Computing the Rabinowitz Floer homology of tentacular hyperboloids
aa r X i v : . [ m a t h . S G ] A ug COMPUTING THE RABINOWITZ FLOER HOMOLOGY OFTENTACULAR HYPERBOLOIDS
A. FAUCK, W. J. MERRY, AND J. WI´SNIEWSKA
Abstract.
We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids Σ ≃ S n + k − × R n − k . Using an embedding of a compactsphere Σ ≃ S k − into the hypersurface Σ, we construct a chain map from theFloer complex of Σ to the Floer complex of Σ . In contrast to the compact case,the Rabinowitz Floer homology groups of Σ are both non-zero and not equal to itssingular homology. As a consequence, we deduce that the Weinstein Conjectureholds for any strongly tentacular deformation of such a hyperboloid. Introduction
Rabinowitz Floer homology is a homology theory for exact contact hypersurfacesin symplectic manifolds. It has various applications in symplectic and contact ge-ometry: it provides obstructions for exact embeddings of contact manifolds, it canbe used to distinguish contact structures, and it gives qualitative information on theReeb flow on a contact manifold. Rabinowitz Floer homology was originally definedby Cieliebak and Frauenfelder in [14] for compact contact type hypersurfaces of ex-act convex symplectic manifolds, and has since been extended to cover more generalcompact hypersurfaces in more general symplectic manifolds. These include certainstable Hamiltonian structures [17], negative line bundles [7], and symplectisationsof hypertight contact manifolds [5]. In a different direction, and more relevantlyfor the present article, the third author constructed in her PhD thesis [41] a classof so-called tentacular
Hamiltonians on R n with non-compact level sets for whichRabinowitz Floer homology is well defined.In the compact case, Rabinowitz Floer homology is eminently computable. For ex-ample, Rabinowitz Floer homology has been completely computed for unit cotangentbundles [16, 2], magnetic cotangent bundles [32], Brieskorn spheres [21], negativeline bundles and certain annulus subbundles thereof [7, 39]. Many of these compu-tations either rely on – or can alternatively be proved by – the intimate relationshipof Rabinowitz Floer homology with symplectic homology [16, 19]. Date : August 19, 2020.
Key words and phrases.
Rabinowitz Floer homology, non-compact hypersurfaces, Weinsteinconjecture.
The non-compact case is rather less tractable. In [35, 34] the class of non-compacthypersurfaces for which the Rabinowitz Floer homology could be defined was ex-tended, and a rather general invariance result was proved. However no explicitcomputations were presented. The goal of the present article is to remedy thisdeficit, by providing a complete calculation of the Rabinowitz Floer homology for aclass of (deformations of) symplectic hyperboloids. As a byproduct, using the afore-mentioned invariance result we establish the validity of the Weinstein Conjecturefor all such non-compact hypersurfaces.Here are the details. Consider a quadratic Hamiltonian H on T ∗ R n of the form H ( z ) := z T Az −
1, where A is a non-degenerate symmetric matrix. The H¨ormanderclassification of symplectic forms [29] tells us that by performing a linear symplecticchange of coordinates, H can be brought into a certain standard form, and moreoverthat these standard forms are classified by the eigenvalues of the Jordan decompo-sition of J A . We are interested in the case where this linear symplectic change ofcoordinates yields a decomposition H ( x, y ) = 12 x T A x − | {z } := H ( x ) + 12 y T A y | {z } := H ( y ) , (1.1)for ( x, y ) ∈ T ∗ R k × T ∗ R n − k , where A is positive definite and J A is hyperbolic. Hereand throughout this paper, we implicitly assume that 1 ≤ k ≤ n − k = 0 , n ). This implies that A has signature ( n − k, n − k ). If wedenote the regular level setsΣ := H − (0) and Σ := H − (0) , (1.2)then Σ is a hyperboloid diffeomorphic to S n + k − × R n − k , whereas Σ is an ellipsoiddiffeomorphic to S k − . Note that Σ is never compact.We say that H is tentacular if it satisfies a number of carefully chosen growthconditions at infinity (see Definition 2.1). For the reader not familiar with theseconditions , for the purposes of this Introduction they may regard “tentacular” tomean: either A is symplectically diagonalisable or all eigenvalues λ i of J A have | Re( λ i ) | >
2. In this case we call Σ a tentacular hyperboloid . Theorem 1.1.
Let H be a tentacular quadratic Hamiltonian of the form (1.1) .Then the Rabinowitz Floer homology of H is given by: RF H ∗ ( H ) := (cid:26) Z ∗ = 1 − n, − k, otherwise.In particular RF H ∗ ( H ) = 0 and RF H ∗ ( H ) = H ∗ + n − (Σ) . Here J stands for the standard complex structure on C n ≃ T ∗ R n . The authors are aware the complement of this set of readers may consist solely of the authors. For k = n − , RF H − n ( H ) = Z ⊕ Z . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 3
The Weinstein Conjecture is obvious for the hyperboloids contained in Theorem1.1. One of the strengths of Floer theoretical calculations, however, is that they areinvariant under controlled perturbations. Therefore as an immediate corollary toTheorem 1.1, we obtain:
Corollary 1.2. If H is a Hamiltonian as in Theorem 1.1 and { Σ s } s ∈ [0 , is a smooth -parameter family of compact perturbations of Σ := H − (0) through strongly ten-tacular hypersurfaces, then each Σ s carries a closed characteristics.Remark . The cases k = 0 and k = n : We assume throughout this paper that1 ≤ k ≤ n −
1. Let us briefly comment on the two other cases. For k = 0, theHamiltonian H from (1.1) is of the form H = H −
1, and the proof of Lemma4.5 below shows that the Hamiltonian vector field of H on Σ has no closed orbits.Thus Theorem 1.1 holds trivially as RF H ( H ) agrees by definition with the Morsehomology of Σ (with shifted degree). Meanwhile if k = n then H = H , and thehypersurface is compact. The Rabinowitz Floer homology for such hypersurfacesvanishes by [14, Thm. 1.2]. The Weinstein Conjecture is trivially false for k = 0,while it holds true for k = n , as proved by Viterbo in [40] for compact contact typehypersurfaces in T ∗ R n . Remark . Symplectic homology has also recently been extended to the non-compact setting. In [18], Cieliebak, Eliashberg and Polterovich define and com-pute the symplectic homology for a certain subclass of the symplectic hyperboloidswe consider here. Their computations are consistent with ours, and we conjecturethat the long exact sequence relating symplectic homology and Rabinowitz Floerhomology [16] extends to the non-compact setting. In a slightly different direction,Ganatra, Pardon and Shende [25] have defined the symplectic homology for a classof Liouville manifolds with boundary – Liouville sectors – using the compactnessmethods of Groman [27]. It seems likely that the hypersurfaces we consider herecan also be fitted into this framework. We hope to discuss this elsewhere.
Outlook:
In the compact world, both Rabinowitz Floer homology and symplec-tic homology have been profitably used to study orderability problems for compactcontact manifolds [23, 5, 6, 8, 13]. Cieliebak, Eliashberg and Polterovich initiatedthe study of orderability problems for non-compact contact manifolds in [18] usingsymplectic homology. Our companion computation shows that Rabinowitz Floerhomology is also well suited to this problem. We will return to these questions in asequel to the present paper.In another direction, we note that our results are consistent (as they should be!)with van den Berg, Pasquotto and Vandervorst’s earlier Weinstein Conjecture [11]results for non-compact hypersurfaces in T ∗ R n , which were based on variationalmethods. These results were later extended to cover unit cotangent bundles ofRiemannian manifolds with flat ends [10, 38]. It is an interesting – albeit, formidable A. FAUCK, W. J. MERRY, AND J. WI´SNIEWSKA – problem to try and generalise the Floer-theoretical methods used in the presentarticle to cover this setting.Finally, Miranda and Oms [33] have very recently used the methods from b -symplectic and contact geometry to study the Weinstein Conjecture for certain non-compact hypersurfaces, including examples in the planar restricted circular three-body problem. Sketch of the proof of Theorem 1.1:
The
Rabinowitz action functional fora Hamiltonian H : T ∗ R n → R associates to a pair ( v, η ) of a loop v : S → T ∗ R n and a real number η its action by A H ( v, η ) := Z S v ∗ λ − η Z S H ( v ( t )) dt. Here, λ is a primitive of ω = dp ∧ dq . The critical set Crit( A H ) of A H consists of pairs( v, η ), such that v ( S ) ⊂ H − (0) and ∂ t v = ηX H ( v ), where X H is the Hamiltonianvector field of H . The positive L gradient equation for this functional, whichwe call the Rabinowitz Floer equations for H , is the following Floer equation for v : R × S → T ∗ R n coupled with an ODE for η : R → R : (cid:18) ∂ s v∂ s η (cid:19) = − J ( v, η, t ) (cid:2) ∂ t v − ηX H ( v ) (cid:3) − R H ( v ) dt ! . The standard counting of rigid solutions of this equation defines a boundary operator ∂ : CF ∗ ( H, f ) → CF ∗− ( H, f ) , on a graded Z -vector space CF ∗ ( H, f ) generated by critical points of an auxiliarycoercive Morse function f on Crit( A H ), which generically is a countable union offinite dimensional manifolds. The Z -grading of CF ∗ ( H, f ) is defined by the trans-verse Conley-Zehnder index of periodic orbits plus the signature index of criticalpoints of f . We use the Z -grading convention of [16], which differs from the Z + grading convention of [14] by a factor of 1 /
2. The homology of the chain com-plex ( CF ∗ ( H, f ) , ∂ ) is called the Rabinowitz Floer homology of H and is denoted by RF H ∗ ( H ). For the Hamiltonians H that we consider in Theorem 1.1, the fact that RF H ∗ ( H ) is well defined and independent of the auxiliary data used to constructit is proved in [35, 34].One can also play the same game with the Hamiltonian H on T ∗ R k , thus yieldinganother Rabinowitz Floer homology RF H ∗ ( H ). This construction is rather easieras Σ is compact (and falls under the remit of the setup originally conceived in[15]). However as Σ is displaceable in T ∗ R k , by [15, Thm. 1.3] the RabinowitzFloer homology of H is not particularly interesting RF H ∗ ( H ) = 0 . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 5
All is not lost though: if one restricts to the subcomplex generated by orbits ( v, η )with η > positive Rabinowitz Floer homology anddenote by
RF H + ∗ ( H ) – then we obtain something non-zero: RF H + ∗ ( H ) = ( Z , ∗ = k, , otherwise.The proof of Theorem 1.1 uses the “hybrid problem” technique pioneered by Ab-bondandolo and Schwarz in their seminal paper [3]. The starting observation is thatany periodic orbit of X H on Σ takes the form γ ( t ) = ( γ ( t ) , , where γ is a periodic orbit of X H on Σ . This sets up an inclusion i : Crit( A H ) ֒ → Crit( A H ) , which restricted to the non-constant orbits gives a bijection between Crit( A H ) \ (Σ ×{ } ) and Crit( A H ) \ (Σ ×{ } ). Despite this relationship between their criticalpoints, there is no obvious relation between the space of negative gradient flow linesfor these two functionals, and hence no reason to hope that i induces a chain map.Nevertheless, we prove that for a particular choice of Morse functions f and f : Theorem 1.5.
There exists a chain map ψ : CF ∗ ( H, f ) → CF ∗ ( H , f ) , which induces a sequence of homomorphisms Ψ , such that the following diagram oftwo long exact sequences commutes: . . . / / H ∗ + n − (Σ) / / Ψ (cid:15) (cid:15) RF H ≥ ∗ ( H ) / / (cid:15) (cid:15) RF H + ∗ ( H ) / / Ψ + (cid:15) (cid:15) . . .. . . / / H ∗ + k − (Σ ) / / RF H ≥ ∗ ( H ) / / RF H + ∗ ( H ) / / . . . Moreover Ψ + is an isomorphism, whereas Ψ is a composition of an isomorphismcoming from the retraction Σ ∼ = S n + k − × R n − k → S n + k − and an Umkehr map. The construction of ψ is based on counting solutions of the following hybridproblem: we consider tuples ( v − , η − , v + , η + ) where v + : [0 , ∞ ) × S → T ∗ R n , η + : [0 , ∞ ) → R ,v − : ( −∞ , × S → T ∗ R k , η − : ( −∞ , → R , are solutions of the Rabinowitz Floer equations, ( v + , η + ) for H and ( v − , η − ) for H ,with prescribed asymptotics at ±∞ and satisfying the following coupling conditionat s = 0: identifying T ∗ R n ∼ = T ∗ R k × R n − k × R n − k , we require v + (0 , t ) = ( v − (0 , t ) , ∗ , , η − (0) = η + (0) . A. FAUCK, W. J. MERRY, AND J. WI´SNIEWSKA
This coupling condition can be seen as a Lagrangian boundary condition for R n +2 k -valued maps on a half-cylinder, from which it follows that the hybrid problem isFredholm. Precompactness of the moduli spaces of solutions ( v − , η − , v + , η + ) withfixed asymptotes is established in Section 5.2. The chain map ψ is then defined bycounting rigid solutions of the hybrid problem. To prove the remaining statementsof Theorem 1.5, we show that automatic transversality holds at stationary solutionsof the hybrid problem. Combining this with sharp energy estimates implies that ifwe order the critical points of A H and A H by increasing action, then the matrixrepresentation of ψ is upper triangular, and moreover the diagonal entries are allequal to 1, except for a single 0 coming from the minimum of f on Σ. From this,Theorem 1.5 follows, and hence so does Theorem 1.1. Acknowledgements:
The first and third authors are supported by the SNF grantPeriodic orbits on non-compact hypersurfaces. We thank Kai Cieliebak for pointingout [18] to us, and for helpful remarks during the preparation of this article.2.
Preliminaries
We begin with a brief discussion of Rabinowitz Floer homology, with a special em-phasis on the non-compact framework. To keep the exposition concise we place our-selves throughout in the linear setting of T ∗ R m . Whilst this restriction is at presentnecessary for the construction of Rabinowitz Floer homology for non-compact hy-persurfaces, this is by no means the case for compact hypersurfaces. We refer thereader to the survey article [4] for a leisurely introduction to the various settingsthat (compact) Rabinowitz Floer homology can be defined. Sign Conventions : Let ω = dp ∧ dq denote the standard symplectic form on T ∗ R m = R m × ( R m ) ∗ . We identify T ∗ R m with the complex vector space ( C m , i )via the map ( q, p ) q + ip , and denote by J the corresponding complex structure. Explicitly, this means that J = (cid:18) − Id 0 (cid:19) . (2.1)Let g J := ω ( · , J · ), so that g J is the real part of the standard Hermitian structure on C m , and hence a Riemannian metric on T ∗ R m . Sometimes it will be necessary toinclude the dimension in our notation, in which case we write ω m , J m and so on. Weuse the sign convention that the symplectic gradient/Hamiltonian vector field X H of a Hamiltonian H : T ∗ R m → R is given by ω ( X H , · ) = − dH , so that the Poisson As a general notational guide to the reader, throughout the rest of this article, we work withcompact hypersurfaces in T ∗ R k and non-compact hypersurfaces in T ∗ R n , where 0 < k < n . In thispreliminary section we treat both cases simultaneously, and hence use the letter m instead. OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 7 bracket of two Hamiltonians on T ∗ R m is given by { F, H } := ω ( X F , X H ). Hamiltonians:
We now introduce the class of Hamiltonians that we work with.A vector field Y on T ∗ R m is said to be a Liouville vector field if d ( ω ( Y, · )) = ω . ALiouville vector field Y is said to be asymptotically regular if k DY ( x ) k ≤ c for somepositive constant c and all x ∈ T ∗ R m . Definition 2.1.
