A simple construction of an action selector on aspherical symplectic manifolds
aa r X i v : . [ m a t h . S G ] J a n A SIMPLE CONSTRUCTION OF AN ACTION SELECTOR ONASPHERICAL SYMPLECTIC MANIFOLDS
ALBERTO ABBONDANDOLO, CARSTEN HAUG, AND FELIX SCHLENK
Abstract.
We construct an action selector on aspherical symplectic manifolds that areclosed or convex. Such selectors have been constructed by Matthias Schwarz using Floerhomology. The construction we present here is simpler and uses only Gromov compactness. Introduction
Hamiltonian systems on symplectic manifolds tend to have many periodic orbits. The“actions” of these orbits form an invariant for the Hamiltonian system. The set of actionscan be very large, however. To get useful invariants, one selects for each Hamiltonian functionjust one action value by some minimax procedure: A so-called action selector associates toevery time-periodic Hamiltonian function on a symplectic manifold the action of a periodicorbit of its flow in a continuous way. For this one needs compactness assumptions on either thesymplectic manifold or the support of the Hamiltonian vector field. The mere existence of anaction selector has many applications to Hamiltonian dynamics and symplectic topology: Itreadily yields a symplectic capacity and thus implies Gromov’s non-squeezing theorem, impliesthe almost existence of closed characteristics on displaceable hypersurfaces and in particularthe Weinstein conjecture for displaceable energy surfaces of contact type, often proves the non-degeneracy of Hofer’s metric and its unboundedness, etc., see for instance [3, 4, 9, 20, 27, 30]and Section 6 below.Action selectors were first constructed for the standard symplectic vector space ( R n , ω )by Viterbo [30], and by Hofer–Zehnder [9] who built on earlier work by Ekeland–Hofer [1].For more general symplectic manifolds ( M, ω ), action selectors were obtained, up until now,only by means of Floer homology: For symplectically aspherical symplectic manifolds (namelythose for which [ ω ] | π ( M ) = 0), Schwarz [27] constructed the so-called PSS selector when M is closed, and his construction was adapted to convex symplectic manifolds in [4]. We referto Appendix A of [3] for a short description of these selectors. For some further classes ofsymplectic manifolds and Hamiltonian functions, the PSS selector was constructed in [11, 19,29].In this paper we give a more elementary construction of an action selector for closed orconvex symplectically aspherical manifolds. Our construction uses only results from Sec-tion 6.4 of the text book [9] by Hofer and Zehnder, that rely on Gromov compactness andrudimentary Fredholm theory, but on none of the more advanced tools in the constructionof Floer homology (such as exponential decay, the spectral flow, unique continuation, gluing, Key words and phrases. symplectic manifold, Hamiltonian system, action selector.The research of A. Abbondandolo is supported by the SFB/TRR 191 Symplectic Structures in Geometry,Algebra and Dynamics, funded by the Deutsche Forschungsgemeinschaft.The research of F. Schlenk is supported by the SNF grant 200021-181980/1.2010
Mathematics Subject Classification: or transversality). In this way, the three basic properties of an action selector (spectrality,continuity and local non-triviality) are readily established by rather straightforward proofs,since the only tool at our hands is the compactness property of certain spaces of holomorphiccylinders.After recollecting known results in Section 2, we give the construction of our action selec-tor for closed symplectically aspherical manifolds in Section 3. In Section 4 we adapt thisconstruction to convex symplectically aspherical manifolds. Examples are cotangent bundlesand their fiberwise starshaped subdomains, on which most of classical mechanics takes place.In Section 5 we show that the three basic properties of the action selector imply many furtherproperties, and in Section 6 we illustrate by three examples how any action selector yieldssimple proofs of results in symplectic geometry and Hamiltonian dynamics. In Section 7 wesketch some variations of our construction and address open problems.
Idea of the construction.
In the rest of this introduction we outline the construction of ouraction selector on a closed symplectically aspherical manifold (
M, ω ). Denote by T = R / Z thecircle of length 1. Recall that the Hamiltonian action functional on the space of contractibleloops C ∞ contr ( T , M ) associated to a Hamiltonian function H ∈ C ∞ ( T × M, R ) =: H ( M ) isgiven by A H ( x ) := Z D ¯ x ∗ ( ω ) + Z T H ( t, x ( t )) dt, where ¯ x ∈ C ∞ ( D , M ) is such that ¯ x | ∂ D = x . The critical points of A H are the contractible1-periodic solutions of the Hamiltonian equation˙ x ( t ) = X H ( t, x ( t )) , where the vector field X H is defined by ω ( X H , · ) = dH , and the set of critical values of A H is called the action spectrum of H and denoted by spec ( H ). An action selector should selectan element of spec ( H ) in a monotone and continuous way, with respect to the usual orderrelation and to some reasonable topology on the space of Hamiltonians.A first idea for defining an action selector is to boldly take the smallest action value of a1-periodic orbit, σ ( H ) := min spec ( H ) . Since spec ( H ) is a compact subset of R , this definition makes sense, and yields an invariantwith the spectral property. However, this invariant is not very useful, since it fails to becontinuous and monotone, two crucial properties for applications. To see why, consider radialfunctions H f ( z ) := f ( π | z | ) on R n , where f : [0 , + ∞ ) → R is a smooth function with compact support. For an arbitrary sym-plectic manifold, such functions can be constructed in a Darboux chart and then be extendedby zero to the whole manifold. The critical points of A H are the origin and the (Hopf-)circleson those spheres that have radius r with s = πr and f ′ ( s ) ∈ Z ; at such a critical point x thevalue of the action is(1) A H f ( x ) = f ( s ) − s f ′ ( s ) , see the left drawing in Figure 1. Now take the profile functions f, f + , f − as in the rightdrawing: f ′ ∈ [0 ,
1] and f ′ ( s ) = 1 for a unique s , while f − , f + are C ∞ -close to f and satisfy f − ≤ f ≤ f + and f ′− , f ′ + ∈ [0 , σ ( H f ) is much smallerthan σ ( H f − ) ≈ σ ( H f + ), whence σ is neither continuous nor monotone. Or take g with | g | SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 3 very small and very steep. Then σ ( H g ) is much smaller than σ ( H f ), whence monotonicityfails drastically.PSfrag replacements ss s = πr f ( s ) f ( s ) − sf ′ ( s ) f ff + f − gσ ( H f ) σ ( H f − ) σ ( H f + ) σ ( H g ) Figure 1.
Radial functions and their minimal spectral valuesThe above discussion shows that the continuous, or monotone, selection of an action fromspec ( H ) must be done by some kind of minimax procedure involving more information on theaction functional than the mere knowledge of its critical values. This was done for the Hofer–Zehnder selector by minimax over a uniform minimax family, and for the Viterbo selectorand the PSS selector by a homological minimax. Our minimax will be over certain spaces ofperturbed holomorphic cylinders.PSfrag replacements xyq h (0 , p p Figure 2.
