplectic trivialization of the contact structure ξ over the Reeb orbits α i and β j . Definition 2.3. (cf. [26, § relative first Chern class c τ ( α, β, Z ) ∈ Z as follows. Let S be a compact oriented surface with boundary and let f : S → Y be a smooth map representing the class Z . Let ψ be a section of f ∗ ξ which oneach boundary component is nonvanishing and constant with respect to τ . Define c τ ( α, β, Z ) to be the algebraic count of zeroes of ψ . Definition 2.4. [26, § admissible representative of Z ∈ H ( Y, α, β ) is asmooth map f : S → [ − , × Y where S is a compact oriented surface withboundary; the restriction of f to ∂S consists of positively oriented covers of { } × α i with total multiplicity m i and negatively oriented covers of {− } × β j with totalmultiplicity n j ; the composition of f with the projection [ − , × Y → Y representsthe class Z ; the restriction of f to the interior of S is an embedding; and f istransverse to {− , } × Y . 7 efinition 2.5. [26, § Z, Z ′ ∈ H ( Y, α, β ), define the relative intersectionpairing Q τ ( Z, Z ′ ) ∈ Z as follows. Let S, S ′ be admissible representatives of Z and Z ′ respectively whoseinteriors are transverse and do not intersect near the boundary. Define Q τ ( Z, Z ′ ) = S ) ∩ int( S ′ )) − X i ℓ τ ( ζ + i , ζ + i ′ ) + X j ℓ τ ( ζ − j , ζ − j ′ ) . (2.1)Here ‘ ε > S with { − ε } × Y consists of the union over i of a braid ζ + i in a neighborhood of α i (see § S with {− ε } × Y consists of the union over j of abraid ζ − j in a neighborhood of β j . Likewise, S ′ determines braids ζ + i ′ and ζ − j ′ . Thenotation ℓ τ indicates the linking number in a neighborhood of α i or β j computedusing the trivialization τ ; see [26, § Z = Z ′ , we write Q τ ( α, β, Z ) = Q τ ( Z, Z ) . As explained in [26], the relative first Chern class c τ ( α, β, Z ) and the relativeself-intersection number Q τ ( α, β, Z ) depend only on α , β , Z , and τ . Moreover, ifwe change Z by adding A ∈ H ( Y ) then they behave as follows: c τ ( α, β, Z + A ) − c τ ( α, β, Z ) = h c ( ξ ) , A i , (2.2) Q τ ( α, β, Z + A ) − Q τ ( α, β, Z ) = 2[ α ] · A. (2.3) Remark 2.6. If γ is a third orbit set, if τ is a trivialization of ξ over the Reeb orbitsin α , β , and γ , and if W ∈ H ( Y, β, γ ), then we have the additivity properties c τ ( α, β, Z ) + c τ ( β, γ, W ) = c τ ( α, γ, Z + W ) ,Q τ ( α, β, Z ) + Q τ ( β, γ, W ) = Q τ ( α, γ, Z + W ) . Note also that the definition of c τ makes sense more generally if the α i and β j aretransverse knots. Likewise the definition of Q τ makes sense if the α i and β j areknots and τ is an oriented trivialization of their normal bundles. Let γ : R /T Z → Y be a Reeb orbit and let τ be a symplectic trivialization of γ ∗ ξ . The derivative of the time t Reeb flow from ξ γ (0) to ξ γ ( t ) , with respect to τ , An alternate, equivalent definition of Q τ ( α, β, Z ) is given in [29, § S and S ′ are required tosatisfy additional conditions which force these linking number terms to be zero.
8s a 2 × t ). The family of symplectic matrices { Φ( t ) } t ∈ [0 ,T ] induces a family of diffeomorphisms of S in the universal cover of Diff( S ), whichhas a dynamical rotation number, which we denote by θ τ ( γ ) ∈ R . We call this realnumber the rotation number of γ with respect to τ and denote it by θ τ ( γ ) ∈ R ; itdepends only on γ and the homotopy class of τ . When θ τ ( γ ) / ∈ Z , the eigenvaluesof the linearized return map (1.2) are e ± πiθ τ ( γ ) . Definition 2.7.
Define the
Conley-Zehnder index CZ τ ( γ ) = ⌊ θ τ ( γ ) ⌋ + ⌈ θ τ ( γ ) ⌉ ∈ Z . (2.4) Remark 2.8.
The above definition agrees with the usual Conley-Zehnder indexwhen γ is nondegenerate. When γ is degenerate, it is common to give a differentdefinition of the Conley-Zehnder index, as the the minimum of the Conley-Zehnderindices of nondegenerate perturbations of γ , and this will sometimes differ from ourdefinition by 1. For the purposes of this paper, especially to obtain an estimateas in Proposition 3.1 below (possibly with a different constant), it does not matterwhich of these definitions of the Conley-Zehnder index we use for degenerate Reeborbits. Notation 2.9. If α = { ( α i , m i ) } is an orbit set and if τ is a trivialization of ξ overall of the Reeb orbits α i , defineCZ Iτ ( α ) = X i m i X k =1 CZ τ ( α ki ) . (2.5)Here γ k denotes the k th iterate of γ . Definition 2.10.
Let α and β be orbit sets with [ α ] = [ β ] ∈ H ( Y ), and let Z ∈ H ( Y, α, β ). Define the
ECH index I ( α, β, Z ) = c τ ( α, β, Z ) + Q τ ( α, β, Z ) + CZ Iτ ( α ) − CZ Iτ ( β ) ∈ Z . (2.6)The above agrees with the usual definition of the ECH index, see e.g. [29, § § I ( α, β, Z ) depends only on α , β , and Z , and not on τ . Moreover, it followsfrom (2.2) and (2.3) that if we change Z by adding A ∈ H ( Y ), then I ( α, β, Z + A ) − I ( α, β, Z ) = h c ( ξ ) + 2 PD(Γ) , A i , (2.7)where Γ = [ α ] = [ β ] ∈ H ( Y ) and PD denotes the Poincar´e dual. By Remark 2.6,we have I ( α, β, Z ) + I ( β, γ, W ) = I ( α, γ, Z + W ) . (2.8)9 .3 Embedded contact homology In this subsection assume that the contact form λ is nondegenerate. Let Γ ∈ H ( Y ).We now review how to define the embedded contact homology ECH ∗ ( Y, ξ,
Γ). Moredetails may be found in [29].
Definition 2.11. An ECH generator is an orbit set α = { ( α i , m i ) } such that m i = 1whenever α i is hyperbolic. Definition 2.12.