Let H ∗ denote the set of Hamiltonians H on T ∗ R m such that either (c) dH is compactly supported,or the following three axioms are satisfied:(h1) there exists an asymptotically regular Liouville vector field Z and constants c, c ′ >
0, such that dH ( Z )( x ) ≥ c | x | − c ′ , for all x ∈ T ∗ R m ;(h2) (sub-quadratic growth) sup x ∈ T ∗ R m k D H ( x ) k · | x | < ∞ ;(h3) in the neighbourhood of H − (0) exists a coercive function F , such that forall x ∈ H − (0) either { H, F } ( x ) = 0 or { H, { H, F }} ( x ) > , as | x | → ∞ . Remark . Note that if H ∈ H ∗ and h ∈ C ∞ c ( T ∗ R n ) then also H + h ∈ H ∗ .We say that H − (0) is of contact type if there exists an asymptotically regularLiouville vector field Y such that dH ( Y )( x ) > x ∈ H − (0). Note that thisimplies that H − (0) is a smooth hypersurface. Definition 2.3.
Let
H ⊆ H ∗ denote the subset of those Hamiltonians H whichin addition have the property that H − (0) is of contact type. We say that Hamil-tonians H ∈ H satisfying (h1), (h2) and (h3) are strongly tentacular . If we dropthe requirement that the function F in (h3) is coercive then H is called simply tentacular . Remark . The strongly tentacular condition implies that all the periodic orbits of H are contained in a compact set. This may not be the case for tentacular Hamiltoni-ans. Rabinowitz Floer homology is defined for Morse-Bott tentacular Hamiltonianswhose orbits are contained in a compact set. Invariance under compact pertur-bation requires the strongly tentacular condition. This explains why the adjective“strongly” appears in Corollary 1.2 but not in Theorem 1.1.The connection between the definition of strongly tentacular Hamiltonians givenhere and the one presented in the introduction is given by the following result: Proposition 2.5.
Let H be a Hamiltonian of the form (1.1) . Then H belongs to H ,if the Jordan decomposition of J A has m i × m i blocks corresponding to eigenvalues λ i , where each pair ( m i , λ i ) satisfies one of the following conditions:i) m i = 1 and Re( λ i ) = 0 ;ii) m i = 2 and | Re( λ i ) | > √ ; A. FAUCK, W. J. MERRY, AND J. WI´SNIEWSKA iii) m i > and | Re( λ i ) | > .Remark . Proposition 2.5 provides a concrete class of examples for which ourmain results, Theorem 1.1 and Theorem 1.5 are valid. Note that cases i) and iii)correspond to our “definition” of strongly tentacular on page 2. We emphasisehowever that Theorems 1.1 and 1.5 are valid for any tentacular Hamiltonian H ofthe form (1.1) with A positive definite and J A hyperbolic .Below we will outline the construction of Rabinowitz Floer homology groups forHamiltonians H ∈ H . In the case H satisfies (c), this reduces to the originaldefinition presented by Cieliebak and Frauenfelder [14], only specialised to T ∗ R m .Meanwhile for strongly tentacular H , the construction comes from [34]. Complex structures:
Although T ∗ R m comes equipped with a preferred choice J of complex structure, in order to achieve transversality for the various modulispaces used in the definition of Rabinowitz Floer homology, we are forced to workwith generic data. To this end we now introduce a suitable parameter space ofalmost complex structures.Let J denote the space of all compatible almost complex structures J on T ∗ R m .Here compatible means that g J := ω ( · , J · ) is a Riemannian metric on T ∗ R m . Weview J as a pointed space with basepoint J . An easy linear algebra argument (seefor example [12, Prop. 13.1]) shows that J is contractible.Fix an open set V ⊂ T ∗ R m × R , and let J ( V, J ) denote the set of smooth maps( t, η ) J ( · , η, t ) ∈ J , ( t, η ) ∈ S × R , such that J ( x, η, t ) = J , whenever ( x, t ) / ∈ V, and such that sup ( t,η ) ∈ S × R k J ( · , η, t ) k C l < + ∞ , ∀ k ∈ N . (2.2)Here the C l norm is taken with respect to the standard Riemannian metric g J on T ∗ R m . Finally we let J ⋆ denote the union of all spaces J ( V, J ) for V of the formint K × R \ [ − a, a ] for K ⊆ T ∗ R m compact and a >
0, equipped with the colimittopology. We view J ⋆ as another pointed space, with basepoint J . This space isagain easily seen to be contractible. We write J m⋆ , if we have to indicate the dimen-sion of the underlying space T ∗ R m . That is, J A has no purely imaginary eigenvalues. Actually the construction for strongly tentacular Hamiltonians subsumes the compact contacttype hypersurfaces as a special case, see Remark 2.12 below.
OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 9
The Rabinowitz action functional:
The free period action functional—or
Rabinowitz action functional —is defined by A H : C ∞ ( S , T ∗ R m ) × R → R ( v, η ) Z S v ∗ λ − η Z S H ( v ( t )) dt. Here λ is any primitive of ω , for example λ = ( pdq − qdp ). The real number η can be thought of as a Lagrange multiplier version of the area functional fromclassical mechanics. Thus critical points A H are critical points of the area functionalrestricted to the space of loops with H mean value zero. Moreover, since H isinvariant under its own Hamiltonian flow, the mean value constraint can be upgradedto a pointwise constaint, and we obtain: Lemma 2.7.
A pair ( v, η ) is a critical point of A H if and only if t v ( t/η ) is aclosed orbit of X H contained in H − (0) . The Morse-Bott condition:
It is not just the complex structure that needsto be chosen generically. Floer theory also requires us to work with a Hamiltonianwhich satisfies a certain generic non-degeneracy condition. Unlike the case of com-plex structures however, the non-degeneracy condition admits a direct definition.Throughout the rest of this section, we denote by Σ the level set H − (0) of a givenHamiltonian H ∈ H and we denote by Y an asymptotically regular Liouville vectorfield such that dH ( Y ) | Σ > Definition 2.8. (i) We say that the Rabinowitz action functional A H is Morse-Bott if the criticalset of A H is a discrete union of finite dimensional manifolds and for everyconnected component Λ ⊆ Crit( A H ) and every x ∈ Λ T x Λ = ker( ∇ x A H ) . (2.3)(ii) We say that the closed orbits of the flow φ t of X H on Σ are of Morse-Botttype if η is constant on every connected component Λ ⊆ Crit( A H ), and theimage N η of a connected component Λ with period η under the projection( v, η ) v (0) is a closed submanifold of Σ, such that for all x ∈ N η we have T p N η = ker( D p φ η − Id) ∩ T p Σ . (2.4) Remark . The two Morse-Bott conditions are closely related. If A H is a Morse-Bott functional and η is constant on every connected component of Crit( A H ), thenby [34, Lem. 3.3] all closed orbits of the Hamiltonian flow φ t on Σ are of Morse-Botttype. Conversely, if H is defining for Σ, i.e. if dH ( Y ) | Σ ≡
1, and if all periodic For η = 0 this should be interpreted as: t v ( t ) is constant. orbits are of Morse-Bott type then by [20, Lem. 20] the corresponding Rabinowitzaction functional A H is Morse-Bott.The Morse-Bott property of the Rabinowitz action functional is typically achievedby perturbing slightly the Hamiltonian function. However in our case we will cal-culate the Rabinowitz Floer homology of the specific Hamiltonians satisfying (1.1)by hand (so to speak), and therefore in Section 4.3 we will check directly that theseHamiltonians fulfill the Morse-Bott property. The Rabinowitz Floer equation:
Fix J ∈ J ⋆ . The positive L gradientequation ∂ s u = ∇ J A H for u = ( v, η ) is the following Floer equation for v coupledwith an ordinary differential equation for η : (cid:18) ∂ s v∂ s η (cid:19) = − J ( v, η, t ) (cid:2) ∂ t v − ηX H ( v ) (cid:3) − R H ( v ) dt ! . (2.5)A solution u of (2.5) with finite L energy R R k ∂ s u k ds < ∞ , is called a Floertrajectory . Given two distinct components Λ − and Λ + of Crit( A H ), we denote by M (Λ − , Λ + ) the set of all solutions of (2.5) with finite energy and lim s →±∞ u ( s ) ∈ Λ ± .We define ev ± : M (Λ − , Λ + ) → Λ ± to be the evaluation maps:ev − ( u ) := lim s →−∞ u ( s, t ) and ev + ( u ) := lim s → + ∞ u ( s, t ) . (2.6) Definition 2.10.
We say that a couple (
H, J ) ∈ H × J ⋆ is regular if it satisfies thefollowing two conditions:i) The Rabinowitz action functional A H is Morse-Bott, and the closed orbitsof X H on Σ are of Morse-Bott type;ii) For every pair of connected components Λ − , Λ + ⊆ Crit( A H ) the associatedmoduli space M (Λ − , Λ + ) is a smooth finite dimensional manifold withoutboundary.By [34, Lem. 8.7] the set of such regular couples is comeagre in H × J ⋆ . Fromnow on we assume that ( H, J ) is regular. Next, we introduce flow lines with cas-cades following [24]: Consider a Morse function f : Crit( A H ) → R such that f restricts to a coercive function on Σ. Fix a Riemannian metric on Crit( A H ) hav-ing a Morse-Smale gradient flow φ t . For z ∈ Crit( f ), we denote by W sf ( z ) and W uf ( z ) the (un)stable manifolds with respect to φ t . A flow line with k ≥ z − , z + ∈ Crit( f ) belonging to distinct connected components is a tuple( u , ..., u k , t , ..., t k − ), where each u i is a non-stationary Floer trajectory (cf. (2.5)), When H satisfies (c) this is condition is automatic. OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 11 such that φ t i ◦ ev + ( u i ) = ev − ( u i +1 ) , i = 1 , ..., k − , ev − ( u ) ∈ W uf ( z − ) , ev + ( u k ) ∈ W sf ( z + ) . We denote the set of all flowlines with k cascades from z − to z + as M k cas ( z − , z + ).There is a natural R k action on M k cas ( z − , z + ) given by u i ( · ) u i ( a + · ) and wedefine the space of all flow lines with cascades from z − to z + by M ( z − , z + ) := [ k ≥ (cid:16) M k cas ( z − , z + ) . R k (cid:17) . By taking such a union, we obtain another smooth manifold without boundary,which moreover is compact “up to breaking”.If z − and z + are critical points of f belonging to the same component of Crit( A H )we set M ( z − , z + ) to be the quotient W u ( z − ) ∩ W s ( z + ) (cid:14) R arising from the natural R action by translation. Grading:
We use the Z -grading convention of [16], which differs from the Z + grading convention of [14] by a factor of 1/2. Explicitly, for z ∈ Crit( f ) we define µ ( z ) := µ trCZ ( z ) + µ σ ( z ) + 12 , (2.7)where µ trCZ is the transverse Conley-Zehnder index and µ σ is the signature indexdefined by µ σ ( x ) = 12 (cid:0) dim W sf ( x ) − dim W uf ( x ) (cid:1) . (2.8)For a regular pair ( H, J ), one hasdim M ( z − , z + ) = µ ( z + ) − µ ( z − ) − . The compactness up to breaking property alluded to above tells us that when µ ( z + ) − µ ( z − ) = 1 the space M ( z − , z + ) is compact, and hence a finite set. Wedenote by n ( z − , z + ) its parity. The Rabinowitz Floer complex:
We define the Z -vector space CF ( H, f ) asthe set of formal sums of the form P z ∈ S z , where S ⊂ Crit( f ) is a (possibly infinite)set satisfying the Novikov finiteness condition (cid:8) z ∈ S (cid:12)(cid:12) A H ( z ) > a (cid:9) < ∞ ∀ a ∈ R . (2.9)We denote by CF k ( H, f ) ⊂ CF ( H, f ) those sums P S z with µ ( z ) = k for all z ∈ S .We turn CF ∗ ( H, f ) into a chain complex by defining ∂z + := X n ( z − , z + ) z − , where the sum is taken over all critical points z − with µ ( z + ) = µ ( z − ) + 1, andthen extending by linearity. Compactness up to breaking implies that ∂ = 0, and a continuation argument tells us that the resulting Rabinowitz Floer homology
RF H ( H ) is independent of the auxiliary data used to define it. We refer the readerto [14] (when H satisfies (c)) and [34] (when H is strongly tentacular) for details. Remark . If H ∈ H satisfies condition (c) then the Rabinowitz Floer homologygroups only depend on H through its zero level set Σ, and thus we could write RF H ∗ (Σ) instead (although we won’t). For strongly tentacular H this need not bethe case. However for H of the form (1.1) the main result of this article, Theorem1.1, shows that the Rabinowitz homology groups only depend on the “compact part”of the zero level set. Remark . In fact, when Σ is compact, there is considerable freedom in the choiceof the Hamiltonian H realising Σ as its zero level set. In this section for historicalreasons we have concentrated on the case where dH is compactly supported, butone could also use a quadratic Hamiltonian [3]. In particular, for the hypersurfaceΣ from (1.2) we can use the Hamiltonian H from (1.1) to compute its RabinowitzFloer homology. This fact will be used in the proof of Theorem 1.5. This alsoshows that the compact case is subsumed by the strongly tentacular case, i.e. if H ∈ H satisfies condition (c) then there exists a strongly tentacular H ′ ∈ H suchthat H − (0) = ( H ′ ) − (0). Positive Rabinowitz Floer homology:
The action functional A H provides an R − filtration on CF (cid:0) A H , f (cid:1) as follows: For t ∈ R denote the complex generated bycritical points of action ≤ t by CF ≤ t ( H, f ) := (cid:26)P z ∈ S z ∈ CF ( H, f ) (cid:12)(cid:12)(cid:12)(cid:12) sup z ∈ S A H ( z ) ≤ t (cid:27) . (2.10)The boundary operator does not increase the action, i.e. ∂ (cid:16) CF ≤ t ∗ +1 ( H, f ) (cid:17) ⊆ CF ≤ t ∗ ( H, f ) . (2.11)The positive Rabinowitz Floer homology RF H + ( H ) is the homology of the followingquotient complex generated by the critical points with positive action: CF + ( H, f ) := CF ( H, f ) . CF ≤ ( H, f )= (cid:8) x ∈ Crit( f ) (cid:12)(cid:12) A H ( x ) > (cid:9) ⊗ Z . (2.12)The associated boundary operator ∂ + is induced by ∂ on the quotient, which iswell-defined as ∂ reduces the action (cf. (2.11)). Geometrically, ∂ + is defined bycounting flow lines with cascades where both endpoints have strictly positive action.Analogously, one defines RF H − ( H ) as the homology of CF < ( H, f ). OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 13
More generally, CF + ( H, f ) fits into the following short exact sequence of com-plexes induced by action filtration:0 → CF ( H, f ) = CF ≤ /CF < → CF ≥ ( H, f ) = CF/CF < → CF + ( H, f ) = CF/CF ≤ → , (2.13)where the boundary operator of each complex is induced by ∂ and well-defined dueto (2.11). We hence obtain the following long exact sequence in homology · · · → RF H ∗ ( H ) → RF H ≥ ∗ ( H ) → RF H + ∗ ( H ) → RF H ∗− ( H ) → . . . (2.14)where RF H ∗ ( H ) ∼ = H ∗ + n − (Σ), as ∂ counts in the zero-action window only flowlines with no cascades, i.e. Morse flow lines on Σ.The positive Rabinowitz Floer homology is independent of the auxiliary choicesused to define it and invariant under compact perturbations. When H satisfiescondition (c), these facts follow from [14, Cor. 3.8]. The proof for strongly tentacular H goes along similar lines, but the argument is slightly more involved, and has notyet appeared in the literature. Therefore in the next section we supply the fulldetails. Similar statements apply to the other variants RF H ≥ , RF H − , etc.3. Invariance of the positive Rabinowitz Floer homology
The goal of this section is to show the following invariance of
RF H + ( H ): Theorem 3.1.
For a strongly tentacular Hamiltonian H the positive RabinowitzFloer homology RF H + ( H ) does not depend on the almost complex structure J oron the Morse-Smale pair ( f, g ) on Crit( A H ) . Moreover if { H s } is a -parameterfamily of Hamiltonians in the affine space of compactly supported perturbations of H , then RF H + ( H s ) is constant along { H s } . As seen above, the positive Rabinowitz Floer homology
RF H + ( H ) is generatedby non-constant periodic orbits and thus vanishes in the absence of closed charac-teristics.Throughout this section we will denote by ( M, ω ) any exact symplectic mani-fold. In order to prove Theorem 3.1, we need compactness results for homotopiesof Hamiltonians and almost complex structures, which are stronger than the onesproved in [35]. To obtain these results, we recall the notion of uniform continuityof (PO), as introduced in [34]:
Definition 3.2.