A perturbed quadratic form q h To introduce our construction, we first look at a toy model: Consider the quadratic form q ( x, y ) = x − y on R and its perturbations q h = q + h where h is a compactly supported function on R . Here, the indefinite quadratic form q modelsthe symplectic action and the compactly supported function h models the Hamiltonian termin A H , cf. [9, § h = 0, the only critical point of q h is the origin, with critical value 0. If ALBERTO ABBONDANDOLO, CARSTEN HAUG, AND FELIX SCHLENK h consists, for instance, of two little positive bumps, one centered at (1 ,
0) and one at (0 , q h looks as in Figure 2. A continuous selection of critical values h σ ( h )should, in our example, choose again 0, by somehow discarding the four new critical values.In this finite dimensional example, one could define an action selector by the minimaxformula σ ( h ) = inf max Y q h , where the infimum is over the space of all images Y of continuous maps R → R that arecompactly supported perturbations of the embedding y (0 , y ). Monotonicity in h is clearfrom the definition, and spectrality can be proved by standard deformation arguments usingthe negative gradient flow of q h . The definition of the Hofer–Zehnder action selector (see[9, Section 5.3]) is based on a similar idea and uses the fact that the Hamiltonian actionfunctional for loops in R n has a nice negative gradient flow.Alternatively, one can fix a very large number c such that the sublevel { q h < − c } coincideswith the sublevel { q < − c } and define the same critical value σ ( h ) asinf (cid:8) a ∈ R | the image of i a ∗ : H ( { q h < a } , { q < − c } ) → H ( R , { q < − c } ) is non-zero (cid:9) , where the map i a is the inclusion i a : ( { q h < a } , { q < − c } ) ֒ → (cid:0) R , { q < − c } (cid:1) and we are using the fact that H ( R , { q < − c } ) ∼ = Z . Viterbo’s definition of an action selector for compactly supported Hamiltonians on R n usesa similar construction, which is applied to suitable generating functions, see [30]. The Floerhomological translation of this second definition is, in turn, at the basis of Schwarz’s con-struction of an action selector for symplectically aspherical manifolds, see [27], and of all itssubsequent generalizations.Here, we would like to define an action selector σ ( h ) using only spaces of bounded negativegradient flow lines: In the case of the Hamiltonian action functional A H , these will correspondto finite energy solutions of the Floer equation, which have good compactness properties. Afirst observation is that the knowledge of the space of all bounded negative gradient flow linesof q h is not enough for defining an action selector. Indeed, it is easy to perturb q on a smalldisc disjoint from the origin in such a way that the negative gradient flow lines of q h look likein Figure 3: A new degenerate critical point z is created, and the constant orbits at (0 , z are the only bounded negative gradient flow lines. But since q h ( z ) could be eitherpositive or negative, the set { (0 , , z } contains too little information for us to conclude thatthe value of the action selector should be q h (0 ,
0) = 0.If, however, we are allowed to deform the function q h , we can use bounded gradient flowlines to define an action selector that identifies the lowest critical value that “cannot be shakenoff”. More precisely, take a family { h s } s ∈ R of compactly supported functions such that h s = h for s small and h s = 0 for s large, and look at the space U ( h s ) of bounded solutions of thenon-autonomous gradient equation˙ u ( s ) = −∇ q h s ( u ( s )) , s ∈ R . The boundedness of u is equivalent to bounded energy E ( u ) := Z R |∇ q h s ( u ( s )) | ds = lim s →−∞ q h s ( u ( s )) − lim s → + ∞ q h s ( u ( s )) + Z R ∂h s ∂s ( u ( s )) ds < ∞ , SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 5
PSfrag replacements xy z (0 , Figure 3.
The only bounded gradient flow lines are the constant orbits at(0 ,
0) and z .or, since h s = h in the first limit and h s = 0 in the second limit, to the fact that u ( s ) isasymptotic for s → −∞ to the following critical level of q h q − h ( u ) := lim s →−∞ q h ( u ( s ))and for all s large lies on the x -axis and converges for s → + ∞ to the origin (the only criticalpoint of q ). The number min u ∈ U ( h s ) q − h ( u )is the lowest critical value of q h from which a bounded h s -negative gradient flow line starts.In our example from Figure 2, if we take h s = β ( s ) h with a cut-off function β , then U ( h s )contains no flow line u emanating from the two low critical points p or p near (0 , h s that has a negative gradient line u ( s ) thatconverges to p for s → −∞ and to the origin for s → + ∞ . To be sure that we discard allinessential critical values, we therefore set σ ( h ) := sup h s min u ∈ U ( h s ) q − h ( u ) . In the example, it is quite clear that for every deformation h s there exists a flow line in U ( h s )emanating from the critical point (0 , σ ( h ) = 0 as it should be. In general, it is nothard to see that σ ( h ) is a critical value of q h that depends continuously and in a monotoneway on h .The number σ ( h ) is the lowest critical value c of q h such that for every deformation h s of h there exists a bounded flow line u ∈ U ( h s ) starting at a critical level not exceeding c .Equivalently, σ ( h ) is the highest critical value c of q h such that for every critical level c ′ < c there exists a deformation h s of h such that all flow lines of q h s starting at level c ′ areunbounded. That is: the whole critical set strictly below c can be shaken off.Imitating the above construction, and inspired by the proof of the degenerate Arnol’dconjecture in [9, § M, ω ) in the following way. Given H ∈ C ∞ ( T × M ) weconsider s -dependent Hamiltonians K in C ∞ ( R × T × M ) such that K ( s, · , · ) = H for s small ALBERTO ABBONDANDOLO, CARSTEN HAUG, AND FELIX SCHLENK and K ( s, · , · ) = 0 for s large. Following Floer’s interpretation of the L -gradient flow of theaction functional, we consider the space U ( K ) of solutions u ∈ C ∞ ( R × T , M ) of Floer’sequation(2) ∂ s u + J ( u ) (cid:0) ∂ t u − X K ( s, t, u ) (cid:1) = 0that have finite energy E ( u ) = Z R × T | ∂ s u | J < ∞ . Here, J is a fixed ω -compatible almost complex structure on T M and | · | J is the inducedRiemannian norm. The space U ( K ) is C ∞ loc -compact by Gromov’s compactness theorem.Now define the function a − H : U ( K ) → R , a − H ( u ) := lim s →−∞ A H ( u ( s ))and finally define the action selector of H by A J ( H ) := sup K min u ∈ U ( K ) a − H ( u ) , where the supremum is taken over all deformations K of H as above. The number A J ( H )is the smallest essential action of H in the following sense: It is the lowest critical value c of A H (that is, the lowest action of a contractible 1-periodic orbit of H ) such that for everydeformation K of H there exists a finite energy solution of Floer’s equation for K and J thatstarts at a critical level ≤ c . For another characterization of A J ( H ), see Section 3.3.In our finite dimensional model, we could have allowed for a larger class of deformationsof the gradient flow of q h , by looking at families h s that for s large do not depend on s but are not necessarily zero, and by taking the gradient with respect to any family g s ofRiemannian metrics that depend on s on a compact interval. In the symplectic setting, therole of Riemannian metrics is played by ω -compatible almost complex structures. We maythus modify the above definition by looking at functions K with K ( s, · , · ) = H for s smalland K ( s, · , · ) independent of s for s large, and at families J s of ω -compatible almost complexstructures that depend on s on a compact interval. In Sections 2–4, we shall construct anaction selector A ( H ) by using these larger families of deformations. This has the advantagethat A ( H ) is manifestly independent of the choice of J . It will be clear from the analysisof A ( H ) that A J ( H ) is also an action selector, cf. Section 7.1.There are also action selectors relative to closed Lagrangian submanifolds, that have manyapplications in the study of these important submanifolds. Such selectors were first con-structed by Viterbo [30] and Oh [18] for Hamiltonian deformations of the zero-section of cotan-gent bundles, and then in more general settings by Leclercq [12] and Leclercq–Zapolsky [13].Except for Viterbo’s generating function approach, all these constructions are based on La-grangian Floer homology. Our elementary construction of an action selector can also becarried out for closed Lagrangian submanifolds L under the assumption that [ ω ] vanisheson π ( M, L ). We shall focus on the absolute case, however, leaving the necessary adaptationsto the interested reader.2.