Define
ECC ∗ ( Y, λ,
Γ) to be the vector space over Z / α with [ α ] = Γ. This vector space has a relative Z /d grading,where d denotes the divisibility of c ( ξ )+2 PD(Γ) ∈ H ( Y ; Z ); if α and β are two gen-erators, then their grading difference is I ( α, β, Z ) mod d for any Z ∈ H ( Y, α, β ).This makes sense by (2.7) and (2.8).
Remark 2.13.
In the special case where c ( ξ ) ∈ H ( Y ; Z ) is torsion and Γ = 0,the chain complex ECC ∗ ( Y, λ,
0) has a canonical absolute Z -grading defined by I ( α ) = I ( α, ∅ , Z ) ∈ Z for any Z ∈ H ( Y, α, ∅ ). This is well-defined by (2.7). Definition 2.14.
An almost complex structure J on R × Y is λ -compatible if J ∂ s = R , where s denotes the R coordinate; J is invariant under the R action on R × Y by translation of s ; and J ( ξ ) = ξ , rotating positively with respect to dλ .If J is a generic λ -compatible almost complex structure, one defines a differential ∂ J : ECC ∗ ( Y, λ, Γ) −→ ECC ∗− ( Y, λ,
Γ)whose coefficient from α to β is a count of “ J -holomorphic currents” that representclasses Z ∈ H ( Y, α, β ) with ECH index I ( α, β, Z ) = 1. See [29, §
3] for details. Itis shown in [30] that ∂ J = 0. The embedded contact homology ECH ∗ ( Y, λ, Γ , J ) isdefined to be the homology of the chain complex ( ECC ∗ ( Y, λ, Γ) , ∂ J ). A theorem ofTaubes [40], tensored with Z /
2, asserts that there is a canonical isomorphism
ECH ∗ ( Y, λ, Γ , J ) = d HM −∗ ( Y, s ξ + PD(Γ)) ⊗ Z / , (2.9)where the right hand side is a version of Seiberg-Witten Floer cohomology as definedby Kronheimer-Mrowka [36], and s ξ is a spin-c structure on Y determined by ξ . Inparticular, ECH depends only on the triple ( Y, ξ,
Γ), and so we can denote it by
ECH ∗ ( Y, ξ,
Γ). It is also possible to use Z coefficients, as explained in [31, § In a sense, ECH does not depend on the contact structure either; see [29, Rmk. 1.7] forexplanation. Y is connected, there is also a well-defined “ U -map” U : ECH ∗ ( Y, ξ, Γ) −→ ECH ∗− ( Y, ξ, Γ) . (2.10)This is induced by a chain map U J,z : (
ECC ∗ ( Y, λ, Γ) , ∂ J ) −→ ( ECC ∗− ( Y, ξ, Γ) , ∂ J )which counts J -holomorphic currents with ECH index 2 passing through a genericbase point z ∈ R × Y . The assumption that Y is connected implies that the inducedmap on homology does not depend on the choice of base point z ; see [32, § U J,z agrees with a corresponding map on Seiberg-WittenFloer cohomology. We thus obtain a well-defined U -map (2.10). Definition 2.15. A U -sequence for Γ is a sequence { σ k } k ≥ where each σ k is anonzero homogeneous class in ECH ∗ ( Y, ξ,
Γ), and
U σ k +1 = σ k for each k ≥ U -map, which is proved bycombining Taubes’s isomorphism (2.9) with results from Kronheimer-Mrowka [36]: Proposition 2.16. [12, Prop. 2.3] If c ( ξ ) + 2 PD(Γ) ∈ H ( Y ; Z ) is torsion, then a U -sequence for Γ exists. If α = { ( α i , m i ) } is an orbit set, define its symplectic action by A ( α ) = X i m i Z α i λ. Note here that R α i λ agrees with the period of α i , because λ ( R ) = 1.Assume now that λ is nondegenerate. For L ∈ R , define ECC L ∗ ( Y, λ,
Γ) to bethe subspace of
ECC ∗ ( Y, λ,
Γ) spanned by ECH generators α with symplectic action A ( α ) < L . It follows from the definition of “ λ -compatible almost complex structure”that ∂ J maps ECC L to itself; see [29, § filtered ECH to be thehomology of this subcomplex, which we denote by ECH L ∗ ( Y, λ,
Γ). The inclusion ofchain complexes induces a map ı L : ECH L ∗ ( Y, λ, Γ) −→ ECH ∗ ( Y, ξ, Γ) . It is shown in [33, Thm. 1.3] that the filtered homology
ECH L ∗ ( Y, λ,
Γ) and the map ı L do not depend on the choice of J . However, unlike the usual ECH, filtered ECHdoes depend on the contact form λ and not just on the contact structure ξ . Definition 2.17.
If 0 = σ ∈ ECH ∗ ( Y, ξ,
Γ), define the spectral invariant c σ ( Y, λ ) = inf { L | σ ∈ Im( ı L ) } ∈ R .
11n equivalent definition is that c σ ( Y, λ ) is the minimum L such that the class σ can be represented by a cycle in the chain complex ( ECC ∗ ( Y, λ, Γ) , ∂ J ) which isa sum of ECH generators each having symplectic action ≤ L . In particular, bydefinition c σ ( Y, λ ) = A ( α ) for some ECH generator α with [ α ] = Γ.We can change the contact form λ , without changing the contact structure ξ ,by multiplying λ by a smooth function f : Y → R > . As explained in [11, § λ is degenerate, one can still define c σ ( Y, λ ) as a limit ofspectral invariants c σ ( Y, f n λ ) where f n λ is nondegenerate and f n → C .These spectral invariants have the following important properties: Proposition 2.18.
Let Y be a closed connected three-manifold, and let λ be a(possibly degenerate) contact form on Y . Then:(a) If = σ ∈ ECH ∗ ( Y, ξ, Γ) , then c σ ( Y, λ ) = A ( α ) for some orbit set α with [ α ] = Γ .(b) If σ ∈ ECH ∗ ( Y, ξ, Γ) and U σ = 0 , then c Uσ ( Y, λ ) ≤ c σ ( Y, λ ) . (2.11) If there are only finitely many simple Reeb orbits, then the inequality (2.11) isstrict.(c) (“Volume Property”) If c ( ξ ) + 2 PD(Γ) ∈ H ( Y ; Z ) is torsion, and if { σ k } k ≥ is a U -sequence for Γ , then lim k →∞ c σ k ( Y, λ ) k = 2 vol( Y, λ ) . Proof.