We say that H satisfies property (PO) if for any fixed actionwindow, all non-degenerate periodic orbits are contained in a compact subset of M .Moreover, we say that property (PO) is uniformly continuous at H if there existsan open neighbourhood O ( H ) of 0 in C ∞ c ( M ) and an exhaustion of M by compact sets { K n } n ∈ N , such that for every n ∈ N and every h ∈ O ( K n ) = O ( H ) ∩ C ∞ ( K n ),whenever ( v, η ) ∈ Crit( A H + h ) and 0 < (cid:12)(cid:12)(cid:12) A H + h ( v, η ) (cid:12)(cid:12)(cid:12) ≤ n, then v ( S ) ⊆ K n ∩ ( H + h ) − (0). Remark . By [34, Lem. 8.4] every strongly tentacular Hamiltonian satisfies theaxiom of uniform continuity of (PO).Define: Crit + ( A H ) := Crit( A H ) ∩ (cid:0) A H (cid:1) − ((0 , + ∞ )) . Remark . Let H : T ∗ R n → R be a Hamiltonian on ( T ∗ R n , ω n ). If ( v, η ) ∈ Crit + ( A H ), then the following adaptation of [30, Lem. 2.2] to the Rabinowitz Floerframework holds: | η | sup v ( S ) k DX H k ≥ π. Lemma 3.5.
Let H be a Hamiltonian on an exact symplectic manifold ( M, ω ) ,such that Σ := H − (0) is of contact type. If property (PO) is uniformly continuousat H then there exists an open neighbourhood O ( H ) of in C ∞ c ( M ) , such thatfor every compact set K ⊆ M there exists a constant δ > , such that for all h ∈ O ( H ) ∩ C ∞ c ( K )inf n A H + h ( v, η ) (cid:12)(cid:12)(cid:12) ( v, η ) ∈ Crit + ( A H + h ) o > δ. Proof.
We argue by contradiction. Let e O ( H ) be the open neighbourhood fromproperty (PO). Without loss of generality, we can assume that K = K n for some K n in the exhaustion of M . Suppose that there exists a sequence h n ∈ e O ( H ) ∩ C ∞ c ( K )and a sequence ( v n , η n ) ∈ Crit + ( A H + h n ), such that lim n →∞ h n = 0 andlim n →∞ A H + h n ( v n , η n ) = 0 . (3.1)Property (PO) is uniformly continuous at H , hence all the periodic orbits v n arecontained in the compact set K . By requiring certain bounds on the derivatives for h ∈ e O ( H ), we can find an open subset O ( H ) ⊂ e O ( H ) where the following uniformbounds are satisfied: sup h ∈O ( H ) ∩ C ∞ c ( K ) sup K k DX H + h k < + ∞ . (3.2)As Σ is of contact type there exists a Liouville vector field Y , such that dH ( Y ) > U of Σand of a constant δ ′ >
0, such thatinf U ∩ K dH ( Y ) > δ ′ > . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 15
By possibly shrinking O ( H ), we can assume that for all h ∈ O ( H ) ∩ C ∞ c ( K ) wehave ( H + h ) − (0) ∩ K ⊆ U ∩ K andinf U ∩ K d ( H + h )( Y ) > δ ′ . Thus we obtain A H + h n ( v n , η n ) = η n Z S d v n ( H + h n )( Y ) > η n δ ′ n →∞ η n = 0. On the other hand, as ( v n , η n ) ∈ Crit + ( A H + h n ), we get uniform bounds on the derivative of v n v n ( t ) ∈ K, k ∂ t v n ( t ) k ≤ | η n | sup K k X H + h n k ∀ n ∈ N , t ∈ S . This allows us to use the Arzel´a-Ascoli theorem, which yields a convergent subse-quence (which we denote the same)lim n →∞ ( v n , η n ) = ( x, ∈ Σ ×{ } . Let x ∈ V ⊆ M, ϕ : V → R m be a coordinate chart around x . For n big enoughwe can assume v n ( S ) ⊆ V . Then x n := ϕ ◦ v n satisfies the respective Hamiltonianequation ∂ t x n = η n Dϕ − ( X H + h n )( x n ). Hence ∂ tt x n = η n D (cid:0) Dϕ − ( X H + h n ) (cid:1) ( ∂ t x n ).However, from Remark 3.4 we can infer that η n (cid:13)(cid:13) D (cid:0) Dϕ − ( X H + h n ) (cid:1) (cid:13)(cid:13) ≥ π, which together with (3.2) contradicts lim n →∞ η n = 0. (cid:3) Fix a strongly tentacular Hamiltonian H : T ∗ R n → R and let O ( H ) be an openneighbourhood of 0 in C ∞ c ( T ∗ R n ), such that all Hamiltonians from H + O ( H ) arestrongly tentacular and O ( H ) is as in Lemma 3.5. Fix sets K ⊆ V ⊆ T ∗ R n , suchthat K = ∅ is compact and V is open and precompact. Let δ > y ∈ (0 , δ ) and let 0 < ε , ˜ c < ∞ be constants as in [35, Lem. 2.1]depending only on H .Finally, fix h ± ∈ O ( H ) ∩ C ∞ c ( K ) and let R ∋ s ( H s , J s ) ∈ (cid:0) H + O ( H ) (cid:1) × J ( V× ( R \ [ − y , y ]) , J ) , be a smooth homotopy of Hamiltonians and ω -compatible almost complex struc-tures, constant outside of [0 , H s = H + h − for s ≤ H s = H + h + for s ≥
1. In this setting we formulate the following lemma:
Lemma 3.6.
Let ( v ± , η ± ) be a pair of critical points of A H + h ± . If the homotopy { H s , J s } s ∈ R satisfies k ∂ s H s k L ∞ < min ( ε (cid:0) ˜ cε + k J k / L ∞ (cid:1) , δ c (1 + δ ) ) , (3.3) and if u is a solution to the equation ∂ s u = ∇ J s A H s ( u ) with lim s →±∞ u ( s ) = ( v ± , η ± ) ,then A H + h − ( v − , η − ) > δ implies A H + h + ( v + , η + ) > δ .Proof. The following proof is an adjustment of the proof of [14, Cor. 3.8] to thesetting of strongly tentacular Hamiltonians and is based on the results proven in[35, Prop. 3.3]. Abbreviate a := A H + h − ( v − , η − ) , b := A H + h + ( v + , η + ) . By assumption a ≥ δ . Let u = ( v, η ) ∈ C ∞ (cid:0) R , C ∞ ( S , T ∗ R n ) × R (cid:1) be a solution tothe equation ∂ s u = ∇ J s A H s ( u ) with lim s →±∞ u ( s ) = ( v ± , η ± ). Then our settingsatisfies the assumptions of [35, Prop. 3.3].First assume additionally | b | ≤ a . Then b − a ≤ k η k L ∞ ( R ) ≤ (cid:18) ˜ c (cid:16) max {| a | , | b |} + 1 (cid:17) + b − aε k J k L ∞ (cid:19) ≤
87 ˜ c ( a + 1) ≤ a ˜ c (cid:18) δ (cid:19) (3.4)On the other hand, by equation (3.5) from the proof of [35, Prop. 3.3], we have that k ∂ s u k L ( R × S ) ≤ k J k L ∞ ( b − a + k η k L ∞ k ∂ s H s k L ∞ ) . (3.5)Combined with (3.4) and (3.3), we obtain the following estimate: b ≥ a − k η k L ∞ ( R ) k ∂ s H s k L ∞ ≥ a (cid:18) − ˜ c (cid:16) δ (cid:17) k ∂ s H s k L ∞ (cid:19) ≥ a. In particular, A H + h − ( v − , η − ) = b >
0. By assumption h − ∈ O ( H ) ∩ C ∞ c ( K ), henceby Lemma 3.5 we can conclude that A H + h − ( v − , η − ) ≥ δ . This implies the resultprovided | b | ≤ a .Now assume b < − a . Then b − a ≤ k η k L ∞ ( R ) ≤ (cid:18) ˜ c (cid:16) max {| a | , | b |} + 1 (cid:17) + b − aε k J k L ∞ (cid:19) ≤
87 ˜ c (1 − b ) ≤ − b ˜ c (cid:18) δ (cid:19) , (3.6)where the last inequality comes from the assumption − b ≥ a ≥ δ . Combining itwith (3.5) and (3.3), we obtain the following inequality: a ≤ b + k η k L ∞ ( R ) k ∂ s H s k L ∞ ≤ b (cid:18) − ˜ c (cid:16) δ (cid:17) k ∂ s H s k L ∞ (cid:19) ≤ b < − a, which contradicts the assumption a > δ >
0. That excludes b < − a and proves thelemma. (cid:3) OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 17
Proof of Theorem 3.1:
Let us fix a Hamiltonian H ∈ H . By [34, Lem. 8.8] and [34,Lem. 8.9] for every two close enough regular couples ( H + h , J ), ( H + h , J ) with h , h ∈ C ∞ c ( T ∗ R n ) , J , J ∈ J ⋆ and a homotopy Γ = { H + h s , J s } s ∈ R of compactlyperturbations h s of Hamiltonian H and almost complex structures J s ∈ J ⋆ one canconstruct a homomorphism φ Γ : CF ∗ ( H + h , f ) → CF ∗ ( H + h , f ) , which is defined by counting the perturbed flow lines with cascades (cf. [9, Sec.11.1], [34, Lem. 7.2]). Moreover, φ Γ satisfies φ Γ ◦ ∂ = ∂ ◦ φ Γ , thus induces ahomomorphism on the homology levelΦ Γ : RF H ( H + h , J ) → RF H ( H + h , J ) . Moreover, by Lemma 3.6 we know that if the two regular couples ( H + h , J ) , ( H + h , J ) are close enough, then the preimage of CF + ∗ ( H + h , f ) under φ Γ lies in CF + ∗ ( H + h , f ). In other words φ Γ (cid:0) CF ≤ ( H + h , f ) (cid:1) ⊆ CF ≤ ( H + h , f ) . (3.7)This together with (2.11) allows us to infer that the restriction φ Γ+ of φ Γ to CF + ∗ ( H + h , f ) constructed by counting the perturbed flow lines with cascades be-tween critical points with positive action, commutes with the respective boundaryoperators ∂ + and ∂ + and thus induces a homomorphism Φ Γ+ : RF H + ( H + h , J ) → RF H + ( H + h , J ).Now, we show that Φ Γ+ is an isomorphism, using the fact that Φ Γ is an isomor-phism by [9, Prop. 11.2.9]. Via the inverse homotopy Γ − = { H + h − s , J − s } s ∈ R ,one can construct analogously a homomorphism φ Γ − : CF ∗ ( H + h , f ) → CF ∗ ( H + h , f ) . As this homomorphism also satisfies φ Γ − ◦ ∂ = ∂ ◦ φ Γ − and φ Γ − (cid:0) CF ≤ ( H + h , f ) (cid:1) ⊆ CF ≤ ( H + h , f ) , (3.8)it induces a homomorphism Φ Γ − + : RF H + ( H + h , J ) → RF H + ( H + h , J ).Finally, by [9, Prop. 11.2.9] there exists a homomorphism S : CF ∗ ( H + h , f ) → CF ∗ +1 ( H + h , f ) , satisfying φ Γ − ◦ φ Γ − Id = S ◦ ∂ + ∂ ◦ S (3.9)and S (cid:0) CF ≤ ( H + h , f ) (cid:1) ⊆ CF ≤ ( H + h , f ) , (3.10)where (3.10) comes from applying Lemma 3.6 once again this time to the flow-lines with cascades coming from a homotopy of homotopies (cf. [9, Thm. 11.3.11]).By combining (2.11), (3.7), (3.8) and (3.10) we infer that the restriction of S to CF + ( H + h , f ) also satisfies (3.9) with φ Γ+ and φ Γ − + , thus establishing that Φ Γ+ and Φ Γ − + are also isomorphisms on the homology level. This proves that
RF H + ( H ) is invariant under small enough compactly sup-ported perturbations of H and (via h = h =0) is invariant of J or ( f, g ). (cid:3) Tentacular hyperboloids
H¨ormander classification.