Notations, conventions and known results
Let (
M, ω ) be a closed symplectic manifold such that [ ω ] | π ( M ) = 0. We assume through-out that M is connected. We denote by X H the Hamiltonian vector field associated to a SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 7
Hamiltonian H ∈ C ∞ ( M ), that is ω ( X H , · ) = dH. Let T = R / Z be the circle of length 1. The Hamiltonian action functional on the space ofcontractible loops C ∞ contr ( T , M ) associated to a time-periodic Hamiltonian H ∈ C ∞ ( T × M )has the form A H ( x ) := Z D ¯ x ∗ ( ω ) + Z T H ( t, x ( t )) dt, where ¯ x ∈ C ∞ ( D , M ) is an extension of the loop x to the closed disk D , that is ¯ x | ∂ D = x ;here we are identifying ∂ D and T in the standard way. The first integral does not depend onthe choice of the extension ¯ x of x because [ ω ] vanishes on π ( M ). The critical points of A H are precisely the elements of P ( H ), the set of contractible 1-periodic orbits of X H . By theAscoli–Arzel`a theorem, P ( H ) is a compact subset of C ∞ contr ( T , M ).The space C ∞ ( R × T , M ) is endowed with the C ∞ loc -topology, which is metrizable andcomplete. We shall identify C ∞ ( R × T , M ) with C ∞ ( R , C ∞ ( T , M )), and we use the notation u ( s ) = u ( s, · ) ∈ C ∞ ( T , M ) , ∀ s ∈ R . The additive group R acts on C ∞ ( R × T , M ) by translations( σ, u ) τ σ u, where ( τ σ u )( s ) := u ( σ + s ) . Let J be a smooth ω -compatible almost complex structure on M , meaning that g J ( ξ, η ) := ω ( J ξ, η ) , ∀ ξ, η ∈ T x M, ∀ x ∈ M, is a Riemannian metric on M . The associated norm is denoted by | · | J . The L -negativegradient equation for the functional A H is the Floer equation(3) ∂ s u + J ( u ) (cid:0) ∂ t u − X H ( t, u ) (cid:1) = 0 . If u is a solution of (3), then the function s A H ( u ( s, · )) is non-increasing and(4) lim s →−∞ A H ( u ( s, · )) − lim s → + ∞ A H ( u ( s, · )) = E ( u ) := Z R × T (cid:12)(cid:12) ∂ s u (cid:12)(cid:12) J ds dt. The quantity E ( u ) defined above is called energy of the cylinder u . Any x ∈ P ( H ) defines astationary solution u ( s, t ) := x ( t ) of (3), which has zero energy and is called a trivial cylinder.Now let H ∈ C ∞ ( R × T × M, R ) be such that ∂ s H , the partial derivative of H with respectto the first variable, has compact support and set H − ( t, x ) := H ( − s, t, x ) and H + ( t, x ) := H ( s, t, x ) for s large . Further, let J = { J s } be a smooth s -dependent family of ω -compatible almost complexstructures such that ∂ s J has compact support, and set J − ( x ) := J − s ( x ) and J + ( x ) := J s ( x ) for s large . If u solves the s -dependent Floer equation(5) ∂ s u + J s ( u ) (cid:0) ∂ t u − X H ( s, t, u ) (cid:1) = 0 , then the energy identity reads:(6) A H ( s , · , · ) ( u ( s )) − A H ( s , · , · ) ( u ( s )) = Z [ s ,s ] × T (cid:12)(cid:12) ∂ s u (cid:12)(cid:12) J s ds dt − Z [ s ,s ] × T ∂ s H ( s, t, u ( s, t )) ds dt, ALBERTO ABBONDANDOLO, CARSTEN HAUG, AND FELIX SCHLENK for every s < s . It follows that the function s A H ( s, · , · ) ( u ( s )) is non-increasing on aneighborhood of −∞ and on a neighborhood of + ∞ , and thatlim s →−∞ A H − ( u ( s, · )) − lim s → + ∞ A H + ( u ( s, · )) = E ( u ) − Z R × T ∂ s H ( s, t, u ( s, t )) ds dt where the energy E ( u ) is defined as in (4), but with an s -dependent J : E ( u ) = E J ( u ) := Z R × T (cid:12)(cid:12) ∂ s u (cid:12)(cid:12) J s ds dt. Set U ( H, J ) := { u ∈ C ∞ ( R × T , M ) | u is a solution of (5) with E ( u ) < ∞} . We recall that a subset U of C ∞ ( R × T , M ) is said to be bounded if for every multi-index α ∈ N , | α | ≥
1, there holds sup u ∈ U sup ( s,t ) ∈ R × T | ∂ α s ∂ α t u ( s, t ) | J < ∞ . Bounded subsets are relatively compact in the C ∞ loc -topology. The next result is a specialinstance of Gromov compactness. Proposition 2.1.