As noted above, part (a) holds by definition when λ is nondegenerate, andin the degenerate case it follows from a compactness argument for Reeb orbits; cf.[11, Lem. 3.1(a)].If λ is nondegenerate, then since the chain map U J,z counts J -holomorphic curves,it decreases symplectic action like the differential, so strict inequality in (2.11) holds.The not necessarily strict inequality (2.11) in the degenerate case follows by a lim-iting argument. The fact that the inequality (2.11) is strict for degenerate contactforms with only finitely many simple Reeb orbits is proved by a more subtle com-pactness argument for holomorphic curves in [11, Lem. 3.1(b)].Part (c), the most nontrivial part, is a special case of [13, Thm. 1.3]. The equality c Uσ ( Y, λ ) = c σ ( Y, λ ) is possible for degenerate contact forms with infinitely manysimple Reeb orbits. This happens for example for some classes σ when Y is an ellipsoid ∂E ( a, b )with a/b rational. The ECH index and perturbations
The goal of this section is to prove Proposition 3.1 below, which gives an upperbound on how much the ECH index can change when one perturbs the contactform. This is an important ingredient in the proof of Theorems 1.2 and 1.5.To state the proposition, let λ be a contact form on a closed three-manifold Y , and let λ n = f n λ be a sequence of contact forms with f n → C . In thecase of interest, λ will be degenerate, while each of the contact forms λ n will benondegenerate.Fix an orbit set α = { ( α i , m i ) } for λ , and let N be a disjoint union of tubularneighborhoods N i of the simple Reeb orbits α i . Consider a sequence of orbit sets α ( n ) for λ n that converges to α as currents. In particular this implies that if n is sufficiently large, and if we write α ′ = α ( n ), then α ′ is contained in N , andits intersection with N i is homologous to m i α i . There is then a unique W α ∈ H ( Y, α ′ , α ) that is contained in N .Likewise fix an orbit set β = { ( β j , n j ) } for λ along with disjoint tubular neigh-borhoods of the simple Reeb orbits β j , and consider a sequence of orbit sets β ( n ) for λ n that converges to β as currents. Then for k sufficiently large, writing β ′ = β ( n ),we obtain a distinguished W β ∈ H ( Y, β ′ , β ).For fixed large n there is now a bijection H ( Y, α, β ) ≃ H ( Y, α ′ , β ′ )sending Z ∈ H ( Y, α, β ) to Z ′ = Z + W α − W β ∈ H ( Y, α ′ , β ′ ) . Proposition 3.1.
With the notation as above, for fixed orbit sets α and β , if n issufficiently large, then (cid:12)(cid:12) I ( α, β, Z ) − I ( α ′ , β ′ , Z ′ ) (cid:12)(cid:12) ≤ X i m i + X j n j ! . Here I ( α, β, Z ) denotes the ECH index for λ , and I ( α ′ , β ′ , Z ′ ) denotes the ECHindex for λ n . We now reduce Proposition 3.1 to a local statement, Proposition 3.3 below.Let γ be an oriented knot in Y , and let N be a tubular neighborhood of γ with anidentification N ≃ S × D . By a “braid in N with d strands”, we mean an orientedknot in N which is positively transverse to the D fibers and which intersects eachfiber d times. 13 efinition 3.2. • A weighted braid in N with m strands is a finite set of pairs ζ = { ( ζ i , m i ) } where the ζ i are disjoint braids in N with d i strands, the m i are positive integers, and P i m i d i = m . • If τ is an oriented trivialization of the normal bundle of γ , then for i = j thereis a well-defined linking number ℓ τ ( ζ i , ζ j ) ∈ Z , as discussed in § i there is a well-defined writhe w τ ( γ i ) ∈ Z ; see [26, § writhe of the weighted braid ζ by w τ ( ζ ) = X i m i w τ ( ζ i ) + X i = j m i m j ℓ τ ( ζ i , ζ j ) . (3.1)Suppose now that γ is a simple Reeb orbit for λ , and that the normal bundleidentification N ≃ S × D above is chosen so that the Reeb vector field for λ istransverse to the D fibers. If λ ′ = f λ with f sufficiently C close to 1, then theReeb vector field for λ ′ in N is also transverse to the D fibers. Suppose that thisis the case.Let γ ′ = { ( γ ′ k , m k ) } be an orbit set for λ ′ which is contained in N . We canregard γ ′ as a weighted braid with m strands for some positive integer m . Also notethat a trivialization τ of γ ∗ ξ extends to a trivialization of ξ over the entire tubularneighborhood N , and thus canonically induces a homotopy class of trivialization τ ′ of ξ over the Reeb orbits γ ′ k . We can now state: Proposition 3.3.
With the notation as above, if λ ′ is sufficiently C close to λ ,and if γ ′ is sufficiently close to mγ as a current, then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w τ ( γ ′ ) − CZ Iτ ′ ( γ ′ ) + m X l =1 CZ τ ( γ l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m. (3.2) Proof of Proposition 3.1, assuming Proposition 3.3.
By shrinking the tubular neigh-borhoods, we can assume without loss of generality that the chosen tubular neigh-borhood of each orbit α i or β j has an identification with S × D with the Reebflow of λ transverse to the D fibers.In the orbit set α ′ , each pair ( α i , m i ) gets replaced by an orbit set α ′ i which repre-sents a weighted braid with m i strands in the tubular neighborhood of α i . Likewise,each pair ( β j , n j ) gets replaced by an orbit set β ′ j which represents a weighted braidwith n j strands in the tubular neighborhood of β j . Let τ be a homotopy class ofsymplectic trivializations of ξ over the Reeb orbits α i and β j . As in Proposition 3.3,this canonically induces a homotopy class of symplectic trivializations τ ′ over theReeb orbits in the orbit sets α ′ i and β ′ j .Because τ and τ ′ extend to a trivialization of ξ over the tubular neighborhoodscontaining W α and W β , it follows from the definition of the relative first Chern classthat c τ ′ ( α ′ , β ′ , Z ′ ) = c τ ( α, β, Z ) . (3.3)14y Proposition 3.3, if n is sufficiently large then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w τ ( α ′ i ) − CZ Iτ ′ ( α ′ i ) + m i X k =1 CZ τ ( α ki ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m i , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w τ ( β ′ j ) − CZ Iτ ′ ( β ′ j ) + n j X l =1 CZ τ ( β lj ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n j . (3.4)By equations (2.6), (3.3), and (3.4), to complete the proof of Proposition 3.1 it isenough to show that Q τ ′ ( α ′ , β ′ , Z ′ ) = Q τ ( α, β, Z ) + X i w τ ( α ′ i ) − X j w τ ( β ′ j ) . (3.5)To prove (3.5), by Remark 2.6 it is enough to show that Q τ ( α ′ , α, W α ) = X i w τ ( α ′ i ) ,Q τ ( β ′ , β, W β ) = X j w τ ( β ′ j ) . Since the chosen tubular neighborhoods of the Reeb orbits of α i are disjoint, andthe chosen tubular neighborhoods of the Reeb orbits of β j are disjoint, the aboveequations follow from Lemma 3.4 below. Lemma 3.4.