Let A be a non-degenerate, quadratic, symmetricmatrix and consider the non-degenerate quadratic Hamiltonian H on ( T ∗ R n , ω n ) H ( x ) := 12 x T Ax − . (4.1)The hypersurface Σ := H − (0) is diffeomorphic to S l − × R n − l , where ( l, n − l ) isthe signature of A . A hyperboloid is the 0-level set of a quadratic Hamiltonian H as in (4.1) with 1 ≤ l ≤ n − Remark . Note that every hyperboloid is a hypersurface of contact type, as theradial Liouville vector field Y = x∂ x satisfies dH x ( Y ) = H ( x ) + 1 = 1 (cid:12)(cid:12) Σ > X H coincides with the Reeb vector fieldcorresponding to ι Y ω n (cid:12)(cid:12) Σ , as X H ∈ ker ω n (cid:12)(cid:12) Σ and ι Y ω n ( · , X H ) = dH ( Y ) = 1 on Σ.In [34, Sec. 9] the last author together with Pasquotto and Vandervorst introducedthe notion of symplectic hyperboloids , which are equivalence classes of the set ofhyperboloids under the action of the linear symplectic group Sp( R n ). A symplectichyperboloid is called a tentacular hyperboloid if it admits a strongly tentacularrepresentative of the form (4.1). Proposition 2.5 above provides plenty of examplesof tentacular hyperboloids.According to H¨ormander [29, Thm. 3.1] each equivalence class is uniquely deter-mined by the eigenvalues of J A . Observe that if λ is an eigenvalue of J A with acorresponding m × m block in the Jordan decomposition, then its additive inverseand its complex conjugate are also eigenvalues of J A each with a m × m block.By H¨ormander classification, T ∗ R n splits into a direct sum of symplectic sub-spaces S i . In other words, after a symplectic change of coordinates we can assumethat the matrix A is a diagonal block matrix consisting of matrices A i = A | S i .Each matrix A i corresponds to a m × m block with an eigenvalue λ in the Jordandecomposition of J A and it is determined as follows:(a) if Im( λ ) = 0, then S i := T ∗ R m and A i is a 2 m × m block matrix (cid:0) BB T (cid:1) where B = { b j,k } mj,k =1 with b j,k := | λ | if j = k, j = k + 1 , . The invariance for any compact perturbation follows by splitting a given perturbation into asequence of smaller perturbations (see [14], p. 275)
OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 19
Thus B is a m × m matrix with | λ | ’s on the diagonal and with 1’s under thediagonal for m >
1. The signature of A i is ( m, m ) and the correspondingHamiltonian H i : T ∗ R m → R is given by H i ( q, p ) := | λ | m X j =1 q j p j + m − X j =1 q j +1 p j . (b) if Re( λ ) = 0 , Im( λ ) = 0, then S i := T ∗ R m and A i is an 4 m × m blockmatrix A i := (cid:0) BB T (cid:1) where B = { b j,k } mj,k =1 with b j,k := | Re( λ ) | if j = k, | Im( λ ) | if j ∈ N and k = j − , −| Im( λ ) | if k ∈ N and j = k − , k = j + 2 , . The signature of A i is (2 m, m ) and H i : T ∗ R m → R is given by H i ( q, p ) := m − X j =1 q j p j +2 + | Re( λ ) | m X j =1 q j p j + | Im( λ ) | m X j =1 (cid:0) q j p j − − q j − p j (cid:1) . (c) if Re( λ ) = 0, then S i := T ∗ R m and for some γ = ± A i is an 2 m × m blockmatrix (cid:0) B B P (cid:1) , where B = { b j,k } mj,k =1 with b j,k := γ | Im( λ ) | if j = k = m +12 ∈ N , − j = k = m +22 ∈ N , | Im( λ ) | if j = k and j + k = m + 1 , − j = k and j + k = m + 2 , . and B P is the reflection of B with respect to the anti-diagonal. That is, B P = { b δ ( j,k ) } mj,k =1 with δ ( j, k ) = ( m +1 − j, m +1 − k ). The signature of A i is ( m, m ) if m is even and ( m + γ, m − γ ) if m is odd, and the correspondingHamiltonian H i : T ∗ R m → R is equal to H i ( q, p ) := γ | Im( λ ) | m X j =1 (cid:0) q j q m +1 − j + p j p m +1 − j (cid:1) − m − X j =1 (cid:0) q j +1 q m +1 − j + p j p m − j (cid:1) . The H¨ormander classification determines a unique (up to permutation of theblocks) representative of the equivalence class of non-degenerate matrices underthe action of Sp( R n ). In particular, after a linear symplectic change of coordi-nates, every quadratic Hamiltonian of the form (4.1) can be represented as a sumof Hamiltonians H i : S i → R ,H ( x ) = X i H i ( x i ) − , H i ( x i ) := 12 x Ti A i x i , x i ∈ S i , with A i either of type (a), (b) or (c). Remark . Observe, that the Hamiltonian H i ( x i ) = x Ti A i x i for a matrix A i either of type (a) or (b) vanishes on the 0-section χ = R m × { } ⊆ T ∗ R m = S i .In other words, if a quadratic Hamiltonian H : T ∗ R m → R as in (4.1) comes froma non-degenerate, symmetric matrix A and all the eigenvalues of the matrix J m A have non-zero real part, then there exists a Lagrangian subspace of T ∗ R m on whichthe Hamiltonian H + 1 vanishes.A classical result [31, Lem. 2.43] shows that a positive definite matrix is sim-plectically diagonalizable. More precisely, by applying a linear symplectic changeof coordinates to a positive definite matrix, we can obtain a diagonal matrix withcouples of positive real numbers µ , µ , . . . , µ k , µ k , µ j ≤ µ j +1 on the diagonal. Inthe words of the H¨ormander classification: a positive definite matrix corresponds to k block matrices of type (c) and dimension 2. Therefore, we can conclude that if A is a positive definite matrix, then all the eigenvalues of J A are purely imaginaryand equal ± iµ j , j = 1 , . . . , k .Given x ∈ R n , write x = ( x ′ , x ′′ ) with respect to the splitting R n = R k ⊕ R n − k .Define the symplectic splitting T ∗ R n ∼ = T ∗ R k × T ∗ R n − k via the symplectomorphism σ : (cid:0) T ∗ R n , ω n (cid:1) → (cid:0) T ∗ R k × T ∗ R n − k , ω k ⊕ ω n − k (cid:1) σ ( q, p ) := (cid:0) ( q ′ , p ′ ) , ( q ′′ , p ′′ ) (cid:1) . (4.2)Then σ (Σ) ∩ (cid:0) T ∗ R k × { } (cid:1) = Σ × { } . Remark . From the arguments presented above we can deduce the followingproperties of the matrices and Hamiltonians satisfying (1.1):(i) The eigenvalues of J k A are purely imaginary;(ii) Without loss of generality we can assume that A is a diagonal matrix withcouples of positive real numbers µ , µ , . . . , µ k , µ k , µ j ≤ µ j +1 on the diago-nal, which correspond to eigenvalues ± iµ j of J k A ;(iii) The matrix A has signature ( n − k, n − k );(iv) Define Σ := H − (0) , Σ := H − (0) andΣ := Σ ∩ σ − ( T ∗ R k × S ) , (4.3)where S denotes the n − k -dimensional subspace of T ∗ R n − k spanned by theeigenvectors of A corresponding to positive eigenvalues. Then there existdiffeomorphisms φ : Σ → S k − and φ : Σ → S n + k − × R n − k , such that φ (Σ ) = S n + k − × { } .(v) Without loss of generality, we can assume that the Hamiltonian H satisfies H ( x,
0) = 0 for all x ∈ R n − k , i.e. H vanishes on the 0-section χ = R n − k ×{ } ⊆ T ∗ R n − k . In particular, σ − (Σ × χ ) ⊆ Σ . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 21 (vi) For S defined as above we have S ∩ χ = { } . Remark . We emphasise (cf. Remark 2.6) that all points of Remark 4.3 only usethat H is of the form (1.1) with A positive definite and J A hyperbolic.The embedding T ∗ R k ∼ = T ∗ R k × { } n − k ) ֒ → T ∗ R n induces an inclusion j : Σ ֒ → Σ , (4.4)and a retraction r : Σ → Σ . Recall that the Umkehr map j ! : H ∗ (Σ ) → H ∗ + k − n (Σ )is defined as the composition j ! := ( − ⌢ [Σ ]) ◦ j ∗ ◦ ( − ⌢ [Σ ]) − . For later purposes, we observe that the composition j ! ◦ r ∗ : H ∗ (Σ) → H ∗ + k − n (Σ ) , (4.5)is an isomorphism for ∗ = n + k − Periodic orbits.
In this subsection we discuss the properties of non-degenerateperiodic orbits of quadratic Hamiltonians H ( x, y ) = H ( x ) + H ( y ) on T ∗ R n = T ∗ R k ⊕ T ∗ R n − k satisfying (1.1). In particular, we establish an action and degree-preserving 1-to-1 correspondence between periodic orbits of X H on Σ = H − (0) andperiodic orbits of X H on Σ = H − (0). Lemma 4.5.
There is an action preserving -to- correspondence of the periodicorbits of X H on Σ with the periodic orbits of X H on Σ . Explicitly, for η = 0 v : S → Σ , ∂ t v = ηX H ( v ) , if and only if σ ( v ) = ( v , , where v : S → Σ , ∂ t v = ηX H ( v ) . Moreover A H ( v ) = A H ( v ) = η , for the Rabinowitz action functionals of H on ( T ∗ R n , ω n ) and H on ( T ∗ R k , ω k ) respectively.Proof. For a Hamiltonian H as in (4.1) the Hamiltonian flow of X H is given by φ t ( x ) = exp( t J n A ) x. Consequently, ( v, η ) ∈ Crit( A H ) if and only if v (0) ∈ Σ and φ η ( v (0)) = v (0) or,equivalently, v (0) ∈ ker (cid:0) exp( η J n A ) − Id (cid:1) . By assumption the matrix A can bepresented with respect to the symplectic splitting (4.2) as a block matrix: A := (cid:18) A A (cid:19) , where A and A are as in (1.1). As a result,ker (cid:0) exp( η J n A ) − Id (cid:1) = ker (cid:0) exp( η J k A ) − Id (cid:1) ⊕ ker (cid:0) exp( η J n − k A ) − Id (cid:1) . A simple calculation shows that det (cid:0) exp( η J n − k A ) − Id (cid:1) = 0 if and only if e ηµ = 1for some eigenvalue µ of J n − k A . However, by assumption all the eigenvalues of J n − k A have non-zero real parts, hence e ηµ = 1 for all η = 0.Consequently, ( v, η ) ∈ Crit( A H ) and η = 0 if and only if v (0) ∈ Σ and σ ◦ v (0) =( w ,
0) for some w ∈ ker (cid:0) exp( η J k A ) − Id (cid:1) . If we define v ( t ) := φ ηt ( w ) then( v , η ) ∈ Crit( A H ), as σ (Σ) ∩ (cid:0) T ∗ R k ×{ } (cid:1) = Σ × { } .Finally, recall from Remark 4.1 that Y = x∂ x is a Liouville vector field for ω n and satisfies dH x ( Y ) = 1 for all x ∈ Σ. Hence, we have for ( v, η ) ∈ Crit (cid:0) A H (cid:1) A H ( v, η ) = Z λ n ( ∂ t v ) = η Z ω n ( Y, X H ) = η Z dH ( Y ) = η, (4.6)where λ n := ι Y ω n is the Liouville form for Y . The same arguments for Y = x∂ x on T ∗ R k show that A H ( v , η ) = η . This concludes the proof. (cid:3) Corollary 4.6.
Denote by {± iµ l } kl =1 , µ l > the eigenvalues of J k A . Then CritVal (cid:0) A H (cid:1) = CritVal (cid:0) A H (cid:1) = k [ l =1 πµ l Z . Proof.
By Lemma 4.5 it suffices to prove the second equality. By the proof of Lemma4.5, we know that ( v , η ) ∈ Crit (cid:0) A H (cid:1) if and only if v ( t ) = exp( tη J k A ) w and w ∈ Σ ∩ ker (cid:0) exp( η J k A ) − Id (cid:1) . (4.7)As ker (cid:0) exp( η J k A ) − Id (cid:1) = { } if and only if there exists an eigenvalue ± iµ l of J k A , such that ηµ l ∈ π Z , and as A H ( v , η ) = η , the corollary follows. (cid:3) Remark . We define the following subspaces of the critical setsΛ η := (cid:8) ( v, y ) ∈ Crit (cid:0) A H (cid:1) (cid:12)(cid:12) y = η (cid:9) , Λ η := (cid:8) ( v, y ) ∈ Crit (cid:0) A H (cid:1) (cid:12)(cid:12) y = η (cid:9) . (4.8)In particular Λ = Σ × { } and Λ = Σ × { } . Meanwhile for η = 0, we have byLemma 4.5 A H (Λ η ) = A H (Λ η ) = η andΛ η = (cid:8) ( σ − ( v , , η ) (cid:12)(cid:12) ( v , η ) ∈ Λ η (cid:9) ∼ = Λ η . As Σ is diffeomorphic to S k − , we obtain from (4.7) for η ∈ CritVal( A H ) that Λ η is diffeomorphic to S m − , where m := dim ker (cid:0) exp( η J k A ) − Id (cid:1) .Next, we show that the correspondence between the periodic orbits of X H on Σand the periodic orbits of X H on Σ is not only action- but also degree-preserving. Proposition 4.8. If ( v, η ) , η = 0 is a periodic orbit of X H on Σ ⊆ T ∗ R n with σ ( v ) = ( v , and ( v , η ) a periodic orbit of X H on Σ ⊆ T ∗ R n , then µ trCZ ( v, η ) = µ trCZ ( v , η ) . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 23
Here µ trCZ denotes the transverse Conley-Zehnder index, which we will denote inthe following proof by µ ξ CZ , as it is only evaluated on the contact structure ξ . Proof.
Let H be a quadratic Hamiltonian as in (4.1) and H − (0) = Σ. The radialLiouville vector field Y = x∂ x defines a contact structure ξ = ker (cid:0) ι Y ω (cid:12)(cid:12) T Σ (cid:1) on Σby Remark 4.1. Writing L ω for the symplectic complement of a subspace L , we havefor x ∈ Σ: ξ x = { v ∈ T x Σ | ω n ( Y, v ) = 0 } = { v ∈ T x T ∗ R n | ω n ( Y, v ) = 0 , dH ( v ) = 0 } = (span { X H , Y } ) ω . Consequently, ξ ω = span { X H , Y } . If L X H denotes the Lie-derivative along X H , weobtain by Cartan’s formula and Remark 4.1 that L X H ι Y ω = d ( ι X H ι Y ω n ) + ι X H d ( ι Y ω n ) = d ( dH ( Y )) + ι X H ω n = d ( H +1) − dH = 0 . Consequently, the Hamiltonian flow φ t of X H preserves ι Y ω n . As φ t preserves also dH and ω n , we find that the vector fields X H , Y , the bundles ξ, ξ ω and the splitting T ( T ∗ R n ) = T R n = ξ ⊕ ξ ω over Σ are also preserved by φ t .Let γ be a closed characteristic on Σ. Then we have γ ∗ T R n = γ ∗ ξ ⊕ γ ∗ ξ ω . As the Hamiltonian flow preserves this splitting, it follows by the product property[37] that the Conley-Zehnder index of γ is the sum: µ T R n CZ ( γ ) = µ ξ CZ ( γ ) + µ ξ ω CZ ( γ ) . As ( X H , Y ) ◦ γ provides a trivialisation of γ ∗ ξ ω and as φ t preserves ( X H , Y ), we findthat Dφ t | ξ ω expressed in this trivialisation is a path of identity matrices and hence µ ξ ω CZ ( γ ) = µ CZ (Id) = 0 = ⇒ µ ξ CZ ( γ ) = µ T R n CZ ( γ ) . The Hamiltonian flow of a Hamiltonian as in (4.1) is given by φ t ( x ) = exp( t J n A ) x and Dφ t = exp( t J n A ) . For a closed characteristic γ with period η we can take Φ( t ) = Id for all t as thetrivialization Φ : [0 , η ] × R n → γ ∗ ( T R n ). With respect to this trivialization µ ξ CZ ( γ ) = µ T R n CZ ( γ ) = µ CZ (exp( t J n A )) . Naturally, R n = R k ⊕ R n − k ) is a splitting into symplectic subspaces. By assump-tion, the matrix A can be written as a block matrix with respect to this splitting: A := (cid:18) A A (cid:19) , where A and A are as in (1.1). This way the Conley-Zehnder index of γ is a sum: µ ξ CZ ( γ ) = µ CZ (cid:0) exp( t J k A ) (cid:1) + µ CZ (cid:0) exp( t J n − k A ) (cid:1) . By Lemma 4.5, γ is also a characteristic on Σ := H − (0). Repeating the argumentspresented above yields that ξ := ker( ι Y ω k ) is a contact structure on Σ and that µ ξ CZ ( γ ) = µ CZ (cid:0) exp( t J k A ) (cid:1) . Thus it suffices to show that µ CZ (cid:0) exp( t J n − k A ) (cid:1) = 0. The Conley-Zehnder indexof the path of matrices exp( t J n − k A ) , t ∈ [0 , η ] is by definition µ CZ (cid:0) exp( t J n − k A ) (cid:1) = 12 sgn( A ) + 12 sgn (cid:18) A (cid:12)(cid:12)(cid:12) ker (cid:0) exp( η J n − k A ) − Id (cid:1)(cid:19) + X t ∈ (0 ,η ) , det(exp( t J n − k A ) − Id)=0 sgn (cid:18) A (cid:12)(cid:12)(cid:12) ker (cid:0) exp( t J n − k A ) − Id (cid:1)(cid:19) (4.9)First observe that the first term vanishes, since by assumption A has signature 0.To analyse the rest of the terms, we will calculate the crossings , i.e. those t ∈ R , suchthat det (cid:0) exp( t J n − k A ) − Id (cid:1) = 0. A simple calculation shows that t is a crossingof exp( t J n − k A ) if and only if e tµ = 1 for some eigenvalue µ of J n − k A . However,by assumption all the eigenvalues of J n − k A have non-zero real parts, hence e tµ = 1for all t = 0. Therefore, exp( t J n − k A ) has no other crossings than 0 and all termsin (4.9) vanish giving µ CZ (cid:0) exp( t J n − k A ) (cid:1) = 0. (cid:3) Morse-Bott property.
In this section we show that the Morse-Bott condi-tions (2.3) and (2.4) are satisfied for Hamiltonians H and H as in (1.1). Lemma 4.9.