Let H = { H s } and J = { J s } be as above. (i) The set U ( H, J ) is a compact subset of C ∞ ( R × T , M ) . (ii) For u ∈ U ( H, J ) the set α - lim( u ) := n lim n →∞ τ s n u | s n → −∞ is such that τ s n u converges o is a non-empty subset of U ( H − , J − ) and consists of trivial cylinders of the form v ( s, t ) := x ( t ) , for some x ∈ P ( H − ) with action A H − ( x ) = lim s →−∞ A H − ( u ( s )) . Outline of the proof.
Statement (i) is proved in Corollary 1 and Proposition 11 in Section 6.4of [9] for the case that H and J do not depend on s . That proof readily generalizes to oursituation. Statement (ii) can be obtained by adapting Propositions 8 and 9 in [9, § § §
3] for moredetails.One starts by proving that for every c ≥ U c ( H, J ) := { u ∈ U ( H, J ) | E ( u ) ≤ c } is compact. The uniform boundedness of the first derivatives requires a bubbling-off analysis,that uses the assumption that [ ω ] vanishes on π ( M ) and the uniform bound on the energy.Once uniform bounds on the first derivatives have been established, the bounds on all higherderivatives follow from elliptic bootstrapping. This shows that U c ( H, J ) is bounded in C ∞ ( R × T , M ). By the lower semicontinuity of the energy, that is u n → u in C ∞ ( R × T , M ) = ⇒ E ( u ) ≤ lim inf n →∞ E ( u n ) , the set U c ( H, J ) is also closed in C ∞ ( R × T , M ), and hence compact. SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 9
Statement (i) will thus follow from the fact that U ( H, J ) = U c ( H, J ) when c is largeenough. In order to prove the latter fact, we need to address statement (ii). Let u ∈ U ( H, J ).That the set α -lim( u ) is not empty follows from the fact that the set { τ s u | s ∈ R } is relatively compact in C ∞ ( R × T , M ), by the same argument sketched above. Now assumethat v = lim n →∞ τ s n u with s n → −∞ . Since v n := τ s n u solves the equation ∂ s v n + ( τ s n J )( v n ) (cid:0) ∂ t v n − X τ sn H ( s, t, v n ) (cid:1) = 0 , and since τ s n H converges to H − and τ s n J converges to J − , the limit v is a solution of the s -independent Floer equation defined by H − and J − . Moreover, since Z [ − T,T ] × T | ∂ s v | J − ds dt = lim n →∞ Z [ − T,T ] × T | ∂ s v n | τ sn J ds dt ≤ lim inf n →∞ E τ sn J ( v n )for every T > E τ sn J ( v n ) = E J ( u ) for all n , we have E J − ( v ) ≤ lim inf n →∞ E τ sn J ( v n ) = E J ( u ) . Hence v ∈ U ( H − , J − ), and it remains to show that v is a trivial cylinder for H − . Considerthe function a H − : C ∞ ( R × T , M ) → R , a H − ( w ) = A H − ( w (0)) . Since H ( s, · , · ) = H − for s ≤ − S , where S is a sufficiently large number, the function s a H − ( τ s u ) = A H − ( u ( s ))is non-increasing on the interval ( −∞ , − S ]. Since u has finite energy, this function is alsobounded, and hence converges to some real number a for s → −∞ . From the continuityof a H − we deduce that a H − ( v ) = a for all v ∈ α -lim( u ). The latter set is clearly invariantunder the action of τ s , so we have that a H − ( τ s v ) = a H − ( v ) = a = lim s →−∞ A H − ( u ( s ))for all s ∈ R . The energy identity for v then forces v to be a trivial cylinder v ( s, t ) = x ( t ) ofaction A H − ( x ) = a . This concludes the proof of (ii).By the energy identity (6), each u ∈ U ( H, J ) has then the uniform energy bound(7) E ( u ) ≤ max x ∈ P ( H − ) A H − ( x ) − min x ∈ P ( H + ) A H + ( x ) + L k ∂ s H k ∞ , where L is the length of an interval outside of which ∂ s H ( · , t ) vanishes for all t ∈ T . Thisshows that if c is at least the quantity on the right-hand side of inequality (7), then U ( H, J ) = U c ( H, J ), and concludes the proof of (i). (cid:3)
The other crucial fact that we need is the following result, which implies in particular that U ( H, J ) is not empty.
Proposition 2.2.
Let H = { H s } and J = { J s } be as above. For every z ∈ R × T and every m ∈ M there is at least one u ∈ U ( H, J ) such that u ( z ) = m .Outline of the proof. The proof uses arguments from [9, § T >
0, we can glue two disks to the cylinder [ − T, T ] × T and obtain a sphere S T . The Floerequation for the pair ( H, J ) on [ − T, T ] × T can be extended to the two capping disks byhomotoping the Hamiltonian to zero and by extending J by J − respectively J + (see [9, p.231]). This leads to spaces U T ( H, J, z, m ) of solutions u of this Floer equation on S T with the property that u ( z ) = m . By the same argument sketched in the proof of Proposition 2.1,this space is compact in C ∞ ( S T , M ). It suffices to show that U T ( H, J, z, m ) is not empty forall large T , since then any sequence u n ∈ U T n ( H, J, z, m ) with T n → ∞ has a subsequencewhich converges on compact sets to some u ∈ U ( H, J ) such that u ( z ) = m , again by theusual compactness argument.The space of solutions U T ( H, J, z, m ) can be seen as the set of zeroes of a smooth section ofa suitable smooth Banach bundle π : E → B . Here, B is the Banach manifold of W ,p mapsfrom S T to M mapping z to m , where 2 < p < ∞ , and the fiber of E at u ∈ B is a Banachspace of L p sections. By homotoping the Hamiltonian H to zero and the S T -dependent ω -compatible almost complex structure J to an S T -independent one J , we obtain a smooth1-parameter family of smooth sections S : [0 , × B → E such that U T ( H, J, z, m ) is the set of zeros of S (1 , · ), while the zeros of S (0 , · ) are J -holomorphic spheres u : S T → M such that u ( z ) = m . The assumption that [ ω ] vanisheson π ( M ) guarantees that the only zero of S (0 , · ) is the map that is constantly equal to m .The usual compactness argument implies that the inverse image S − (0 E ) of the zero-section 0 E of E under S is compact in [0 , × B . Moreover, the Fredholm results from [9, Appendix 4]imply that for each ( t, u ) in S − (0 E ) the fiberwise differential of S ( t, · ) at u is a Fredholmoperator of index 0. Finally, the fiberwise differential of S (0 , · ) at the unique zero u ≡ m isan isomorphism (see [9, Appendix 4, Theorem 8]). Therefore, the section S satisfies all theassumptions of Theorem A.1 in the appendix, from which we conclude that S (1 , · ) has atleast one zero. (cid:3) Remark 2.3.