Let ζ = { ( ζ i , m i ) } be a weighted braid with m strands as in Defini-tion 3.2. Let W be the unique relative homology class in H ( N, ζ , ( γ, m )) . Then Q τ ( ζ , ( γ, m ) , W ) = w τ ( ζ ) . (3.6)Here τ defines a trivialization of the vertical tangent bundle of N → γ which theninduces a trivialization of the normal bundle of each braid ζ i . Proof of Lemma 3.4.
We can make an admissible representative S for W , see Defi-nition 2.4, whose intersection with { − ε } × N consists of m i parallel (with respectto τ ) copies of each ζ i , and which shrinks radially towards γ as the [ − ,
1] coor-dinate on [ − , × N goes down to −
1. We can make another such admissiblerepresentative S ′ , disjoint from S , whose intersection with { − ε } × N is parallelto the first and which likewise shrinks radially towards γ . Then in equation (2.1),the intersection number term vanishes. The first linking number term in (2.1) alsovanishes, as it is a sum of linking numbers of braids in neighborhoods of the ζ i ; foreach i , the braid from S and the braid from S ′ , with respect to τ , are trivial andparallel, and thus have linking number zero. The second linking number term in(2.1) is a linking number in a neighborhood of γ and equals w τ ( ζ ).15 .2 The structure of the braids To prove Proposition 3.3, let γ be a simple Reeb orbit of λ , let N be a tubularneighborhood of γ as in Definition 3.2, and let τ be a trivialization of γ ∗ ξ . Let θ denote the rotation number θ τ ( γ ) ∈ R .Suppose first that θ is irrational. Then when λ ′ is sufficiently C close to λ ,there is a unique Reeb orbit γ ′ for λ ′ close (as a current) to γ , and for n large theonly possibility for the orbit set γ ′ is that it is the singleton set γ ′ = { ( γ ′ , m ) } . Inthis case Proposition 3.3 holds because w τ ( γ ′ ) = 0 and the left hand side of (3.2) iszero.The nontrivial case of Proposition 3.3 is when the rotation number θ is rational.In this case we need to investigate the braids that can arise in γ ′ . The idea in whatfollows is to first analyze the case where the rotation number is an integer, and thenreduce the general case to this one by taking an appropriate cover of a neighborhoodof γ .We start with the case where the rotation number is an integer. Here there is asimple picture. Each braid has just one strand and the linking number of any twois given by the rotation number: Lemma 3.5.
With the above notation, suppose that the rotation number θ = a ∈ Z .Let λ n = f n λ where f n → in C . Then:(a) For a fixed positive integer d , if { α n } is a sequence where each α n is a simpleReeb orbit for λ n in N which is a braid with d strands, with α n converging ascurrents to dγ as n → ∞ , then d = 1 , and in particular the writhe w τ ( α n ) = 0 for n large enough.(b) Given two sequences of simple Reeb orbits { α n } and { β n } as in (a) with α n = β n for each n , if n is sufficiently large, then the linking number ℓ τ ( α n , β n ) = a . The above lemma is proved in § a/b that is not an integer. Here there is asimilarly nice picture. Each new simple Reeb orbit that can appear can be treatedfor our purposes like an ( a, b ) torus braid, see also Remark 3.7. More precisely: Lemma 3.6.
With the above notation, suppose that the rotation number is θ = a/b where a, b are relatively prime integers with b > . Let λ n = f n λ where f n → in C . Then:(a) For n sufficiently large, there is a unique simple Reeb orbit γ ′ for λ n that isclose to γ as a current.(b) For a fixed integer d > , if { α n } is a sequence where each α n is a simpleReeb orbit for λ n in N which is a braid with d strands, with α n convergingas currents to dγ as n → ∞ , then for n sufficiently large we have d = b , thewrithe w τ ( α n ) = a ( b − , and the linking number ℓ τ ( γ ′ , α n ) = a . c) Given two sequences of Reeb orbits { α n } and { β n } as in (b) with α n = β n foreach n , if n is sufficiently large, then the linking number ℓ τ ( α n , β n ) = ab . Remark 3.7.
In Lemma 3.6(b), we expect that one can further show that if n issufficiently large then α n is an ( a, b ) torus braid around γ ′ ; however we do not needthis. Proof of Lemma 3.6, assuming Lemma 3.5.
Part (a) holds because the Reeb orbit γ is nondegenerate.To prove part (b), we first note that by the same argument as for (a), we musthave that d ≥ b , because for 0 < d < b the d th iterate of γ has rotation number da/b / ∈ Z .Now let e N denote the b -fold cyclic cover of the tubular neighborhood N , withthe pullback of the contact form λ n . There is a unique simple Reeb orbit e γ ′ in e N whose projection to N is a b -fold cover of γ ′ . In addition, by lifting the Reeborbit α n to a Reeb trajectory in e N and extending it by the Reeb flow if needed,we obtain a simple Reeb orbit f α n in e N whose projection to N is a cover of α n . ByLemma 3.5(a), if n is sufficiently large, then f α n is a braid with one strand in e N ,hence α n has at most b strands, so d = b . By Lemma 3.5(b) we have ℓ τ ( e γ ′ , f α n ) = a in e N , and it follows that ℓ τ ( γ ′ , α n ) = a .We now compute the writhe w τ ( α n ). There are b possibilities for the Reeb orbit f α n in the previous paragraph, which we denote by η l for l ∈ Z /b , ordered so thatthe Z /b action on e N by deck transformations sends η l to η l +1 . The writhe w τ ( α n )is a signed count of crossings of two strands of α n . Each such crossing correspondsto a crossing of some η l with some η l ′ for l = l ′ , as well as crossings of η l + p with η l ′ + p for p = 1 , . . . , b − η l with η l ′ is one half the signed count ofcrossings of η l with η l ′ . Thus we obtain w τ ( α n ) = 1 b X l = l ′ ℓ τ ( η l , η l ′ )= 1 b · b ( b − · a = a ( b − . Here we are using Lemma 3.5(b) to get that ℓ τ ( η l , η l ′ ) = a when l = l ′ .We now prove (c). Similarly to the previous calculation, each crossing counted bythe linking number ℓ τ ( α n , β n ) corresponds to b crossings of some lift of α n (extendedto a simple Reeb obit) with some lift of β n (extended to a simple Reeb orbit).Thus the linking number we want is 1 /b times the sum of the linking number ofeach of the b extended lifts of α n with each of the b extended lifts of β n , which is(1 /b ) · b · a = ab . Proof of Proposition 3.3.