For Hamiltonians H and H on ( T ∗ R k , ω k ) or ( T ∗ R n , ω n ) as in (1.1) the periodic orbits of X H and X H are of Morse-Bott type and the Rabinowitz actionfunctionals A H and A H are Morse-Bott.Proof. We only give the proof for H , the proof for H is analogous.First, we show that Crit( A H ) is a discrete union of connected manifolds Λ. Let A = (cid:0) A A (cid:1) be the block matrix such that H ( x, y ) = (cid:0) xy (cid:1) T A (cid:0) xy (cid:1) −
1. Let µ j bethe eigenvalues of the matrix J k A and let Λ η be the set of pairs ( v, η ) such that v is an η -periodic orbit of X H (see (4.8)). Recall that Λ η is diffeomorphic to a sphere S m − or Σ and that we have by Corollary 4.6 and Remark 4.7 thatCrit (cid:0) A H (cid:1) = [ η ∈ S kj =1 2 πµj Z Λ η . Next, recall from Remark 4.1 that quadratic Hamiltonians H are defining for thehypersurface Σ = H − (0), as the Liouville vector field Y = x∂ x satisfies dH ( Y ) | Σ =1. This allows us to apply [20, Lem. 20] and conclude that A H is Morse-Bott if theperiodic orbits of X H are of Morse-Bott type.To prove this last property consider the projection ( v, η ) v (0) and for η ∈ Crit( A H ) denote by N η the image of Λ η under this projection. It suffices to show OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 25 for all η that N η ⊆ Σ is a closed submanifold and that for p ∈ N η it holds that T p N η = ker (cid:0) d p H (cid:1) ∩ ker (cid:0) D p φ η − Id (cid:1) , where φ η denotes the time η flow of X H . Both conditions follow from (4.7): N η = π A H (Λ η ) = Σ = H − (0) ∩ ker (cid:0) exp( η J n A ) | {z } = Dφ η − Id (cid:1) = ⇒ T p N η = ker( d p H ) ∩ ker (cid:0) D p φ η − Id (cid:1) . (cid:3) The hybrid problem
In the previous section we have shown that there is a 1-to-1, action and degreepreserving correspondence between periodic orbits of X H on Σ and of X H on Σ ,which will allows us to define a chain map ψ : CF ∗ ( H, f ) → CF ∗ ( H , f ) . However, to ensure that ψ induces a homomorphismΨ : RF H ∗ ( H ) → RF H ∗ ( H ) , we need to verify that it commutes with the boundary operators, which is the subjectof this section.Let Λ ⊆ Crit( A H ) and Λ ⊆ Crit( A H ) be connected components and fix J ∈ J k⋆ and J ∈ J n⋆ , such that the pairs ( H , J ) and ( H, J ) are regular in the sense ofDefinition 2.10. To construct ψ , we consider the following moduli spaces of pairs ofhalf Floer trajectories of the Hamiltonians H and H : Definition 5.1.
An element of M hyb (Λ , Λ) is a pair ( u , u ), such that u = ( v , η )and u = ( v, η ) with v : ( −∞ , × S → T ∗ R k , η : ( −∞ , → R ,v : [0 , ∞ ) × S → T ∗ R n , η : [0 , ∞ ) → R , satisfying the Rabinowitz Floer equations ∂ s u − ∇ J A H ( u ) = 0 , ∂ s u − ∇ J A H ( u ) = 0 , (5.1)together with the limit conditionslim s →−∞ u ( s ) ∈ Λ , lim s → + ∞ u ( s ) ∈ Λ , (5.2)and the coupling conditions σ ( v (0 , t )) = (cid:0) v (0 , t ) , ( ∗ , n − k ) (cid:1) , η (0) = η (0) , (5.3)where σ is the symplectomorphism defined in (4.2). Below we will analyse the moduli space M hyb (Λ , Λ). In subsection 5.1 we ex-plore what happens when A H (Λ) = A H (Λ ); in subsection 5.2 we prove uniform L ∞ -bounds on the elements of M hyb (Λ , Λ) and in subsection 5.3 we show that M hyb (Λ , Λ) has the structure of a smooth manifold.5.1.
Stationary solutions.
For ( u , u ) ∈ M hyb (Λ , Λ) we define its energy as E ( u , u ) := Z −∞ k ∂ s u k ds + Z ∞ k ∂ s u k ds ≥ . (5.4) Lemma 5.2.
For ( u , u ) ∈ M hyb (Λ , Λ) , its energy satisfies E ( u , u ) = A H (Λ) − A H (Λ ) . In particular, if M hyb (Λ , Λ) = ∅ , then A H (Λ) ≥ A H (Λ ) .Proof. At first, we have by (5.1) and (5.2) that E ( u , u ) = Z −∞ k∇ J A H ( u ( s )) k ds + Z ∞ k∇ J A H ( u ( s )) k ds = Z −∞ dds A H ( u ( s )) ds + Z ∞ dds A H ( u ( s )) ds = A H ( u (0)) − A H (Λ ) + A H (Λ) − A H ( u (0)) . Secondly, we have for ( v , η ) ∈ C ∞ ( S , χ ) × R that A H ( v , η ) = Z S λ ( ∂ t v ) − η Z S H ( v ) = 0 . The first integral vanishes, as the primitive λ = pdq of ω n − k vanishes on the 0-section χ := R n − k × { } ⊆ T ∗ R n − k . The second integral vanishes, as H vanisheson χ by (v) of Remark 4.3. Using (5.3), we have u (0) = ( σ − ( v (0) , v (0)) , η ) and u (0) = ( v (0) , η ) with v (0) ∈ C ∞ ( S , χ ), and therefore we find A H ( u (0)) = A H ( u (0)) = ⇒ E ( u , u ) = A H (Λ) − A H (Λ ) . (cid:3) We formulate the next lemma using the notation from (4.8):
Lemma 5.3.
For η ∈ CritVal( A H ) = CritVal( A H ) , η = 0 the moduli space M hyb (Λ η , Λ η ) consists of stationary solutions, of the form u ( s, t ) := ( v ( t ) , η ) , ∀ ( s, t ) ∈ ( −∞ , × S ,u ( s, t ) := ( σ − ( v ( t ) , , η ) , ∀ ( s, t ) ∈ [0 , + ∞ ) × S , where ( v, η ) ∈ Λ η . For η = 0 , M hyb (Λ , Λ ) consists of stationary solutions of the form u ( s, t ) := ( x, , ∀ ( s, t ) ∈ ( −∞ , × S ,u ( s, t ) := ( σ − ( x, y ) , , ∀ ( s, t ) ∈ [0 , + ∞ ) × S , where ( x, y ) ∈ Σ × χ . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 27
Proof. If A H (Λ) = A H (Λ ), then E ( u , u ) = 0 by Lemma 5.2 for every ( u , u ) ∈M hyb (Λ , Λ). As a result M hyb (Λ , Λ) consists of stationary solutions of the form u ( s, t ) := ( v ( t ) , η ) , ∀ ( s, t ) ∈ ( −∞ , × S ,u ( s, t ) := ( σ − ( v ( t ) , v ( t )) , η ) , ∀ ( s, t ) ∈ [0 , + ∞ ) × S , where by (5.2) and (5.3) it holds( v , η ) ∈ Λ , ( σ − ( v , v ) , η ) ∈ Λ , v ( t ) ∈ χ ∀ t ∈ S . On the other hand, by Lemma 4.5 we know that if A H (Λ) = A H (Λ ) = 0, thenΛ = (cid:8) ( σ − ( v , , η ) (cid:12)(cid:12) ( v , η ) ∈ Λ (cid:9) . Combining the two facts above proves the first claim. Similarly, we obtain thesecond claim by observing that Σ × χ ⊆ Σ × H − (0) ⊆ σ (Σ). (cid:3) Bounds.
One of the crucial steps in constructing the homomorphism between
RF H ( H ) and RF H ( H ) is to establish L ∞ -bounds on M hyb (Λ , Λ).To formulate the L ∞ -bounds in Proposition 5.4 and Lemma 6.3 we introducethe following notation: for a compact subset N ⊆ Σ and a connected componentΛ ⊆ Crit( A H ) we denote C ( A H , N ) := n x ∈ Crit( A H ) (cid:12)(cid:12)(cid:12) |A H ( x ) | > x ∈ N × { } o , N hyb (Λ , N ) := (cid:26) ( u , u ) ∈ M hyb (Λ , Σ ×{ } ) (cid:12)(cid:12)(cid:12) lim s → + ∞ u ( s ) ∈ N (cid:27) , For a pair of connected components (Λ , Λ) ⊆ Crit( A H ) × (cid:0) Crit( A H ) (cid:15) (Σ ×{ } ) (cid:1) wedenote N hyb (Λ , Λ) := M hyb (Λ , Λ).
Proposition 5.4.
Consider a compact subset N ⊆ Σ and a pair of connected com-ponents (Λ , Λ) ⊆ Crit( A H ) × C ( A H , N ) , such that a ≤ A H (Λ ) ≤ A H (Λ) ≤ b .Then the corresponding moduli space N hyb (Λ , Λ) admits uniform L ∞ -bounds, whichdepend only on a, b and N .Proof. First observe that by adapting the result in [35, Prop. 6.2] to the hybridproblem we obtainsup n k u ± ( ± s ) k L ( S ) × R (cid:12)(cid:12)(cid:12) ( u − , u + ) ∈ N hyb (Λ , Λ) , s ≥ o < + ∞ . (5.5)Moreover, there exists ε >
0, such thatsup (cid:26) ( u − , u + ) ∈ N hyb (Λ , Λ) , s ≤ , k u − ( s ) k L ∞ × R k∇ J A H ( u − ( s )) k < ε (cid:27) < + ∞ , (5.6)sup (cid:26) ( u − , u + ) ∈ N hyb (Λ , Λ) , s ≥ , k u + ( s ) k L ∞ × R k∇ J A H ( u + ( s )) k < ε (cid:27) < + ∞ . (5.7)and all the bounds depend only on a, b and N . Next, we use a maximum principle argument to prove uniform bounds on thefragment of the Floer trajectories, where the action derivation is greater than ε .For ( u − , u + ) = (( v − , η − ) , ( v + , η + )) ∈ N hyb (Λ , Λ) define a function r : ( −∞ , × S → R ,r ( s, t ) := 14 (cid:0) k v − ( s, t ) k + k v + ( − s, t ) k (cid:1) . Using the function F defined as k · k we can rewrite r as r ( s, t ) = F ◦ v − ( s, t ) + F ◦ v + ( − s, t ) . The function F is plurisubharmonic, which means that − dd C F = ω .By (5.6) and (5.7) we know that there exists c >
0, such that for every ( s, t ) ∈ r − ([ c , + ∞ )) we have k∇ J A H ( u − ( s )) k ≥ ε , k∇ J A H ( u + ( − s )) k ≥ ε . (5.8)On the other hand, by Lemma 5.2 we have the following bound: b − a ≥ E ( u − , u + ) = Z −∞ k∇ J A H ( u − ( s )) k ds + Z ∞ k∇ J A H ( u + ( s )) k ds. Therefore, for every connected component Ω ⊆ r − ([ c , + ∞ )) there exist s , s ≥ ⊆ [ s , s ] × S and | s − s | ≤ b − a ε . Since (5.5) and (5.8) are both satisfied, we can use [35, Thm. 7.1] and conclude thatthere exists a function f : ( −∞ , × S → R , such that △ r ≥ f and the L -norm of f is bounded on Ω and the bound depends only on a, b and N .Now, if ∂ Ω ∩ ( { }× S ) = ∅ , then we can apply the Aleksandrov Maximum Prin-ciple [26, Thm. 9.1] to deduce that there exists c >
0, such thatsup Ω r ≤ sup ∂ Ω r + c k f k L (Ω) = c + c k f k L (Ω) < + ∞ , (5.9)and the bound depends only on a, b and N .However, if ∂ Ω ∩ (cid:0) { }× S (cid:1) = ∅ then we need to check an additional assumption.More precisely we would like to show that ∂ s r ≥ ∂ Ω ∩ ( { }× S ) to be able toapply the Aleksandrov Maximum Principle for half cylinders [2, Thm. 2.8] for (5.9).Denote σ ( v + ) =: ( v , v ) ∈ C ∞ (cid:16) R × S , T ∗ R k (cid:17) × C ∞ (cid:16) R × S , T ∗ R n − k (cid:17) , where σ is the splitting from (4.2). Then (with a slight abuse of notation of F ) ∂ s r = dF ( ∂ s v − ) − dF ( ∂ s v ) − dF ( ∂ s v ) . (5.10) OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 29
By the coupling condition (5.3) we have η − (0) = η + (0) and ∂ t v − (0 , t ) = ∂ t v (0 , t ).Since J ∈ J k⋆ and J ∈ J n⋆ , we can assume without loss of generality that for( s, t ) ∈ r − ([ c , + ∞ )), one has J ( u − ( s, t ) , t ) ≡ J k , J ( u + ( − s, t ) , t ) ≡ J n . Therefore by (5.1) we have ∂ s v (0) = ∇ J k A H ( v (0)) = ∇ J k A H ( v − (0)) = ∂ s v − (0) . As a result, the first and second term in (5.10) cancel each other.To calculate the last term let us make the following observation: dF ( ∂ s v (0)) = dF (cid:0) ∇A H ( v (0)) (cid:1) = − d C F (cid:0) ∂ t v (0) − η + (0) X H ( v (0)) (cid:1) . (5.11)By (5.3) we have that v (0) lies in the 0-section χ and d C F (cid:12)(cid:12) T χ = 12 ( qdp − pdq ) (cid:12)(cid:12) T χ = 0 , thus the first term in (5.11) vanishes. For the second term of (5.11) observe that d C F ( X H ) = h∇ H , ∇ F i = H . As H also vanishes on χ , the whole third term in (5.10) vanishes and we concludethat ∂ s r = 0 on { } × S . This allows us to apply the Aleksandrov MaximumPrinciple for half cylinders and prove (5.9) also in case where ∂ Ω ∩ ( { }× S ) = ∅ . (cid:3) The Index Computation.
Fix two connected components of the respectivecritical sets Λ ⊆ Crit( A H ) and Λ ⊆ Crit( A H ). In this section we establish thatthe hybrid moduli problem is Fredholm, and compute the virtual dimension of thethe moduli space M hyb (Λ , Λ).Let us introduce an anti-symplectic involution on ( T ∗ R k , ω k ) by ρ : T ∗ R k → T ∗ R k , ( q, p ) ( q, − p ) . (5.12)The standard symplectic structure J k satisfies − Dρ ◦ J k ◦ Dρ = J k . (5.13)Therefore, we have the associated diffeomorphism: J k⋆ ∋ J
7→ − Dρ ◦ J ◦ Dρ ∈ J k⋆ . Finally, observe that H ◦ ρ = H , which gives us X H ( x ) = − Dρ [ X H ( ρ ( x ))] . (5.14)As a result we obtain an automorphism of Crit( A H ) defined by:Crit( A H ) ∋ ( v, η ) ( ρ ◦ v, − η ) ∈ Crit( A H ) . (5.15)Using the notation from (4.8), this automorphism maps Λ η to Λ − η . Theorem 5.5.
The hybrid moduli problem is Fredholm, and the space M hyb (Λ , Λ) has virtual dimension vir dim M hyb (Λ , Λ) = µ trCZ (Λ) − µ trCZ (Λ ) + 12 (dim Λ + dim Λ) . Proof.
We prove the theorem in five steps. Denote by y := A H (Λ ) ∈ CritVal( A H ), so that by Corollary 4.6 we haveΛ = Λ y . In this first step, we show that M hyb (Λ y , Λ) is a Fredholm problem. Webegin by identifying M hyb (Λ y , Λ) with a closely related set M hyb (Λ − y , Λ) definedas follows:Let L ⊆ T ∗ R k × T ∗ R n = R n + k ) denote the set of elements of the form L = n ( a, − b, a, b, c, n − k ) (cid:12)(cid:12)(cid:12) a, b ∈ R k , c ∈ R n − k o . In other words, L is the preimage of ∆ T ∗ R k × χ under ρ ⊕ σ − , where ρ is theanti-symplectic map defined in (5.12), σ is the symplectic splitting defined in (4.2),∆ T ∗ R k stands for the diagonal in ( T ∗ R k ) and χ is the 0-section of T ∗ R n − k . Then L is a Lagrangian submanifold of ( T ∗ R k × T ∗ R n , ω k ⊕ ω n ).For J ∈ J k⋆ , J ∈ J n⋆ we define, using (5.13), an element b J ∈ J n + k⋆ by b J := (cid:18) − Dρ ◦ J ◦ Dρ J (cid:19) . (5.16)An element of M hyb (Λ − y , Λ) is a pair ( w, ξ ) with w : [0 , + ∞ ) × S → T ∗ R k × T ∗ R n , ξ : [0 , + ∞ ) → R , (5.17)satisfying the equations ∂ s w + b J ∂ t w − ξ · X ( w ) = 0 , (5.18) ∂ s ξ − X ( w ) = 0 , (5.19)where ( ξ , ξ ) · X ( w , w ) := b J (cid:18) ξ X H ( w ) ξ X H ( w ) (cid:19) , (5.20) X ( w ) := (cid:18) − R S H ( w ) − R S H ( w ) (cid:19) , (5.21)together with the limit conditionslim s → + ∞ w ( s ) ∈ Λ − y × Λ , (5.22)and the coupling conditions w (0 , t ) ∈ L, ∀ t ∈ S , and ξ (0) = − ξ (0) . (5.23) The fact that “hyb” is occurs both as a subscript and a superscript is not a typo! Theyrepresent formally different spaces.
OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 31
There is a natural identification of elements (( v , η ) , ( v, η )) ∈ M hyb (Λ y , Λ) withelements ( w, ξ ) ∈ M hyb (Λ − y , Λ) given by w ( s, t ) := (cid:18) ρ ◦ v ( − s, t ) v ( s, t ) (cid:19) , ξ ( s ) := (cid:18) − η ( − s ) η ( s ) (cid:19) . (5.24)Indeed, under this identification conditions (5.2) and (5.3) become conditions (5.22)and (5.23). Furthermore, we compute that ∂ s w = (cid:18) − Dρ [ ∂ s v ] ∂ s v (cid:19) , ∂ t w = (cid:18) Dρ [ ∂ t v ] ∂ t v (cid:19) , ∂ s ξ = (cid:18) ∂ s η ∂ s η (cid:19) . so that by (5.13), (5.14), (5.16), and (5.24) we have ∂ s w + b J ∂ t w = (cid:18) − Dρ ( ∂ s v + J ∂ t v ) ∂ s v + J ∂ t v (cid:19) = (cid:18) − η Dρ ◦ J X H ( v ) ηJ X H ( v ) (cid:19) = (cid:18) η Dρ ◦ J ◦ Dρ [ X H ( w )] ηJ X H ( w ) (cid:19) = b J (cid:18) ξ X H ( w ) ξ X H ( w ) (cid:19) . In this way the Rabinowitz Floer equations (5.1) become equivalent to (5.18)–(5.21).So M hyb (Λ y , Λ) is a Fredholm problem if and only if M hyb (Λ − y , Λ) is a Fredholmproblem.Fix r >
2. Let B (Λ − y , Λ) denote the Banach manifold of pairs ( w, ξ ) as in (5.17)satisfying (5.22) and (5.23). For ( w, ξ ) ∈ B (Λ − y , Λ), we do not require equations(5.18)–(5.21) to hold. Instead, ( w, ξ ) should be locally of class W ,r and convergefor s → + ∞ exponentially to an element ( x , x ) ∈ Λ − y × Λ. The tangent space of B (Λ − y , Λ) at ( w, ξ ) can be identified with T ( w,ξ ) B (Λ − y , Λ) ∼ = W ,rδ,L ⊕ W ,rδ, ∆ ⊕ T x Λ ⊕ T x Λ , where W ,rδ,L := n z ∈ W ,rδ (cid:16) [0 , ∞ ) × S , R k × R n (cid:17) (cid:12)(cid:12)(cid:12) z (0 , t ) ∈ L o , W ,rδ, ∆ := n ζ ∈ W ,rδ (cid:0) [0 , ∞ ) , R (cid:1) (cid:12)(cid:12)(cid:12) ζ (0) = − ζ (0) o . Here, δ indicates that we are working with weighted Sobolev spaces with weight γ ( s ) = e δs (as our asymptotic operators will not be bijective). Explicitly, f ∈ W ,rδ if and only if f · γ ∈ W ,r . Let E be the Banach bundle over B (Λ − y , Λ) whose fiberat ( w, ξ ) is given by E ( w,ξ ) := L rδ (cid:16) [0 , ∞ ) × S , R k × R n (cid:17) × L rδ (cid:0) [0 , ∞ ) , R (cid:1) . Define a section ¯ ∂ : B (Λ − y , Λ) → E , (5.25)by the left-hand side of the equations (5.18) and (5.19). Then M hyb (Λ − y ; Λ) =¯ ∂ − (0). Since (5.23) is a Lagrangian boundary condition and (5.18) is a perturbed Cauchy-Riemann equation on a half-cylinder, coupled with a pair of ordinary dif-ferential equations (5.19) for ξ , the proof that the linearisation D ¯ ∂ : T ( w,ξ ) B (Λ − y , Λ) → E ( w,ξ ) , is a Fredholm operator is a routine argument (see for instance [3, Sec. 5.4]), and wewill not go over the details here. Computing the index however is somewhat less standard, for two reasons:(i) In contrast to standard index computations in Rabinowitz Floer homology,here there are two Lagrange multipliers to worry about.(ii) We are on a half cylinder, and thus arguments using the spectral flow arenot directly applicable.This computation is very similar to [1, Theorem 4.12]. In order to keep the expositionreasonably clean, we perform the calculation only in the case where both A H (Λ − y )and A H (Λ) are non-zero. The other cases are easier and are left to the reader.In this step, we formulate an appropriate local model for the problem M hyb (Λ − y , Λ)by linearising equations (5.18)–(5.21) at a solution ( w, ξ ) ∈ M hyb (Λ − y , Λ), and ex-press the virtual dimension of M hyb (Λ − y , Λ) in terms of the index of this linearisedoperator.Write m := k, m := n and let P i ∈ W , ∞ (cid:0) [0 , + ∞ ) × S , Mat(2 m i ) (cid:1) , for i = 0 , , + ∞ ] × S in such a way that for each i = 0 , s → + ∞ ess sup ( s,t ) ∈ [ s , ∞ ) × S (cid:16) | ∂ s P i ( s, t ) | + | ∂ t P i ( s, t ) − ∂ t P i ( t, ∞ ) | (cid:17) = 0 . Assume in addition that the limit matrices S i ( t ) := P i (+ ∞ , t ) are symmetric: S ( t ) ∈ Sym( R k ) , S ( t ) ∈ Sym( R n ) . (5.26)Next, for i = 0 , β i ∈ W , ∞ ([0 , ∞ ) × S , R m i ) , denote two vector-valued paths which extend to the compactification [0 , + ∞ ] × S in such a way that for each i = 0 , s → + ∞ ess sup ( s,t ) ∈ ( s , ∞ ) × S (cid:16) | ∂ s β i ( s, t ) | + | ∂ t β i ( s, t ) − ∂ t β i ( t, + ∞ ) | (cid:17) = 0 . Abbreviate b i ( t ) := β i (+ ∞ , t ), and assume that b i ∈ W , ( S , R m i ) ∩ range( J m i ∂ t + S i ) , i = 0 , . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 33
We are now ready to introduce an appropriate local model. Consider the operator D : W ,rδ,L × W ,rδ, ∆ → E defined by D z z ζ ζ := ∂ s z + J k ∂ t z + P z + ζ β ∂ s z + J n ∂ t z + P z + ζ β ζ ′ + R S h z , β i dtζ ′ + R S h z , β i dt (5.27)Note that D is the restriction to W ,rδ,L ⊕ W ,rδ, ∆ of the linearisation D ¯ ∂ of the prob-lem (5.25) at a solution ( w, ξ ) ∈ M hyb (Λ − y ; Λ) of (5.18)–(5.21), when viewed in asuitable symplectic trivialisation. Explicitly, for H = H + H satisfying (1.1) wehave S = − η J k D v X H , b ( t ) = − J k [ X H ]( v ( t )) , (5.28) S = − η J n D v X H , b ( t ) = − J n [ X H ]( v ( t )) , (5.29)for some ( v , η ) ∈ Λ − y , ( v, η ) ∈ Λ. Consequently, we havevir dim M hyb (Λ − y , Λ) = ind D ¯ ∂ = ind D + dim Λ + dim Λ . (5.30) Our aim is to homotope the operator D from (5.27) through Fredholm operatorsinto a new operator of product form D new ( z, ζ ) = ( D ( z ) , D ( ζ )) . (5.31)The operator D new decouples the perturbed Cauchy-Riemann equation satisfied by z and the ordinary differential equation satisfied by ζ . This will allow us to computethe index of D :ind D = ind D new since the index is constant along deformations,= ind D + ind D by additivity. (5.32)The operator D new is constructed using methods from [14, App. C]. The index of D can be computed using the methods from [3]. Meanwhile the index of D canbe computed by hand.In order to construct the homotopy we first need to compute the perturba-tion term defined in [14, Def. C.3] as follows: for i = 0 , δ > z i ∈ W , ( S , R m i ) such that( J m i ∂ t + S i + δ Id) z i = b i , (5.33)and define τ i := Z S h z i ( t ) , b i ( t ) i dt. (5.34) We somewhat ambiguously switch between row and column notation, favouring whichever iscleaner for any given equation.
Since the operator J m i ∂ t + S i is self-adjoint, the number τ i does not depend on thechoice of z i . We will show that in our setting, for δ > τ , τ > . (5.35)First observe, that by Lemma 4.5 every periodic orbit ( v, η ) ∈ Crit( A H ) , η = 0 is ofthe form ( σ − ( v , , η ) with ( v , η ) ∈ Crit( A H ). As a result dH ( v ) = dH ◦ σ − ( v ,
0) = dH ( v ) . Therefore, if ( z , z ) is a solution to (5.33) and ( z , z ) := σ ( z ) ∈ W , ( S , T ∗ R k ) × W , ( S , T ∗ R n − k ), then by (5.29) we get τ = Z S d v H ( z ) dt = Z S d v H ( z ) dt. Therefore, without loss of generality we can assume that z = 0. Consequently, by(4.7), (5.28) and (5.29), equation (5.33) translates to( J k ∂ t + η i A + δ Id) z i = A exp[ tη i J k A ] v i (0) , (5.36)where ( v , η ) ∈ Λ − y , ( σ − ( v , , η ) ∈ Λ ⊆ Crit( A H ).It is easy to check that z i ( t ) := 1 δ A exp[ tη i J k A ] v i (0) , solves (5.36) and as a result for i = 0 , τ i = 1 δ Z S k A exp[ tη i J k A ] v i (0) k dt > , which proves (5.35). In this step we will construct the homotopy between the operator D from (5.27)and a new operator D new of the form as in (5.31).If we set Q = (cid:18) J k ∂ t + P J n ∂ t + P (cid:19) , β = (cid:18) β β (cid:19) , B (cid:18) z z (cid:19) = (cid:18)R S h z , β i dt R S h z , β i dt (cid:19) , then the operator D can be written as D = ∂ s + (cid:18) Q βB (cid:19) . We now consider the homotopy { D θ } θ ∈ [0 , of operators given by D θ = ∂ s + (cid:18) Q (1 − θ ) β (1 − θ ) B c ( θ ) (cid:19) , OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 35 where c : [0 , → R is defined by c ( θ ) := (cid:18) θτ θτ (cid:19) . (5.37)By [14, Thm. C.5] the operators D θ are all Fredholm of the same index. We define D new := D = ( D , D ) , where D : W ,rδ,L → L rδ (cid:16) [0 , ∞ ) × S , R n +2 k (cid:17) , z ∂ s z + Qz, D : W ,rδ, ∆ → L rδ (cid:0) [0 , ∞ ) , R (cid:1) , (cid:18) ζ ζ (cid:19) (cid:18) ζ ′ + τ ζ ζ ′ + τ ζ (cid:19) . In this final step we compute the index of D and D . For this denote by Ψ i for i = 0 , : [0 , → Symp( R k , ω k ) , ( Ψ ′ ( t ) = J k S ( t )Ψ ( t ) , Ψ (0) = Id k , Ψ : [0 , → Symp( R n , ω n ) , ( Ψ ′ ( t ) = J n S ( t )Ψ ( t ) , Ψ (0) = Id n , Denote by µ CZ (Ψ i ) ∈ Z the (full) Conley-Zehnder index of these paths, and letnull(Ψ i ) := dim (ker Ψ i (1) − Id) denote the nullity. Then the Fredholm index of D is given by [3, Theorem 5.25] asind D = X i =0 (cid:18) µ CZ (Ψ i ) −
12 null(Ψ i ) (cid:19) + m ( L ) , (5.38)where m ( L ) stands for the correction term coming from the boundary conditions.Actually (5.38) is much simpler than the general statement in [3, Thm. 5.25], sincewe only have a single boundary condition (rather than a set of jumping boundaryconditions). The correction term m ( L ) is computed by [3, Thm. 5.25] to be m ( L ) := dim T ∗ R n + k −
12 dim ∆ T ∗ R n + k − dim L + dim (∆ T ∗ R n + k ∩ ( L × L )) , where ∆ T ∗ R n + k is the diagonal in ( T ∗ R n + k ) . In this case,dim ∆ T ∗ R n + k = 2( n + k ) , dim (∆ T ∗ R n + k ∩ ( L × L )) = dim L = n + k, hence m ( L ) = 0. On the other hand, by Proposition 4.8 we have µ CZ (Ψ ) = µ CZ (Λ − y ) = − µ CZ (Λ y ) = − µ trCZ (Λ ) ,µ CZ (Ψ ) = µ CZ (Λ) = µ trCZ (Λ) . Moreover, by Lemma 4.5 and (4.7) we havedim Λ = dim Λ − y = dim (ker (Ψ (1) − Id) ∩ Σ ) = null(Ψ ) − , dim Λ = dim (ker (Ψ (1) − Id) ∩ (Σ × { } )) = null(Ψ ) − , which gives ind D = µ trCZ (Λ) − µ trCZ (Λ ) −
12 (dim Λ + dim Λ) − . (5.39)Now we would like to compute ind D . Note thatind D = 2 ind e D − , where the − ζ (0) = − ζ (0) and e D is anoperator of the form e D : W ,rδ ([0 , + ∞ ) , R ) → L rδ ([0 , + ∞ ) , R ) ,f f ′ + τ f. Note that dim coker e D = 0. However, since we are working with weighted Sobolevspaces on a positive half-cylinder, the dimension of ker e D = 1, since in both cases τ > D = 1. Combining this with (5.30) and (5.39)we have vir dim M hyb (Λ , Λ) = ind D new + dim Λ + dim Λ= µ trCZ (Λ) − µ trCZ (Λ ) + 12 (dim Λ + dim Λ) . This completes the proof of Theorem 5.5. (cid:3)
Automatic transversality.