Note that Propositions 2.1 and 2.2 imply that for any H ∈ C ∞ ( T × M )the Hamiltonian vector field X H has 1-periodic orbits. Indeed, Proposition 2.2 implies that U ( H, J ) is not empty and Proposition 2.1 (ii) then gives the existence of a 1-periodic orbit.
Remark 2.4.
By arguing as in [9, § z ∈ R × T , denote byev z : C ∞ ( R × T , M ) → M, ev z ( u ) := u ( z )the evaluation map at z . Denote by ˇ H ∗ the Alexander–Spanier cohomology functor with Z -coefficients. Then the restriction of ev z to U ( H, J ) induces an injective homomorphismin cohomology:(8) (cid:0) ev z | U ( H,J ) (cid:1) ∗ : H ∗ ( M ) ∼ = ˇ H ∗ ( M ) → ˇ H ∗ (cid:0) U ( H, J ) (cid:1) . See [7, § z to U ( H, J ) is surjective,i.e., Proposition 2.2 holds. In the case of an s -independent Hamiltonian, the injectivity ofthe map (8) leads to the proof of the degenerate Arnol’d conjecture for closed symplecticallyaspherical manifolds, see [9, Chapter 6].3. Construction of an action selector
Let H ∈ C ∞ ( T × M ) be a Hamiltonian. We would like to define an action selector for H .3.1. The definition.
Denote by K ( M ) the set of functions K ∈ C ∞ ( R × T × M ) such that ∂ s K has compact support and by J ω ( M ) the set of smooth families J = { J s } of ω -compatiblealmost complex structures on M such that ∂ s J has compact support. Let K ( H ) be the subsetof those K ∈ K ( M ) for which K − = H , and abbreviate D ( H ) = K ( H ) × J ω ( M ). For( K, J ) ∈ D ( H ) let U ( K, J ) be the space of finite energy solutions of Floer’s equation (5)
SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 11 defined by K and J . Let ( K, J ) ∈ D ( H ) and assume that ∂ s K and ∂ s J are supported in[ s − , s + ] × T × M . If u ∈ U ( K, J ), then on ( −∞ , s − ] the function s A H ( u ( s )) is non-increasing and bounded. Therefore, the function a − H : U ( K, J ) → R , a − H ( u ) := lim s →−∞ A H ( u ( s )) = sup s ∈ ( −∞ ,s − ] A H ( u ( s )) , is well-defined. Being the supremum of a family of continuous functions, the function a − H islower semi-continuous. As such, it has a minimum on the compact space U ( K, J ). Definition 3.1.
Let H ∈ C ∞ ( T × M ) and ( K, J ) ∈ D ( H ) . We set A − ( K, J ) := min u ∈ U ( K,J ) a − H ( u ) , A ( H ) := sup ( K,J ) ∈ D ( H ) A − ( K, J ) . It follows from Proposition 3.2 below that A ( H ) is finite.PSfrag replacements ss − s + K − = H K + K Figure 4.
A function K deforming H , for ( t, x ) fixed3.2. First properties.
Denote byspec ( H ) := { A H ( x ) | x ∈ P ( H ) } the set of critical values of A H . This set is compact, since P ( H ) is compact in C ∞ ( T , M )and A H is continuous on C ∞ ( T , M ).Note that the number A − ( K, J ) belongs to spec ( H ). Indeed, take u ∈ U ( K, J ) suchthat a − H ( u ) = A − ( K, J ). By Proposition 2.1 (ii), we find v in α -lim( u ), and v is of the form v ( s, t ) = x ( t ) with x ∈ P ( H ) and A H ( x ) = a − H ( u ). Hence A − ( K, J ) is a critical value of A H .Since spec ( H ) is compact, the supremum A ( H ) is also a critical value of A H . Therefore,we have proved the following result. Proposition 3.2 ( Spectrality). A ( H ) belongs to spec ( H ) . Two very simple properties of the action selector A are: A ( H ) = 0 if H ≡ , (9) A ( H + r ) = A ( H ) + R T r ( t ) dt ∀ r ∈ C ∞ ( T ) , H ∈ C ∞ ( T × M ) . (10)Indeed, the first property follows from the fact that for the Hamiltonian H ≡
0, the set P ( H )consists of all the constant loops, which have action zero. The second property follows fromthe identities K ( H + r ) = K ( H ) + r and a − H + r = a − H + R T r ( t ) dt . Less trivial is the followingcrucial result: Proposition 3.3 ( Monotonicity). If H , H ∈ C ∞ ( T × M ) are such that Z T max x ∈ M (cid:0) H ( t, x ) − H ( t, x ) (cid:1) dt ≤ , then A ( H ) ≤ A ( H ) . Proof.
Fix ε >
0. We shall prove that(11) sup ( K ,J ) ∈ D ( H ) min U ( K ,J ) a − H ≥ sup ( K ,J ) ∈ D ( H ) min U ( K ,J ) a − H − ε, and the claim will follow from the arbitrariness of ε . Proving (11) is equivalent to showingthat for every ( K , J ) in D ( H ) there exists ( K , J ) in D ( H ) such that(12) min U ( K ,J ) a − H ≥ min U ( K ,J ) a − H − ε. Up to a translation, we may assume that(13) K ( s, t, x ) = H ( t, x ) and J ( s, t, x ) = J − ( t, x ) , ∀ s ≤ . Let ϕ ∈ C ∞ ( R ) be a real function such that ϕ ′ ≥ ϕ ( s ) = 0 for s ≤ ϕ ( s ) = 1 for s ≥
1. For λ ∈ R we define K λ ∈ K ( H ) by(14) K λ ( s, t, x ) := ϕ ( s − λ ) K ( s, t, x ) + (cid:0) − ϕ ( s − λ ) (cid:1) H ( t, x ) . We claim that there exists λ ≤ − K , J ) = ( K λ , J ). Arguingby contradiction, we assume that for every λ ≤ − u λ in U ( K λ , J ) such that(15) a − H ( u λ ) < min U ( K ,J ) a − H − ε. Let ( λ n ) ⊂ ( −∞ , −
1] be such that λ n → −∞ . By Proposition 2.1 (i), U ( K , J ) is compact.Arguing by a diagonal sequence argument, we see that after replacing ( λ n ) by a subsequence,( u λ n ) converges to some u in U ( K , J ).PSfrag replacements s λ λ + 1 H H K Figure 5.