As explained above, we can assume that θ = a/b where a, b are relatively prime integers with b >
0. When a/b / ∈ Z , the orbit set γ ′ consists17f the orbit γ ′ from Lemma 3.6(a) with multiplicity m for some m ≥
0, togetherwith orbits γ ′ k for k = 0 with multiplicities m k >
0. When a/b ∈ Z , the same is trueexcept that we do not necessarily have a unique γ ′ and we can take m = 0. Sinceeach γ ′ k for k = 0 is close to a b -fold cover of γ , we have m + b X k =0 m k = m. (3.7)By equation (3.1) and Lemmas 3.5 and 3.6, if λ ′ is sufficiently C close to λ and if γ ′ is sufficiently close to mγ as a current, then we have w τ ( γ ′ ) = a ( b − X k =0 m k + 2 am X k =0 m k + ab X = k = k ′ =0 m k m ′ k . (3.8)Now we consider Conley-Zehnder indices. By equation (2.5) we haveCZ Iτ ′ ( γ ′ ) = m X l =1 CZ τ ′ (cid:0) ( γ ′ ) l (cid:1) + X k =0 m k X l =1 CZ τ ′ (cid:0) ( γ ′ k ) l (cid:1) . (3.9)For a positive integer l ≤ m , if λ ′ is sufficiently close to λ , then with respect to τ ,the Reeb orbit ( γ ′ ) l has rotation number close to ( a/b ) l , and each Reeb orbit ( γ ′ k ) l for k = 0 has rotation number close to al . Then by (2.4) and (3.9) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) CZ Iτ ′ ( γ ′ ) − m X l =1 alb − X k =0 m k X l =1 al (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m + X k =0 m k . It follows from this and (3.7) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) CZ Iτ ′ ( γ ′ ) − ab ( m + m ) − a X k =0 ( m k + m k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m. (3.10)Finally, by (2.4) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X l =1 CZ τ ( γ l ) − ab ( m + m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m. Then by (3.7) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X l =1 CZ τ ( γ l ) − ab ( m + m ) − a (2 m + 1) X k =0 m k − ab X k =0 m k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m. (3.11)Combining (3.8), (3.10), and (3.11) gives the desired estimate (3.2).18 .3 Perturbations of degenerate flows To conclude the proof of Proposition 3.3, we now prove Lemma 3.5.As in the statement of the lemma, let γ be a simple Reeb orbit of λ of period T , and let λ n = f n λ , where f n → C . Let ϕ t and ϕ tn denote the time t flowsof the Reeb vector fields for λ and λ n , respectively. Let p ∈ γ , and let P γ : ξ p → ξ p denote the linearized return map (1.2). Lemma 3.8.
Let { ( p n , T n ) } n =1 ,... be a sequence in Y × (0 , ∞ ) satisfying:(c1) φ T n n ( p n ) = p n → p .(c2) φ T n /jn ( p n ) = p n for all integers j ≥ and all n .(c3) T n → T ∞ ∈ [0 , ∞ ) .Then one of the following alternatives holds:(a1) T ∞ = T .(a2) T ∞ = T d for some integer d ≥ , and the eigenvalues of P γ that are roots ofunity of degree d generate multiplicatively all roots of unity of order d .Proof. This is a special case of a result of Bangert [3, Prop. 1] for C flows.In the situation of Lemma 3.5, more can be said: Corollary 3.9.
Suppose that the eigenvalues of P γ are real and positive. Let { ( p n , T n ) } be a sequence satisfying conditions (c1), (c2), and (c3) of Lemma 3.8.Then alternative (a2) does not hold.Proof. The only root of unity that can be an eigenvalue of P γ is 1, hence the set ofeigenvalues of P γ does not generate multiplicatively the group of roots of unity oforder d when d ≥ Proof of Lemma 3.5.
Part (a) follows from Corollary 3.9.To prove part (b), fix a diffeomorphism Φ from the tubular neighborhood N of γ to ( R /T Z ) × C such that γ corresponds to ( R /T Z ) × { } , the Reeb vector field R n of λ n is transverse to the C fibers for n sufficiently large (assume that n is thislarge below), and the derivative of Φ in the normal direction along γ agrees withthe trivialization τ . We omit the diffeomorphism Φ from the notation below anddenote points in N using the coordinates ( t, z ) ∈ ( R /T Z ) × C .By part (a), by taking n large enough we can assume that α n and β n have thesame period as γ . After reparametrization, the Reeb orbit α n is given by a map R /T Z −→ ( R /T Z ) × C ,t ( t, ˆ α n ( t ))19here ˆ α n : R /T Z → C . Likewise the Reeb orbit β n is described by a map ˆ β n : R /T Z → C . We have ℓ τ ( α n , β n ) = wind( ˆ α n − ˆ β n ) , (3.12)where the right hand side denotes the winding number of the loopˆ α n − ˆ β n : R /T Z −→ C ∗ . We now compute the right hand side of (3.12). There is a convex neighborhood U of 0 in C such that if n is sufficiently large (which we assume below), then thefollowing two conditions hold: First, ˆ α n (0) , ˆ β n (0) ∈ U . Second, for each t ∈ [0 , T ]there is a well-defined map ψ tn : U → C such that for z ∈ U , the flow of theReeb vector field R n starting at (0 , z ) first hits { t } × C at the point ( t, ψ tn ( z )). Inparticular, it follows from the definition thatˆ α n ( t ) = ψ tn ( ˆ α n (0)) , ˆ β n ( t ) = ψ tn ( ˆ β n (0)) . (3.13)Now consider the derivative of ψ tn , which we denote by Dψ tn : U × C −→ C . By (3.13), we may apply the fundamental theorem of calculus to the function s → ψ tn (cid:16) s ˆ α n (0) + (1 − s ) ˆ β n (0) (cid:17) to obtainˆ α n ( t ) − ˆ β n ( t ) = Z Dψ tn (cid:16) s ˆ α n (0) + (1 − s ) ˆ β n (0) , ˆ α n (0) − ˆ β n (0) (cid:17) ds. (3.14)By the convergence of λ n , if n is sufficiently large (which we assume below), thenthe amount that Dψ tn ( s ˆ α n (0) + (1 − s ) ˆ β n (0) , · ) rotates any vector as compared to Dψ tn (0 , · ) can be made arbitrarily small. It follows that the integrand in (3.14), andhence ˆ α n ( t ) − ˆ β n ( t ), has positive inner product with Dψ tn (0 , ˆ α n (0) − ˆ β n (0)) . Thus,the right hand side of (3.12) differs by less than 1 / π ) of the path[0 , T ] −→ C ∗ ,t Dψ tn (cid:16) , ˆ α n (0) − ˆ β n (0) (cid:17) . (3.15)The rotation number of the linearized Reeb flow along γ differs from the rotationnumber of any individual vector by less than 1 /
2. Hence, by again applying conver-gence of the λ n as above, if n is sufficiently large then the rotation number of thepath (3.15) differs by less than 1 / a . Since the right hand side of (3.12) is aninteger which differs by less than 3 / a , it must equal a .20 Two simple Reeb orbits implies nondegenerate
We now prove Theorem 1.2. Throughout this section, assume that Y is a closedconnected three-manifold, and λ is a contact form on Y with exactly two simpleReeb orbits, γ and γ , of periods T and T respectively. Lemma 4.1.