If the space M hyb (Λ , Λ) contains no stationarysolutions, transversality can be achieved for generic choices of almost complex struc-tures. This is a standard – albeit, difficult – argument, which involves no new ideasnot already present in the proof that the moduli spaces defining the RabinowitzFloer complex are generically transverse (see for instance [1, Thm. 4.11] or [41,Thm. 2] for the non-compact case). We therefore omit the proof.The case where M hyb (Λ , Λ) admits stationary solutions is somewhat less stan-dard, however. In the presence of stationary solutions it is not possible to obtain Sanity check: The same computation works for any Lagrangian L . Thus the correction termis always 0 when there is but a single Lagrangian boundary condition. This can also be proveddirectly, without appealing to the machinery of [3]. OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 37 regularity by perturbing J , and one must therefore prove transversality “by hand”.This is the content of the following section.Fix η ∈ CritVal( A H ) = CritVal( A H ) and let Λ η ⊆ Crit( A H ) and Λ η ⊆ Crit( A H )be the corresponding connected components of the respective critical sets. Then M hyb (Λ η , Λ η ) consists entirely of stationary solutions by Lemma 5.3 andfor η = 0 : M hyb (Λ η , Λ η ) ∼ = Λ η ∼ = Λ η , for η = 0 : M hyb (Λ η , Λ η ) ∼ = Σ × χ , (5.40)where χ ⊂ T ∗ R n − k denotes the 0-section. Proposition 5.6. (i)
The problem M hyb (Λ η , Λ η ) is always transversly cut out. (ii) The kernel of the linearisation D ¯ ∂ of the problem M hyb (Λ − η , Λ η ) (see (5.25) )at a stationary solution ( w, ξ ) ∈ M hyb (Λ − η , Λ η ) has dimension (dim Λ +dim Λ ) .Proof of Proposition 5.6. The proof presented below is an adjustment of [1, Lem.4.14] to our setting. Note that part (i) is an immediate corollary of part (ii). Indeed,by the correspondence (5.24), we have that M hyb (Λ η , Λ η ) is transversely cut out ifand only if M hyb (Λ − η , Λ η ) is transversely cut out. The latter holds if and onlyif the operator D ¯ ∂ is surjective at ( w, ξ ). Now, D ¯ ∂ is a Fredholm operator ofindex (dim Λ + dim Λ ) by Theorem 5.5 (see also (5.30)), as µ trCZ (Λ ) = µ trCZ (Λ) byProposition 4.8. Part (ii) tells us that dim ker D ¯ ∂ = (dim Λ + dim Λ ). Thus D ¯ ∂ is surjective, as required.Fix ( w, ξ ) ∈ M hyb (Λ − η , Λ η ) and let ( z, ζ ) ∈ ker (cid:0) D ( w,ξ ) ¯ ∂ (cid:1) be arbitrary, i.e. ( z, ζ )is a solution to the linearised equation D ( w,ξ ) ¯ ∂ ( z, ζ ) = 0. In the following, we write w = ( w , w , w ) and z = ( z , z , z ) with respect to the splitting T ∗ R k × T ∗ R n = T ∗ R k × T ∗ R k × T ∗ R n − k . Denote ( x , x ) := lim s → + ∞ ( w, ξ )( s ) ∈ Λ − η × Λ η . Then( z, ζ ) ∈ W ,rδ,L ⊕ W ,rδ, ∆ ⊕ T x Λ − η ⊕ T x Λ η and satisfies explicitly dds ( z , ζ )+ ∇ w ,ξ ) A H ( z , ζ ) = 0 , dds ( z , z , ζ )+ ∇ w ,w ,ξ ) A H ( z , z , ζ ) = 0 , (5.41)the coupling conditions ζ (0) = − ζ (0) , ∀ t ∈ S : z (0 , t ) = Dρ [ z (0 , t )] , z (0 , t ) ∈ T χ , (5.42)and the asymptotic conditionslim s → + ∞ ( z, ζ )( s ) = ( y , y ) ∈ T x Λ − η × T x Λ η . (5.43)To prove part (ii), we show that any solution ( z, ζ ) of D ( w,ξ ) ¯ ∂ ( z, ζ ) = 0 is constant.Then, we have by (5.43) that ( z, ζ ) = ( y , y ) ∈ T x Λ − η ⊕ T x Λ η , which implies with the coupling condition (5.42) and with (5.40) that the space of such ( z, ζ ) hasdimension (dim Λ + dim Λ ).First recall that the second derivative of the functional A H at a loop ( v, η ), is asymmetric, bilinear operator on W , ( S , v ∗ T ∗ R n ) × R ) given by: d v,η ) A H (( ξ, σ ) , ( ξ, σ )) = Z ω ( ξ, ∂ t ξ ) − η Z Hess v H ( ξ, ξ ) − σ Z dH ( ξ ) . (5.44)We define a function ϕ : [0 , + ∞ ) → R in the following way: ϕ ( s ) := (cid:13)(cid:13)(cid:0) z ( s, · ) , ζ ( s ) (cid:1) − ( y , y ) (cid:13)(cid:13) L ( S ) . By assumption ϕ ∈ L rδ ([0 , + ∞ ) , R ). As A H and A H are Morse-Bott (cf. (2.3)), wehave ( y , y ) ∈ T x Λ − η × T x Λ η = ker ∇ x A H ⊕ ker ∇ x A H and hence ϕ ′ ( s ) = 2 (cid:16) d w ,ξ ) A H (( z , ζ ) , ( z , ζ )) + d w ,w ,ξ ) A H (( z , z , ζ ) , ( z , z , ζ )) (cid:17) ,ϕ ′′ ( s ) = 4 (cid:18)(cid:13)(cid:13)(cid:13) ∇ w ,ξ ) A H ( z , ζ ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ∇ w ,w ,ξ ) A H ( z , z , ζ ) (cid:13)(cid:13)(cid:13) (cid:19) ≥ . Thus on one hand ϕ ′′ ( s ) ≥ ϕ is convex and on the other hand lim s →∞ ϕ ( s ) = 0.That implies that either ϕ ′ (0) < ϕ ′ (0) = 0 and ϕ is constant, equal to 0everywhere. We will show that ϕ ′ (0) = 0, which implies that ( z, ζ ) is constantlyequal to ( y , y ). First, we show that d w ,ξ ) A H (( z , ζ ) , ( z , ζ )) (cid:12)(cid:12) s =0 = 0 . (5.45)Recall from the coupling conditions (5.23) and (5.42) that w (0 , t ) ∈ χ and z (0 , t ) ∈ T χ for all t ∈ S . Writing (5.45) in the form (5.44) as 3 integrals, we find that thefirst integral vanishes, as χ is a Lagrangian submanifold of (cid:0) T ∗ R n − k , ω n − k (cid:1) . Onthe other hand, for every x ∈ χ we can identify T x χ with R n − k × { } and thereforeby (1.1) for every y ∈ T x χ we haveHess x H ( y, y ) = y T A y = 2 H ( y ) = 0 , since χ ⊆ H − (0) by Remark 4.3. Consequently, the second integral in (5.44)vanishes. Analogously, T χ ⊆ T H − (0) = ker( dH ), hence the third integral in(5.44) also is 0. This proves (5.45). Next we will show that (cid:16) d w ,ξ ) A H (( z , ζ ) , ( z , ζ )) + d w ,ξ ) A H (( z , ζ ) , ( z , ζ )) (cid:17) (cid:12)(cid:12)(cid:12) s =0 = 0 . Observe that by the coupling conditions (5.23) and (5.42) we have for all t ∈ S that w (0 , t ) = ρ ◦ w (0 , t ) and z (0 , t ) = Dρ [ z (0 , t )] and hence d w ,ξ ) A H (( z , ζ ) , ( z , ζ )) (cid:12)(cid:12)(cid:12) s =0 = d ρ ◦ w , − ξ ) A H (( Dρ [ z ] , − ζ ) , ( Dρ [ z ] , − ζ )) (cid:12)(cid:12)(cid:12) s =0 Thus, the first corresponding integrals in (5.44) have opposite signs, since ρ isanti-symplectic. The second integrals in (5.44) have opposite signs, since ξ (0) = Observe that ( w , ξ ) ≡ x and ( w , w , ξ ) ≡ x for all s ∈ [0 , ∞ ). OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 39 − ξ (0) and H ◦ ρ = H and thus Hess H ( Dρ · , Dρ · ) = Hess H . The third in-tegrals in (5.44) have opposite signs, since ζ (0) = − ζ (0) and for every ( x, y ) ∈ T R k , dH ρ ( x ) ( Dρ [ y ]) = dH x ( y ). Consequently ϕ ′ (0) = 0. (cid:3) Computation of the Rabinowitz Floer homology
Building the isomorphism.
The main goal of this section will be the proofof Theorem 1.5. Throughout this section we will consider the following setting:let H : T ∗ R n → R and H : T ∗ R k → R be Hamiltonians satisfying (1.1) and let J ∈ J k⋆ , J ∈ J n⋆ be two 2-parameter families of ω -compatible almost complexstructures, such that the couples ( H , J ) and ( H, J ) are regular in the sense ofDefinition 2.10.By Lemma 4.5, the embedding T ∗ R k ∼ = T ∗ R k × { } n − k ) ֒ → T ∗ R n induces aninclusion i : Crit( A H ) ֒ → Crit( A H ) , (6.1)such that its restriction to Crit( A H ) (cid:15) (Σ ×{ } ) is a diffeomorphism. By Lemma4.5 and Proposition 4.8 the inclusion i : Crit( A H ) ֒ → Crit( A H ) is both action anddegree preserving .By Remark 4.7 for each connected component Λ ⊆ Crit( A H ) (cid:15) (Σ ×{ } ) thereexists m ∈ { , . . . , k } , such that Λ is diffeomorphic to S m − . By Remark 4.3, wehave that Σ ×{ } ⊆ Crit( A H ) is diffeomorphic to S k − , while Σ ×{ } ⊆ Crit( A H )is diffeomorphic to S n + k − × R n − k . Moreover this diffeomorphism can be chosen suchthat if Σ ⊂ Σ is the sphere of dimension n + k − S n + k − × { } then Σ ⊂ Σ . As in (4.4) we denote by j : Σ ֒ → Σ the inclusion. Thus theinclusion i from (6.1) satisfies i ( x,
0) = ( j ( x ) , , ∀ ( x, ∈ Σ × { } . Therefore, we can choose a Morse-Smale pair ( f, g ) on Crit( A H ), such that:i) f is coercive;ii) Crit( f ) ∩ Λ = { z − , z + } for each connected component Λ ⊂ Crit( A H );iii) ( f , g ) is a Morse-Smale pair on Crit( A H ), such that ( f , g ) := ( f ◦ i, i ∗ g )on Crit( A H ) (cid:15) (Σ ×{ } );iv) for A H ( z ± ) = 0 we have W sf ( z ± ) ⊆ Σ ×{ } , z + ∈ i (Σ ) and z − / ∈ i (Σ );v) Crit( f ) ∩ (Σ ×{ } ) = { x − , x + } and i ( x + ) = z + and i ( x − ) / ∈ Crit( f ).We denote by x − or z − always the minimum of f or f on a connected componentΛ of Crit( A H ) or Λ of Crit( A H ) respectively. In the following, let x ± or z ± be thetwo critical points of f or f belonging to the same components Λ or Λ respectively.From the assumptions above, we can conclude the following:a) Crit( f ) ∩ Λ = { x ± } for all connected components Λ ⊂ Crit( A H ). This is not to be confused with the inclusion j from (4.4). with respect to the transverse Conley-Zehnder index. b) The restriction i : Crit( f ) (cid:15) (Σ ×{ } ) → Crit( f ) (cid:15) (Σ ×{ } ) is a bijection with i ( x ± ) = z ± .c) The signature index of x ± on Λ ∼ = S m − is µ σ ( x − ) = − m + 12 , µ σ ( x + ) = m − . (6.2)d) For η = 0, the signature index of z ± on Λ ∼ = S m − is given by µ σ ( z − ) = − m + 12 , µ σ ( z + ) = m − . (6.3)e) For η = 0, we have z ± ∈ Σ ×{ } ⊂ Λ = Σ ×{ } and z + is the maximum of f | Σ on Σ ∼ = S n + k − . The signature index of z ± in this case is given by µ σ ( z − ) = − n + 12 , µ σ ( z + ) = k − . (6.4)We will construct the isomorphism from Theorem 1.5 via moduli spaces of cascadeswith solutions to the hybrid problem (5.1)-(5.3) defined as follows:For a pair ( x, z ) ∈ Crit( f ) × Crit( f ) and m ∈ N we denote by M m hyb ( x, z ) the setconsisting of sequences (cid:0) { u l } ml =1 , { t l } m − l =1 (cid:1) , such that exactly one u l in the sequenceis a solution to the hybrid problem, whereas all other u l are Floer trajectories of A H for l < l or of A H for l > l . We require φ t l ◦ ev + ( u l ) = ev − ( u l +1 ) l = 1 , ..., m − , ev − ( u ) ∈ W uf ( x ) and ev + ( u m ) ∈ W sf ( z ) , where t l ≥ φ t l is for l < l the time t l gradient flow of ( f , g )on Crit( A H ) and for l ≥ l the time t l gradient flow of ( f, g ) on Crit( A H ). Remark . For m > R m − acts by time shift on Floer trajectories of M m hyb ( x, z ). We consider the quotients of M m hyb ( x, z ) by this action and define M hyb ( x, z ) := M ( x, z ) ∪ [ m> (cid:0) M m hyb ( x, z ) (cid:14) R m − (cid:1) . M hyb ( x, z ) carries the structure of a smooth manifold by Theorem 5.5 and Propo-sition 5.6 together with the standard Floer-theoretical results (cf. [22, Prop. 1b],[9, Thm. 9.2.3], [21, Sec. 2.4], [24, Cor. A.15], [34, Thm. 4.2], [1, Sec. 2.3]). Itsdimension is µ ( z ) − µ ( x ).If A H ( x ) = η = A H ( z ), Λ := Λ η and Λ := Λ η , we can describe M hyb ( x, z ) moreexplicitly. As the action decreases along non-trivial Floer trajectories, we find thatin this case M m hyb ( x, z ) = ∅ for m >
1. Thus M hyb ( x, z ) = M ( x, z ) consistsentirely of stationary solutions. More precisely, we find that M hyb ( x, z ) is given by OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 41 the fibre product M hyb ( x, z ) / / (cid:15) (cid:15) M hyb (Λ , Λ) ev (cid:15) (cid:15) W uf ( x ) × W sf ( z ) ι / / Λ × Λwhere ι is the inclusion and ev : M hyb (Λ , Λ) → Λ × Λ the evaluation map:ev( u , u ) = (cid:0) lim s →−∞ u ( s ) , lim s → + ∞ u ( s ) (cid:1) . Hence M hyb ( x, z ) = ev − (cid:0) W uf ( x ) × W sf ( z ) (cid:1) is a manifold of dimensiondim M hyb ( x, z ) = dim W uf ( x ) + dim W sf ( z ) + dim M hyb (Λ , Λ) − dim Λ − dim Λ= dim W uf ( x ) + dim W sf ( z ) −
12 (dim Λ + dim Λ)= µ σ ( z ) − µ σ ( x ) . (6.5)provided the following intersection is transverse in Λ × Λ: W uf ( x ) × W sf ( z ) ∩ ev (cid:0) M hyb (Λ , Λ) (cid:1) . (6.6)For η = 0, we have Λ ∼ = Λ, so that ev (cid:0) M hyb (Λ , Λ) (cid:1) = ∆ is the diagonal in Λ × Λ by Lemma 5.3. On the other hand, by our assumptions on f and f we have that i ( W uf ( x )) = W uf ( i ( x )) and consequently the transversality of (6.6) is equivalentto the transversality of W uf ( i ( x )) ⋔ W sf ( z ), which follows from the Morse-Smaleassumption on the flow of ∇ f .When η = 0, we have by Lemma 5.3 that Y := ev (cid:0) M hyb (Λ , Λ (cid:1) ∼ = ∆ × χ , where ∆ is the diagonal in Λ × Λ ⊆ Λ × Λ and χ is the zero section in T ∗ R n − k .In particular the transversality of (6.6) is equivalent to the transversality of σ − (cid:0) W uf ( x ) × χ (cid:1) ∩ W sf ( z ) , (6.7)in Σ. By Remark 4.3, we have σ − (Σ × χ ) ∩ Σ = σ − (Σ × χ ) ∩ Σ ∩ σ − ( T ∗ R k × S )= Σ ∩ σ − (Σ × ( χ ∩ S )) = Σ ∩ σ − (Σ ×{ } ) = i (Σ ) . By assumption z − / ∈ i (Σ ) and W sf ( z − ) = { z − } , which in view of the above gives: σ − (cid:0) W uf ( x ± ) × χ (cid:1) ∩ W sf ( z − ) = ∅ and (cid:0) W uf ( x ± ) × W sf ( z − ) (cid:1) ∩ Y = ∅ . Thus (6.6) is transverse for ( x, z ) = ( x ± , z − ).On the other hand, by assumption W sf ( z + ) = Σ \ { z − } ⊇ i (Σ ). Thus σ − (cid:0) W uf ( x ± ) × χ (cid:1) ∩ W sf ( z + ) ⊇ i ( W uf ( x ± )) = ∅ . Take w ∈ σ − (cid:0) W uf ( x ± ) × χ (cid:1) ∩ W sf ( z + ) and denote σ ( w ) = ( w , w ). Then T w W sf ( z + ) = T w Σ = T w Σ ∩ Dσ − ( T w T ∗ R k × T w S ) ,T w σ − (cid:0) W uf ( x ± ) × χ (cid:1) = Dσ − (cid:0) T w W uf ( x ± ) × T w χ (cid:1) . By Remark 4.3 dim S = n − k = dim χ and S ∩ χ = { } , hence span { S, χ } = R n − k ) and consequently the intersection σ − ( W uf ( x ± ) × χ ) ∩ W sf ( z + ) is transverse. Remark . In all the above cases for A H ( z ) = A H ( x ) and µ σ ( z ) = µ σ ( x ), we findthat M hyb ( x, z ) = { ( x, z ) } contains exactly one element.In order to use M hyb ( x, z ) for the definition of ψ , one has to show that it iscompact modulo breaking. This follows mostly by standard techniques via Gromovcompactness. However, as we are on a non-compact manifold, we need to assurethat all moduli spaces M hyb ( x, z ) are uniformly bounded in the L ∞ -norm. Lemma 6.3.