The function K λ , for ( t, x ) fixedWe fix a number s ≤
0. If λ n ≤ s −
1, then by (13) and the action-energy identity (6), a − H ( u λ n ) ≥ A H (cid:0) u λ n ( λ n ) (cid:1) = A H (cid:0) u λ n ( s ) (cid:1) + Z [ λ n ,s ] × T (cid:12)(cid:12) ∂ σ u λ n (cid:12)(cid:12) J − dσ dt − Z [ λ n ,s ] × T ϕ ′ ( σ − λ n )( H − H )( t, u λ n ) dσ dt. By the hypothesis of the proposition and the fact that ϕ ′ is non-negative we obtain theinequality Z [ λ n ,s ] × T ϕ ′ ( σ − λ n )( H − H )( t, u λ n ) dσ dt ≤ Z [ λ n ,s ] × T ϕ ′ ( σ − λ n ) max x ∈ M ( H − H )( t, x ) dσ dt ≤ , and hence the previous inequality gives us a − H ( u λ n ) ≥ A H (cid:0) u λ n ( s ) (cid:1) . By taking the limit for n → ∞ , we deduce thatlim inf n →∞ a − H ( u λ n ) ≥ A H (cid:0) u ( s ) (cid:1) , SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 13 and by taking the supremum over all s ≤ n →∞ a − H ( u λ n ) ≥ a − H ( u ) . Together with (15), this implies the chain of inequalities a − H ( u ) ≤ lim inf n →∞ a − H ( u λ n ) ≤ min U ( K ,J ) a − H − ε, which is the desired contradiction because u ∈ U ( K , J ). (cid:3) Monotonicity and property (10) imply the following form of continuity.
Proposition 3.4 ( Lipschitz continuity).
For all H , H ∈ C ∞ ( T × M ) we have Z T min x ∈ M (cid:0) H ( t, x ) − H ( t, x ) (cid:1) dt ≤ A ( H ) − A ( H ) ≤ Z T max x ∈ M (cid:0) H ( t, x ) − H ( t, x ) (cid:1) dt. In particular, the action selector A is 1-Lipschitz with respect to the sup-norm on C ∞ ( T × M ) : (cid:12)(cid:12) A ( H ) − A ( H ) (cid:12)(cid:12) ≤ k H − H k ∞ . Proof.
Set c − ( t ) = min x ∈ M (cid:0) H ( t, x ) − H ( t, x ) (cid:1) , c + ( t ) = max x ∈ M (cid:0) H ( t, x ) − H ( t, x ) (cid:1) . Then H ( t, x ) + c − ( t ) ≤ H ( t, x ) ≤ H ( t, x ) + c + ( t ) , ∀ t ∈ T , x ∈ M. Applying Proposition 3.3 and (10) we obtain A ( H ) + Z T c − ( t ) dt ≤ A ( H ) ≤ A ( H ) + Z T c + ( t ) dt as we wished to prove. (cid:3) An equivalent definition.
By now, we know that our action selector A is spectral,monotone, and continuous. These properties already imply many further properties, seeProposition 5.4 below, and results like the unboundedness of Hofer’s metric, see Section 6.3.For most applications of an action selector, such as the non-squeezing theorem or the (almost)existence of closed characteristics, one also needs that the selector is negative on functionsthat are non-positive and somewhere negative. To prove this property for our selector A weshall describe A by a minimax in which the space U ( K, J ) is replaced by a certain space ofsolutions of the Floer equation for H .Recall that ( τ σ u )( s ) := u ( σ + s ). Given ( K, J ) ∈ D ( H ), consider the set U ess ( K, J ) := n u ∈ C ∞ ( R × T , M ) | u = lim n →∞ τ s n u n where s n → −∞ and ( u n ) ⊂ U ( K, J ) o . Example 3.5.
Assume that neither H nor J depend on s . Then U ess ( H, J ) = U ( H, J ) .Proof. The inclusion U ess ( H, J ) ⊂ U ( H, J ) holds because if u n belongs to U ( H, J ) thenalso ( τ s n u n ) does, and hence also u = lim n →∞ τ s n u n is in the same space, since U ( H, J ) isclosed. Moreover, the inclusion U ( H, J ) ⊂ U ess ( H, J ) holds because for u ∈ U ( H, J ) wehave u n := τ n u ∈ U ( H, J ) and lim n →∞ τ − n ( u n ) = u . ✷ As we shall see in Proposition 3.6, U ess ( K, J ) is a compact τ -invariant subspace of U ( H, J − ).The space U ess ( K, J ) is therefore the space of those cylinders in U ( H, J − ) which are essential with respect to K , in the sense that they survive through the homotopy K . We shall provethat the action selector A ( H ) = sup ( K,J ) ∈ D ( H ) min U ( K,J ) a − H can be expressed as(16) A ( H ) = sup ( K,J ) ∈ D ( H ) min U ess ( K,J ) a H , where a H is the continuous function a H : C ∞ ( R × T , M ) → R , a H ( u ) := A H ( u (0)) . We begin with the following result.
Proposition 3.6.
The set U ess ( K, J ) is a compact τ -invariant subspace of U ( H, J − ) . Forevery z ∈ R × T and m ∈ M there exists u ∈ U ess ( K, J ) such that u ( z ) = m .Proof. The inclusion U ess ( K, J ) ⊂ U ( H, J − ) is shown in the same way as the inclusion α -lim( u ) ⊂ U ( H, J − ) in Proposition 2.1 (ii): Let u = lim τ s n u n be an element of U ess ( K, J ).Since v = τ s n u n solves the equation ∂ s v + ( τ s n J )( v ) (cid:0) ∂ t v − X τ sn K ( s, t, v ) (cid:1) = 0 , and since τ s n K converges to K − = H and τ s n J converges to J − , the map u is a solution ofthe s -independent Floer equation defined by H and J − . Moreover, E J − ( u ) ≤ lim inf n →∞ E τ sn J ( τ s n u n ) = lim inf n →∞ E J ( u n ) ≤ sup v ∈ U ( K,J ) E J ( v ) < + ∞ , where the finiteness of the last supremum follows from (7). Therefore, U ess ( K, J ) ⊂ U ( H, J − ).If σ ∈ R , then τ σ u = lim n →∞ τ s n + σ u n is in U ess ( K, J ), which is therefore τ -invariant. If v h = lim n →∞ τ s hn u hn , where lim n →∞ s hn = −∞ , ∀ h ∈ N , and ( v h ) converges to v ∈ C ∞ ( R × T , M ), then a standard diagonal argument implies theexistence of a diverging sequence ( n h ) ⊂ N such thatlim h →∞ dist (cid:0) τ s hnh u hn h , v h (cid:1) = 0 , lim h →∞ s hn h = −∞ , where dist is a distance on the metrizable space C ∞ ( R × T , M ). Therefore, τ s hnh u hn h convergesto v , which hence belongs to U ess ( K, J ). This shows that U ess ( K, J ) is a closed subspaceof U ( H, J − ). Since U ( H, J − ) is compact, so is U ess ( K, J ).Finally, given z = ( s, t ) ∈ R × T , m ∈ M and n ∈ N , by Proposition 2.2 we can find u n ∈ U ( K, J ) such that u n ( s − n, t ) = m . By compactness, a subsequence of τ − n ( u n )converges to some u ∈ U ess ( K, J ). Since τ − n ( u n )( z ) = u n ( s − n, t ) = m, we conclude that u ( z ) = m . (cid:3) SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 15
Remark 3.7.