The classes [ γ i ] ∈ H ( Y ) and c ( ξ ) ∈ H ( Y ; Z ) are torsion.Proof. We use a similar argument to the proof of [12, Thm. 1.7].Since every oriented three-manifold is spin, we can choose Γ ∈ H ( Y ) such that c ( ξ ) + 2 PD(Γ) = 0 ∈ H ( Y ; Z ). By Proposition 2.16, there exists a U -sequence { σ k } ≥ for Γ. Write c k = c σ k ( Y, λ ) ∈ R .By Proposition 2.18(a), we have c k = m ,k T + m ,k T for some nonnegative integers m ,k and m ,k , and furthermore m ,k [ γ ] + m ,k [ γ ] = Γ ∈ H ( Y ) . (4.1)By Proposition 2.18(b), the sequence { c k } is strictly increasing. It then followsfrom (4.1) that there are infinitely many integral linear combinations of [ γ ] and [ γ ]that have the same value in H ( Y ). Thus the kernel of the map Z −→ H ( Y ) , ( m , m ) m [ γ ] + m [ γ ] (4.2)has rank at least 1.In fact, the kernel of the map (4.2) must have rank at least 2; otherwise c k would grow at least linearly in k , contradicting the sublinear growth in the VolumeProperty in Proposition 2.18(c). It follows that [ γ ] and [ γ ] are torsion. Since c ( ξ ) + 2 PD(Γ) = 0, we deduce that c ( ξ ) is also torsion. If m , m are nonnegative integers, we use the notation γ m γ m to indicate the orbitset { ( γ , m ) , ( γ , m ) } , with the element ( γ i , m i ) omitted when m i = 0. Write α = γ m γ m . If [ α ] = 0, then it follows from Remark 2.13 and Lemma 4.1 that I ( α ) ∈ Z is defined. We now give an explicit computation of I ( α ), following [32, § Definition 4.2.
Define the linking number ℓ ( γ , γ ) := ℓ ( γ l , γ l ) l l ∈ Q (4.3)21here l and l are positive integers such that l i [ γ i ] = 0 ∈ H ( Y ), and on the righthand side ℓ denotes the usual integer-valued linking number of disjoint nullhomolo-gous loops. Definition 4.3.
For i = 1 ,
2, define the
Seifert rotation number φ i ∈ R as follows.Let τ be a trivialization of ξ over γ i . Let θ i,τ = θ τ ( γ i ) ∈ R denote the rotationnumber of γ i with respect to τ . Let l i be a positive integer such that l i [ γ i ] = 0.Define Q i,τ := Q τ ( γ l i i ) l i ∈ Q , (4.4)where Q τ ( γ l i i ) is shorthand for Q τ ( γ l i i , ∅ , Z ) for any Z ∈ H ( Y, γ l i i , ∅ ). Note that Q i,τ does not depend on Z by (2.3), and it does not depend on l i either because Q τ isquadratic in the relative homology class. Finally, define φ i := Q i,τ + θ i,τ ∈ R . (4.5)The number φ i does not depend on the choice of trivialization τ , by the change oftrivialization formulas in [26, § Remark 4.4.
When γ i is nullhomologous, one can alternately describe φ i as follows.Let Σ be a Seifert surface spanned by γ i . There is a distinguished homotopy class oftrivialization τ ′ of ξ over γ i , the “Seifert framing”, for which the normal vector to Σhas winding number zero around γ i . We have Q i,τ ′ = 0 by [26, Lem. 3.10]. It thenfollows that φ i = θ τ ′ ( γ i ), In the general case when γ i is rationally nullhomologous,one can similarly describe φ i as the rotation number with respect to a rationalframing of γ i determined by a rational Seifert surface. Lemma 4.5. If m [ γ ] + m [ γ ] = 0 ∈ H ( Y ) , then I ( γ m γ m ) = φ m + φ m + 2 ℓ ( γ , γ ) m m + O ( m + m ) . (4.6) Proof.
Let l i be a positive integer with l i [ γ i ] = 0. Similarly to (4.4), define c i,τ := c τ ( γ l i i ) l i ∈ Q , where c τ ( γ l i i ) is shorthand for c τ ( γ l i i , ∅ , Z ) for any Z ∈ H ( Y, γ l i i , ∅ ). Then c i,τ doesnot depend on Z by (2.2) since c ( ξ ) is torsion, and it is independent of the choiceof l i because c τ is linear in the relative homology class Z .It follows from the definition of the ECH index and the facts that c τ and Q τ arelinear and quadratic in the relative homology class (see [32, § I ( γ m γ m ) = X i =1 ( m i c i,τ + m i Q i,τ ) + 2 m m ℓ ( γ , γ ) + X i =1 m i X k =1 ( ⌊ kθ i,τ ⌋ + ⌈ kθ i,τ ⌉ ) , Q i,τ and θ i,τ are as in (4.5). Plugging in the approximation X i =1 m i X k =1 ( ⌊ kθ i,τ ⌋ + ⌈ kθ i,τ ⌉ ) = X i =1 m i θ i,τ + O ( m + m )then gives (4.6). Lemma 4.6.
We have φ i = T i vol( Y, λ ) , ℓ ( γ , γ ) = T T vol( Y, λ ) . Proof.