For every pair ( x, z ) ∈ Crit( f ) × Crit( f ) all Floer and hybrid tra-jectories from the set M hyb ( x, z ) are uniformly bounded in the L ∞ -norm and thebound depends only on x and y .Proof. Let a := A H ( x ) and b := A H ( z ). Denote the moduli spaces of Floer trajec-tories of A H and A H in the action window [ a, b ] as follows: M ( a, b ) := [ Λ ± ⊆ Crit ( A H ) A H ( Λ ± ) ∈ [ a,b ] M (Λ − , Λ + ) , M ( a, b ) := [ Λ ± ⊆ Crit ( A H ) A H ( Λ ± ) ∈ [ a,b ] M (Λ − , Λ + ) . By Corollary 4.6 we know that the set CritVal( A H ) ∩ [ a, b ] = CritVal( A H ) ∩ [ a, b ]is finite. Combined with [35, Thm. 1] we infer that M ( a, b ) and M ( a, b ) are finiteunions of sets whose images in R k +1 and R n +1 respectively are bounded, thus theirimages are bounded.Using the notation from section 5.2 we denote for a compact subset N ⊆ Σthe moduli spaces of the solutions to (5.1) and (5.3) connecting components of thecritical set of A H in action window [ a, b ] with components of the critical set of A H in the same action window as follows: M ( a, b ) := [ Λ ⊆ Crit ( A H ) A H (Λ ) ∈ [ a,b ] [ Λ ⊆C ( A H ,N ) A H (Λ) ∈ [ a,b ] N hyb (Λ , Λ) . Again, the set M ( a, b ) is a finite union of sets, which by Proposition 5.4 are allbounded in L ∞ -norm, thus its image is also bounded in L ∞ -norm and the bounddepends only on a, b and N . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 43
Fix m ∈ N and take (cid:0) { u l } ml =1 , { t l } m − l =1 (cid:1) ∈ M m hyb ( x, z ). Then by definition of M m hyb ( x, z ) there exists l ∈ { , . . . , m } , such that u l = ( u − , u + ) is a solution tothe hybrid problem (5.1) and (5.3). By Lemma 5.2 we have that a ≤ A H ◦ ev − ( u − ) ≤ A H ◦ ev + ( u + ) ≤ b. From the above inequality we can conclude that for all l ∈ { , . . . , l − } thecorresponding u l is a Floer trajectory of A H , with a ≤ A H ◦ ev − ( u l ) ≤ A H ◦ ev + ( u l ) ≤ A H ◦ ev − ( u − ) ≤ b, and therefore u l ∈ M ( a, b ), whereas for all l ∈ { l + 1 , . . . , m } we have u l ∈ M ( a, b )as these u l are Floer trajectories of A H with b ≥ A H ◦ ev + ( u l ) ≥ A H ◦ ev − ( u l ) ≥ A H ◦ ev + ( u + ) ≥ a. Now we consider three cases, choosing the compact set N appropriately to eachcase:1. If 0 / ∈ [ a, b ] then none of the cascades passes through Σ; thus we can choose N = ∅ and for u l to be in M ( a, b ).2. If b = 0 then ev + ( u m ) ∈ Σ ×{ } and ev + ( u m ) ∈ W sf ( z ). In particu-lar, ev + ( u m ) ∈ f − (cid:0) ( −∞ , f ( z )] (cid:1) . Therefore, for b = 0 we take N := f − (cid:0) ( −∞ , f ( z )] (cid:1) . Due to coercivity of f , this N is compact. That ensuresev + ( u l ) ∈ C (cid:0) A H , N (cid:1) and u l ∈ M ( a, b ).3. If 0 ∈ [ a, b ) then take N := K ( b ) to be the shade (cf. [34, Sec. 4.1]), i.e. e K ( b ) := [ Λ ⊆ Crit( A H ) , A H (Λ) ∈ (0 ,b ] ev − ( M (Σ × { } , Λ)) ,K ( b ) := f − (cid:16)(cid:0) − ∞ , sup ˜ K ( b ) f (cid:3)(cid:17) . By [34, Lem. 4.1] the set K ( b ) is a compact subset of Σ ×{ } . Moreover,if ev + ( u l ) ∈ Σ ×{ } , then ev − ( u l +1 ) ∈ e K ( b ) and consequently ev + ( u + ) = φ − t l ◦ ev − ( u l +1 ) ∈ K ( b ). Thus taking N := K ( b ) ensures u l ∈ M ( a, b ).In all the above cases we have chosen N to be a compact set, such that u l ∈ M ( a, b ). Consequently all elements of M m hyb ( x, z ) are bounded in the L ∞ -normand the bounds depend only on x and z . (cid:3) Proof of Theorem 1.5:
Let H = H + H and f and f be as described at thebeginning of this section. Let CF ∗ ( H , f ) and CF ∗ ( H, f ) be the chain complexesassociated to the quadruples ( H , J , f , g ) and ( H, J, f, g ), respectively, as definedin Section 2. Now we define the chain map ψ : CF ∗ ( A H , f ) → CF ∗ ( A H , f ).For a given pair of points ( x, z ) ∈ Crit( f ) × Crit( f ) such that µ ( x ) = µ ( z ) wefind by Remark 6.1 that M hyb ( x, z ) is a 0-dimensional manifold. It is compact by Lemma 6.3, and hence a finite set. Denote its parity by n ( x, z ) := M hyb ( x, z ) ∈ Z . (6.8)Define the homomorphism ψ : CF ∗ ( H, f ) → CF ∗ ( H , f ) as the linear extension of: ψ ( z ) := X x ∈ Crit( f ) ,µ ( x )= µ ( z ) n ( x, z ) x. For ψ to be well defined, it has to satisfy the Novikov finiteness condition (2.9), i.e.that for all z ∈ Crit( f ) and a ∈ R holds (cid:8) x ∈ Crit( f ) (cid:12)(cid:12) n ( x, z ) = 0 and A H ( x ) ≥ a (cid:9) < + ∞ . (6.9)By Lemma 5.2 the condition n ( x, z ) = 0 implies A H ( x ) ≤ A H ( z ). In other words,using the notation from (2.10), for every t ∈ R ψ (cid:0) CF ≤ t ∗ ( H, f ) (cid:1) ⊆ CF ≤ t ∗ ( H , f ) . (6.10)On the other hand, all connected components of Crit( A H ) are compact and f isMorse, hence Crit( f ) ∩ (cid:0) A H (cid:1) − (cid:0) [ a, A H ( z )] (cid:1) is finite and (6.9) is satisfied.Lemma 6.3 and Theorem 5.5 together with the standard gluing and compactnessarguments (cf. [9, Thm. 4.2, Thm. 11.1.16], [24, Thm. A.11], [1, Sec. 4.2]) implythat ψ commutes with the respective boundary operators, i.e. ψ ◦ ∂ = ∂ ◦ ψ , andthus induces a homomorphismΨ : RF H ∗ ( H ) → RF H ∗ ( H ) . Recall the short exact sequence (2.13), which was induced by action filtration:0 → CF ( H, f ) → CF ≥ ( H, f ) → CF + ( H, f ) → . Note that there is an analogous sequence for ( H , f ). As ψ reduces action (cf.(6.10)) and commutes with ∂ and ∂ , we find that it induces maps on the filteredchain complexes which fit into the following commutative diagram of complexes:0 / / CF ∗ ( H, f ) ψ (cid:15) (cid:15) / / CF ≥ ∗ ( H, f ) (cid:15) (cid:15) / / CF + ∗ ( H, f ) ψ + (cid:15) (cid:15) / / / / CF ∗ ( H , f ) / / CF ≥ ∗ ( H , f ) / / CF + ∗ ( H , f ) / / . By naturality (cf. [28, Sec. 2.1]), we hence obtain the following commutative diagramof long exact sequences in homology: −→ H ∗ + n − (Σ) Ψ (cid:15) (cid:15) / / RF H ≥ ∗ ( H ) (cid:15) (cid:15) / / RF H + ∗ ( H ) Ψ + (cid:15) (cid:15) / / H ∗ + n − (Σ) −→ Ψ (cid:15) (cid:15) −→ H ∗ + k − (Σ ) / / RF H ≥ ∗ ( H ) / / RF H + ∗ ( H ) / / H ∗ + k − (Σ ) −→ . OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 45
It remains to show that Ψ + is an isomorphism and that Ψ : H n + k − (Σ) → H k − (Σ )is an isomorphism and vanishes otherwise:Recall that by Lemma 4.5 we have a natural bijection between Crit + ( A H ) andCrit + ( A H ). Moreover, we defined f to be the pullback of f under this bijection,such that their critical points are in 1-to-1 correspondence, i.e. ( v, η ) ∈ Crit + ( f )if and only if ( σ − ( v, , η ) ∈ Crit + ( f ). This correspondence allows us to representthe homomorphism ψ + as an infinite matrix with entries n ( x , i ( x )) := ( M hyb ( x , i ( x )) if µ ( x ) = µ ( i ( x ))0 otherwisedefined as in (6.8) for x , x ∈ Crit( A H ). In fact by (6.10) this matrix is uppertriangular. We would like to investigate its diagonal.Fix x ∈ Crit( f ) ∩ Crit + ( A H ). By Proposition 4.8 we know that the inclu-sion i : Crit( A H ) ֒ → Crit( A H ) preserves the Conley-Zehnder index, i.e. µ CZ ( x ) = µ CZ ( i ( x )). On the other hand, since Crit + ( A H ) is diffeomorphic to Crit + ( A H ) with f being the pullback of f under this diffeomorphism, we infer that µ σ ( x ) = µ σ ( i ( x )).Consequently, µ ( x ) = µ ( i ( x )) and n ( x, i ( x )) = M hyb ( x, i ( x )) = 1 by Remark 6.2.We can conclude that the matrix representing ψ + is upper triangular with 1’s onthe diagonal and therefore ψ + is an isomorphism and induces on homology level theisomorphism Ψ + .For Ψ recall from the beginning of this section, that we have four action zerocritical points { x ± } ∈ Λ ⊆ Crit( A H ) and { z ± } ∈ Λ ⊆ Crit( A H ). All four pointscorrespond to constant orbits of constant flows. Hence their Conley-Zehnder indexis zero. From the calculations of the signature indexes (6.2) and (6.4), we infer that µ ( z + ) = µ ( x + ) = k, µ ( z − ) = − n + 1 , µ ( x − ) = − k + 1 . Thus, we find that n ( x + , z + ) = M hyb ( x + , i ( x + )) = 1 by Remark 6.2 and thatfor any other pair ( x, z ) ∈ { ( x − , z − ) , ( x − , z + ) , ( x + , z − ) } we have n ( x, z ) = 0, as µ ( x ) = µ ( z ). Hence, Ψ k : H n + k − (Σ) → H k − (Σ ) is an isomorphism and Ψ ∗ = 0for ∗ 6 = k .Finally, recall from (4.5) that the map H ∗ (Σ) → H ∗ + k − n (Σ ) given by composingthe Umkehr map of the inclusion j : Σ ֒ → Σ together with a retraction r : Σ → Σ is an isomorphism for ∗ = n + k − agrees with the j ! ◦ r ∗ . (cid:3) Remark . Analogously, we can define Ψ :
RF H − ( H ) → RF H − ( H ) and showthat it gives an isomorphism. This can also be seen directly from definition of ψ , using the Morse-theoretic description ofthe Umkehr map from [2, App. B] or [3, App. A, p1716-1717]. Computation.
In this section we prove Theorem 1.1. In other words, we willcalculate the full Rabinowitz Floer homology of H using the isomorphism of thepositive Rabinowitz Floer homologies proven in Theorem 1.5 and the following longexact sequences (see (2.14)): · · · → H ∗ + n − (Σ) → RF H ≥ ∗ ( H ) → RF H + ∗ ( H ) → . . . (6.11) · · · → RF H −∗ ( H ) → RF H ∗ ( H ) → RF H ≥ ∗ ( H ) → . . . (6.12)We begin by collecting the properties of the Rabinowitz Floer homology for thecompact hypersurface Σ that we need . Proposition 6.5.
For the compact hypersurface Σ ⊆ R k we have RF H + ∗ ( H ) = (cid:26) Z ∗ = k + 1 , ∗ 6 = k + 1 . RF H −∗ ( H ) = (cid:26) Z ∗ = − k, ∗ 6 = − k. (6.13) Proof.
Since Σ is compact, it is displaceable in R k . Thus by [14, Thm. 1.2] wehave RF H ( H ) = 0. Thus also the symplectic homology SH ∗ (Σ ) and cohomology SH ∗ (Σ ) vanish by [36, Thm. 13.3]. Therefore the positive symplectic homology SH + ∗ (Σ ) agrees with H k − −∗ ( R k ) by [16, Lem. 2.1]. Then by [16, Thm. 1.4] wehave RF H + ∗ ( H ) ∼ = SH + ∗ (Σ ). Now (6.13) follows from (6.11) and (6.12). (cid:3) That, together with the isomorphism from Theorem 1.5 will allow us to calculatethe full Rabinowitz Floer homology of Σ:
Proof of Theorem 1.1:
Recall from Remark 4.3 that Σ ≃ S n + k − × R n − k , hence H ∗ (Σ) = ( Z ∗ = 0 , n + k − , ⇒ H ∗ + n − (Σ) = ( Z ∗ = 1 − n, k, . . . / / H ∗ + n − (Σ) / / Ψ (cid:15) (cid:15) RF H ≥ ∗ ( H ) / / (cid:15) (cid:15) RF H + ∗ ( H ) / / Ψ + (cid:15) (cid:15) . . .. . . / / H ∗ + k − (Σ ) / / RF H ≥ ∗ ( H ) / / RF H + ∗ ( H ) / / . . . By Theorem 1.5 the map Ψ + : RF H + ∗ ( H ) → RF H + ∗ ( H ) is an isomorphism. Hence,we obtain from (6.13) and (6.14) that RF H ≥ ∗ ( H ) = 0 for all ∗ / ∈ { − n, k + 1 , k } . We remind the reader (Remark 2.11) that in the compact case the Rabinowitz Floer homologyonly depends on H through its zero level set Σ . Therefore instead of RF H ∗ ( H ) we couldwrite RF H ∗ (Σ ), and similarly for the other variants. For consistency with the non-compact case,however, we will not do this. This can also easily be proved directly.
OMPUTING RFH OF TENTACULAR HYPERBOLOIDS 47
In case ∗ = 1 − n we have RF H +2 − n ( H ) = → H (Σ) = Z → RF H ≥ − n ( H ) → RF H +1 − n ( H ) = , which implies RF H ≥ − n ( H ) = Z . In case ∗ ∈ { k + 1 , k } we have0 → RF H ≥ k +1 ( H ) / / (cid:15) (cid:15) RF H + k +1 ( H ) / / Ψ + (cid:15) (cid:15) H n + k − (Σ) / / Ψ (cid:15) (cid:15) RF H ≥ k ( H ) → (cid:15) (cid:15) → RF H ≥ k +1 ( H ) = / / RF H + k +1 ( H ) = Z / / H k − (Σ ) = Z / / RF H ≥ k ( H ) = → RF H + k +1 ( H ) → H k − (Σ ) , Ψ + and Ψ are isomorphisms(cf. Theorem 1.5), implying that the map RF H + k +1 ( H ) → H n + k − (Σ) is also anisomorphism. Consequently, RF H ≥ ∗ ( H ) = 0 for ∗ ∈ { k + 1 , k } . We conclude that RF H ≥ ∗ ( H ) := (cid:26) Z ∗ = 1 − n, . Having determined
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Department of Mathematics, Humboldt-Universit¨at zu Berlin,Germany.
E-mail address : [email protected] Department of Mathematics, ETH Z¨urich, Switzerland.
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