Actually, one can show that the space U ess ( K, J ) satisfies the property ofRemark 2.4: The restriction of ev z to U ess ( K, J ) induces an injective homomorphism incohomology: (cid:0) ev z | U ess ( K,J ) (cid:1) ∗ : H ∗ ( M ) ∼ = ˇ H ∗ ( M ) → ˇ H ∗ (cid:0) U ess ( K, J ) (cid:1) . See [7, Proposition 4.3.4].Formula (16) is an immediate consequence of the following result.
Proposition 3.8. min U ( K,J ) a − H = min U ess ( K,J ) a H .Proof. Let u ∈ U ( K, J ) be a minimizer of a − H . By Proposition 2.1 (ii) there exists v ∈ α -lim( u ) with v ( s, t ) = x ( t ) for some x ∈ P ( H ), and a H ( v ) = A H ( x ) = a − H ( u ) . Since v ∈ α -lim( u ) ⊂ U ess ( K, J ), we concludemin U ess ( K,J ) a H ≤ a H ( v ) = a − H ( u ) = min U ( K,J ) a − H . Conversely, let v ∈ U ess ( K, J ) be a minimizer of a H . Then v = lim n →∞ τ s n u n , where s n → −∞ and ( u n ) ⊂ U ( K, J ) . Up to a subsequence, we may assume that ( u n ) converges to some u ∈ U ( K, J ). For everyfixed s belonging to a half-line ( −∞ , s − ] on which ∂ s K vanishes, we have A H ( u ( s )) = lim n →∞ A H ( u n ( s )) ≤ lim n →∞ A H ( u n ( s n )) = lim n →∞ a H ( τ s n u n ) = a H ( v ) . By taking the limit for s → −∞ , we find a − H ( u ) ≤ a H ( v ) , which implies that min U ( K,J ) a − H ≤ a − H ( u ) ≤ a H ( v ) = min U ess ( K,J ) a H . (cid:3) Autonomous Hamiltonians.
Let H ∈ C ∞ ( M ) be an autonomous Hamiltonian. Inthis case, the critical points of H are the constant orbits of X H , and in particular they areelements of P ( H ). In general, the vector field X H can have other non-constant contractibleorbits, but if this does not happen we can often calculate the value of the action selector A . Proposition 3.9.
Let H ∈ C ∞ ( M ) be an autonomous Hamiltonian with exactly two criticalvalues. Assume also that P ( H ) consists only of constant orbits. Then A ( H ) = min M H. Proof.
In this case, A H has exactly two critical values, min H and max H . Hence A ( H ) isone of these two numbers. For every ( K, J ) ∈ D ( H ),(17) min U ess ( K,J ) a H ≤ max U ess ( K,J ) a H ≤ max U ( H,J − ) a H = max P ( H ) A H = max M H and, by Proposition 3.8, the numbermin U ess ( K,J ) a H = A − ( K, J ) belongs to spec ( H ) = { min H, max H } . Assume by contradiction that A ( H ) = sup ( K,J ) ∈ D H ) A − ( K, J )has the value max H . Then we can find ( K, J ) ∈ D ( H ) such that all inequalities in (17) areequalities, and in particular min U ess ( K,J ) a H = max M H. This identity implies that U ess ( K, J ) consists only of constant cylinders defined by the max-imum points of H . Indeed, for every u ∈ U ess ( K, J ) we can then find a periodic orbit x ∈ P ( H ) in the set α -lim( u ) withmax H ≤ a H ( u ) ≤ A H ( x ) . Since A H ( x ) ≤ max H by the assumption, this yields a H ( u ) = max H . By Proposition 3.6,the translates τ s u also belong to U ess ( K, J ), whence a H ( τ s u ) = max H for all s ∈ R . Sincewe also know that τ s u ∈ U ( H, J − ), it follows that u is a trivial cylinder u ( s, t ) = m with H ( m ) = max H .What we have just proved violates the surjectivity of the evaluation map ev z | U ess ( K,J ) fromProposition 3.6. (cid:3) Remark 3.10.
It is easy to construct autonomous Hamiltonians which satisfy the assump-tions of the above proposition. For instance, take a symplectically embedded ball B ⊂ M of radius 3 ε and a Hamiltonian H on M with support in B that on B is a radial function H = f ( π | z | ), where f : R ≥ → R ≤ is negative constant on { r ≤ ε } , vanishes on { r ≥ ε } ,and has positive derivative on { ε < r < ε } . Then the minimum f (0) and the maximum 0are the only critical values of H , and if we further impose that f ′ <
1, we see as in theintroduction that all non-constant periodic orbits of X H have period larger than 1. HenceProposition 3.9 implies that A ( H ) < A , one caneasily show that A ( H ) < H ∈ C ∞ ( T × M ) which is notidentically zero. This is proved in Section 5, in which we investigate the properties of actionselectors axiomatically. Remark 3.11.
In Remark 7.6 we give an explicit formula for A ( H ) for a class of Hamiltoniansdifferent from the one in Proposition 3.9.4. An action selector on convex symplectic manifolds
A compact symplectic manifold (
M, ω ) is called convex if it has non-empty boundary and ifnear the boundary one can find a Liouville vector field Y , namely such that L Y ω = ω , whichis transverse to the boundary and points outwards. We shall also assume that [ ω ] vanisheson π ( M ).Since the boundary is compact, we can find ε > φ tY of Y defines anembedding (1 − ε, × ∂M → M, ( r, x ) φ log rY ( x )onto an open neighborhood U of ∂M . This embedding defines a smooth positive function r : U → R such that r − ( { } ) = ∂M and r − ((1 − ε, U \ ∂M . SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 17
We consider the set H ( M ) of smooth functions H : T × M → R such that X H has compactsupport in T × ( M \ ∂M ). In other words, for every component C i of the boundary ∂M there exists an open neighborhood U i of C i in M and a function h i ∈ C ∞ ( T ) such that H ( t, x ) = h i ( t ) for all ( t, x ) ∈ T × U i .The symbol K ( M ) now denotes the space of functions K ∈ C ∞ ( R × T × M ) such that X K is supported in R × T × ( M \ ∂M ) and K ( s, t, x ) does not depend on s for s ≤ s − andfor s ≥ s + , for some numbers s − , s + depending on K .The set J ω ( M ) now consists of all smooth families J = { J s } of ω -compatible almostcomplex structures on M such that ∂ s J is compactly supported, J ( s, x ) does not depend on s for all x in a neighborhood of ∂M , and the equation(18) dr ◦ J = ı Y ω holds on this neighborhood, where r is the function which is induced by the Liouville vectorfield Y as above.For K ∈ K ( M ) and J ∈ J ω ( M ), let U ( K, J ) be the set of finite energy solutions ofthe Floer equation (5) on M . Being smooth maps defined on an open manifold (namely thecylinder R × T ), the elements u ∈ U ( K, J ) are tangent to the boundary of M where theytouch it. The next result implies, in particular, that the only elements u ∈ U ( K, J ) thattouch the boundary are constant maps.