Both sides of the above equations are invariant under scaling the contactform by a positive constant, so we may assume without loss of generality thatvol(
Y, λ ) = 1.By Proposition 2.16 and Lemma 4.1, we can choose a U -sequence { σ k } k ≥ forΓ = 0. Since the U map has degree −
2, there is a constant C ∈ Z such that foreach positive integer k the class σ k has grading C + 2 k . By Proposition 2.18(a), foreach positive integer k there are nonnegative integers m ,k , m ,k such that c σ k ( Y, λ ) = m ,k T + m ,k T . (4.7)By the Volume Property of Proposition 2.18(c), we have2 k = ( m ,k T + m ,k T ) + o ( k ) . (4.8)Fix k and write α k = γ m ,k γ m ,k . If λ ′ is a sufficiently C close nondegenerateperturbation of λ , then by the same compactness argument that proves Proposi-tion 2.18(a), there is an orbit set α ′ k close to α k as a current such that I ( α ′ k ) = C +2 k (and also R α ′ k λ ′ is close to c σ k ( Y, λ ), although we do not need this). By Proposi-tion 3.1, we have C + 2 k = I ( α k ) + O ( m ,k + m ,k ) . Combining this with Lemma 4.5, we get2 k = φ m ,k + φ m ,k + 2 ℓ ( γ , γ ) m ,k m ,k + O ( m ,k + m ,k ) . (4.9)Putting together (4.8) and (4.9), we obtain( φ − T ) m ,k + ( φ − T ) m ,k + 2( ℓ ( γ , γ ) − T T ) m ,k m ,k = O ( m ,k + m ,k ) . Consequently, if the sequence ( m ,k /m ,k ) k ≥ has an accumulation point S ∈ [0 , ∞ ],then the line in the x, y plane of slope S through the origin is in the null space ofthe quadratic form f ( x, y ) = ( φ − T ) x + ( φ − T ) y + 2( ℓ ( γ , γ ) − T T ) xy.
23o complete the proof of the lemma, it now suffices to show that the sequence( m ,k /m ,k ) k ≥ has at least three accumulation points, as then the quadratic form f must vanish identically. We claim that in fact this sequence has infinitely manyaccumulation points.If the sequence has only finitely many accumulation points S , . . . , S n , then forevery ε >
0, there exists
R > m ,k , m ,k ) is contained in theunion of the disk x + y ≤ R and the cones around the lines of slope S , . . . , S n with angular width ε .Since lim k →∞ c σ k /k = 2, and since the points ( m ,k , m ,k ) are pairwise distinctby Proposition 2.18(b), by equation (4.7), it follows that for large L , the number ofpoints ( m ,k , m ,k ) contained in the triangle T x + T y ≤ L is approximately L / δ > L sufficiently large, the fraction oflattice points in the triangle T x + T y ≤ L, x ≥ , y ≥ m ,k , m ,k ) k ≥ is at least δ . This gives a contradiction if ε in the previousparagraph is chosen sufficiently small. Proof of Theorem 1.2.
The ratio T /T is irrational by [11, Thm. 1.3]. Also, ℓ ( γ , γ )is rational by the definition (4.3). It then follows from Lemma 4.6 that φ and φ are irrational.By (4.5), since Q i,τ is rational, it follows that the rotation number θ i,τ is ir-rational. Then P γ i has eigenvalues e ± πiθ i,τ , so the Reeb orbits γ i are irrationallyelliptic. As explained in § γ i are nondegenerate, so λ is nondegenerate. To finish up, we now prove Theorem 1.5.To prepare for the proof, recall that if Y is a closed oriented three-manifold, if ξ is a contact structure on Y with c ( ξ ) = 0 ∈ H ( Y ; Z ), and if γ is a nullhomologoustransverse knot, then the self-linking number sl( γ ) ∈ Z is defined to be the differencebetween the Seifert framing (see Remark 4.4) and the framing given by a globaltrivialization of ξ . In the notation of § γ ) = Q τ ( γ ) − c τ ( γ ) (5.1)where τ is any trivialization of ξ | γ .Now suppose that γ above is a simple Reeb orbit. Let φ ( γ ) ∈ R denote therotation number of γ with respect to the Seifert framing as in § θ ( γ ) ∈ R The proof is simple: If T /T is rational, so that T and T are both integer multiples of a singlenumber, then Proposition 2.18(b) implies that the spectral invariants associated to a U -sequencegrow at least linearly, contradicting the Volume Property. γ with respect to a global trivialization of ξ . Also letCZ( γ ) = ⌊ θ ( γ ) ⌋ + ⌈ θ ( γ ) ⌉ ∈ Z denote the Conley-Zehnder index of γ with respect to a global trivialization. Itfollows from (5.1) that φ ( γ ) = θ ( γ ) + sl( γ ) . (5.2) Proof of Theorem 1.5.
By Corollary 1.3, γ and γ are the core circles of a genus oneHeegaard splitting of Y . It follows from this topological description that ℓ ( γ , γ ) =1 /p . Part (a) of the theorem then follows from Lemma 4.6.To prove part (b), suppose first that Y = S . We know from Theorem 1.2that λ is nondegenerate and there are no hyperbolic Reeb orbits. Then ξ is tight,because otherwise [20, Thm. 1.4] would give a hyperbolic Reeb orbit. Moreover, itfollows from [23, Thm. 1.3], combined with [20, Thm. 1.4] and the fact that thereare no Reeb orbits with CZ = 2, that one of the simple Reeb orbits, say γ , satisfiessl( γ ) = − γ ) = 3, and is the binding of an open book decompositionwith pages that are disk-like global surfaces of section for the Reeb flow. The returnmap on a page preserves an area form with finite total area, hence it has a fixedpoint by Brouwer’s Translation Theorem. This fixed point corresponds to the simpleReeb orbit γ , which is transverse to the pages of the open book. Since on S \ γ the tangent spaces of the pages define a distribution that is isotopic to ξ keepingtransversality with the Reeb direction, we get sl( γ ) = −
1. Since CZ( γ ) = 3, wehave θ ( γ ) ∈ (1 , φ ∈ (0 , φ φ = 1, so φ >
1. By equation (5.2) again we have θ ( γ ) >
2. It follows that all iterates of γ and γ have θ >
1, so λ is dynamicallyconvex.To prove part (b) in the general case, let ˜ λ denote the pullback of the contactform λ to the universal cover S of Y . It follows from the Heegaard decompositionthat γ and γ each have order p in π ( Y ). Consequently e λ has exactly two simpleReeb orbits e γ and e γ , which project to γ and γ as p -fold coverings. By theprevious paragraph, ( S , e λ ) is dynamically convex and tight, and it follows that( Y, λ ) is dynamically convex and universally tight.
References [1] P. Albers, H. Geiges, K. Zehmisch,
Pseudorotations of the 2-disc and Reeb flows on the 3-sphere , arXiv:1804.07129.[2] K. Baker and J. Etnyre,
Rational linking and contact geometry , arXiv:0901.0380.[3] V. Bangert,
On the lengths of closed geodesics on almost round spheres , Math. Z. (1986),549-558.[4] D. Bechara Senior, U. Hryniewicz and P. A. S. Salom˜ao,
On the relation between action andlinking , arXiv:2006.06266.
5] B. Bramham,
Periodic approximations of irrational pseudo-rotations using pseudoholomorphiccurves , Ann. of Math. (2) (2015), 1033–1086.[6] F. Bourgeois, K. Cieliebak, and T. Ekholm,
A note on Reeb dynamics on the tight 3-sphere ,J. Mod. Dyn. (2007), 597–613.[7] E. Cineli, V. Ginzburg and B. G¨urel, Pseudo-rotations and holomorphic curves , Selecta Math-ematica, to appear.[8] V. Colin, P. Dehornoy, and A, Rechtman,
On the existence of supporting broken book decom-positions for contact forms in dimension 3 , arXiv:2001.01448.[9] B. Collier, E. Kerman, B. Reiniger, B. Turmunkh, and A. Zimmer,
A symplectic proof of atheorem of Franks , Compositio Math. (2012), 1969–1984.[10] C. Cornwell,
Berge duals and universally tight contact structures , Topol. and its Appl., (2018), 26–43.[11] D. Cristofaro-Gardiner and M. Hutchings,
From one Reeb orbit to two , J. Diff. Geom. (2016), 25–36.[12] D. Cristofaro-Gardiner, M. Hutchings, and D. Pomerleano,
Torsion contact forms in threedimensions have two or infinitely many Reeb orbits , Geom. Topol. (2019), 3601–3645.[13] D. Cristofaro-Gardiner, M. Hutchings, and V. Ramos, The asymptotics of ECH capacities ,Invent. Math (2015), 187–214.[14] J. Etnyre and R. Ghrist,
Tight contact structures via dynamics , Proc. AMS (2015) no.12, 3697–3706.[15] B. Fayad and A. Katok,
Constructions in elliptic dynamics , Ergodic Theory Dynam. Systems (2004), 1477–1520.[16] J. Franks, Geodesics on S and periodic points of annulus homeomorphisms , Invent. Math. (1992), 403–418.[17] V. Ginzburg and B. G¨urel, Hamiltonian pseudo-rotations of projective spaces , Invent. Math. (2018), 1081–1130.[18] B. G¨urel,
Perfect Reeb flows and action-index relations , Geom. Dedicata (2015), 105–120.[19] H.Hofer, K. Wysocki and E. Zehnder,
A characterization of the tight three sphere , Duke MathJ. (1995), no. 1, 159–226.[20] H.Hofer, K. Wysocki and E. Zehnder, Unknotted periodic orbits for Reeb flows on the three-sphere , Topol. Methods Nonlinear Anal., (1996), no. 2, 219–244.[21] H. Hofer, K. Wysocki and E. Zehnder, A characterization of the tight three sphere II.
Commun.Pure Appl. Math. (1999), no. 9, 1139–1177.[22] K. Honda, On the classification of tight contact structures I , Geom. Topol. (2000), 309–368.[23] U. Hryniewicz and P. A. S. Salom˜ao, On the existence of disk-like global sections for Reebflows on the tight -sphere , Duke Math. J., (2011), no. 3, 415–465.[24] U. Hryniewicz, J. Licata and P. A. S. Salom˜ao, A dynamical characterization of universallytight lens spaces , Proc. LMS (2015), no. 1, 213–269.[25] M. Hutchings,
An index inequality for embedded pseudoholomorphic curves in symplectiza-tions , J. Eu. Math. Soc. (2002), 313–361.[26] M. Hutchings, The embedded contact homology index revisited , New perspectives and chal-lenges in symplectic field theory, CRM Proc. Lecture Notes, 49. AMS (2009), 263–297.
27] M. Hutchings,
Taubes’s proof of the Weinstein conjecture in dimension three , Bull. AMS (2010), 73–125.[28] M. Hutchings, Quantitative embedded contact homology , J. Diff. Geom. (2011), 231–266.[29] M. Hutchings, Lectures notes on embedded contact homology , Contact and symplectic topology,389–484, Bolyai Soc. Math. Stud. , Springer, 2014.[30] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched coveredcylinders I , J. Symplectic Geom. (2007), 43–137.[31] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched coveredcylinders II , J. Symplectic Geom. (2009), 29–133.[32] M. Hutchings and C. H. Taubes, The Weinstein conjecture for stable Hamiltonian structures ,Geom. Topol. (2009), 901–941.[33] M. Hutchings and C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions,II , Geom. Topol. (2013), 2601–2688.[34] K. Irie, Dense existence of periodic Reeb orbits and ECH spectral invariants , J. Mod. Dyn. (2015), 357–363.[35] A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems , Izv. Akad.Nauk SSSR Ser. Mat. (1973), 539–576.[36] P. B. Kronheimer and T. S. Mrowka, Monopoles and three-manifolds , Cambridge UniversityPress, 2008.[37] F. Le Roux and S. Seyfaddini,
The Anosov-Katok method and pseudo-rotations in symplecticdynamics , arXiv:2010.06237.[38] E. Shelukhin,
Pseudo-rotations and Steenrod squares , J. Mod. Dynam. (2020), 289–304.[39] C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture , Geom. Topol. (2007), 2117–2202.[40] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology I , Geom.Topol. (2010), 2497–2581.[41] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology V , Geom.Topol. (2010), 2961–3000.[42] W. Wang, X. Hu, and Y. Long, Resonance identity, stability and multiplicity of closed char-acteristics on compact convex hypersurfaces , Duke Math. J. (2007), 411–462.
Dan Cristofaro-Gardiner
University of California, Santa Cruz, andSchool of Mathematics, Institute for Advanced Study, Princeton NJ, USA [email protected]
Umberto Hryniewicz
RWTH Aachen, Jakobstrasse 2, Aachen 52064, Germany [email protected]
Michael Hutchings
University of California, Berkeley [email protected]
Hui Liu
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, P.R. China [email protected]@whu.edu.cn