Lemma 4.1.
Let V ′ ⊂ U be an open neighborhood of ∂M on which the vector field X K ( s, t, · ) vanishes for every ( s, t ) ∈ R × T and the almost complex structure J := J ( s, · ) is independentof s and satisfies (18) . Let δ > be so small that the closure of the open set V := { x ∈ U | r ( x ) > − δ } is contained in V ′ . Then the image of any u ∈ U ( K, J ) that is not constant is containedin M \ V .Proof. The argument is well known, but we reproduce it here for the sake of completeness.Let u be an element of U ( K, J ) and set Ω ′ := u − ( V ′ ). The conditions on V ′ imply that u | Ω ′ is a J -holomorphic map and ρ := r ◦ u : Ω ′ → R is a subharmonic function. The open setΩ := u − ( V ) satisfies Ω ⊂ Ω ′ . We wish to prove that if the open set Ω is not empty, then u is a constant map.The subharmonic function ρ takes the value 1 − δ on ∂ Ω, and it is strictly larger than thisvalue on Ω. By the maximum principle, Ω cannot have bounded components.Without loss of generality, we may assume that s is unbounded from below on Ω. We claimthat in this case Ω ′ contains a subset of the form ( −∞ , S ) × T , for some S ∈ R . If this is notthe case, we can find a sequence ( s n , t n ) ∈ R × T such that s n → −∞ and u ( s n , t n ) ∈ M \ V ′ .As in the proof of Proposition 2.1, one shows that { τ s n u | n ∈ N } is relatively C ∞ loc -compactin C ∞ ( R × T , M ). Up to replacing ( s n , t n ) by a subsequence, we can therefore assume that τ s n u converges in C ∞ loc to a finite energy solution v of Floer’s equation for J − and K − , and asin the proof of Proposition 2.1, we see that v is a trivial cylinder for K − , that is v ( s, t ) = x ( t )is a 1-periodic orbit of X K − . In particular, the sequence of curves u ( s n ) = τ s n u (0) convergesto x ∈ P ( K − ). Since all the solutions of X K − through points in V ′ are constant, x ( T )is disjoint from V ′ . For n large enough the set u ( { s n } × T ) is then contained in M \ V .This implies that { s n } × T is disjoint from Ω. Then the facts that s n → −∞ and that s isunbounded from below on Ω force Ω to have bounded components. Since we have excluded this possibility, we reach a contradiction and conclude that Ω ′ contains a subset of the form( −∞ , S ) × T , for some S ∈ R .Therefore, the biholomorphic map ϕ : R × T → C \ { } , ϕ ( s, t ) = e π ( s + it ) , maps Ω ′ onto an open set of the form e Ω ′ \ { } , where e Ω ′ is an open neighborhood of the originin C . Having finite energy, the J -holomorphic map ˜ u := u ◦ ϕ − extends holomorphicallyto e Ω ′ by the removal of singularity theorem, see [15, Theorem 4.1.2]. Therefore, ˜ ρ := r ◦ ˜ u isa subharmonic function on e Ω ′ . The fact that s is unbounded from below on Ω implies that˜ ρ (0) ≥ − δ .We now define the open subset e Ω of C to be ϕ (Ω) ∪ { } if ˜ ρ (0) > − δ and e Ω := ϕ (Ω)if ˜ ρ (0) = 1 − δ . The subharmonic function ˜ ρ is strictly larger than 1 − δ on e Ω and equal to1 − δ on its boundary. Hence the maximum principle implies that e Ω is unbounded, and so isa fortiori e Ω ′ . By arguing as above and applying the removal of singularity theorem also at ∞ ,we can extend the J -holomorphic map ˜ u to a J -holomorphic map ˆ u which is defined on theopen subset b Ω ′ := e Ω ′ ∪ {∞} of the Riemann sphere C ∪ {∞} such that ˆ ρ := r ◦ ˆ u satisfiesˆ ρ ( ∞ ) ≥ − δ .As before, we set b Ω to be e Ω ∪ {∞} if ˆ ρ ( ∞ ) > − δ and e Ω if ˆ ρ ( ∞ ) = 1 − δ . Then b Ω is anopen subset of the Riemann sphere, and the subharmonic function ˆ ρ is strictly larger than1 − δ on it and equal to 1 − δ on its boundary. The maximum principle now forces b Ω to bethe whole Riemann sphere and ˆ ρ to be constant on it. In particular, ˆ u is a J -holomorphicsphere taking its values in V ⊂ U . The fact that ω = d ( ı Y ω ) is exact on U implies that ˆ u isconstant, and so is u . (cid:3) Proposition 4.2.
The set U ( K, J ) is compact in C ∞ ( R × T , M ) . For every z ∈ R × T and m ∈ M there exists u ∈ U ( K, J ) such that u ( z ) = m .Proof. Compactness is proved as in Proposition 2.1. Let V be an open neighborhood of ∂M satisfying the condition of Lemma 4.1 for the pair ( K, J ) and for all elements of a smoothhomotopy joining (
K, J ) to (0 , J ), where the almost complex structure J does not dependon t and satisfies (18) on U . If m ∈ V , then the constant map taking the value m belongs to U ( K, J ). If m ∈ M \ V , then we can find a map u ∈ U ( K, J ) such that u ( z ) = m arguing asin the proof of Proposition 2.2. Indeed, in this proof we may replace the closed manifold M by the open manifold M \ ∂M because the necessary compactness for the spaces of solutions u of the various Floer equations involved satisfying u ( z ) = m is guaranteed by Lemma 4.1. (cid:3) Thanks to the above result, the action selector A : H ( M ) → R can be defined as in the closed case: A ( H ) := sup ( K,J ) ∈ D ( H ) min u ∈ U ( K,J ) a − H ( u ) , where D ( H ) := K ( H ) × J ω ( M ), with K ( H ) denoting the set of all K ∈ K ( M ) such that K − = H . The same properties that we have proved in the closed case hold also in the presentsetting. Remark 4.3 ( Exhaustions).
Consider a symplectic manifold (
M, ω ) that can be exhaustedby compact convex symplectic manifolds: M = S ∞ i =1 M i where M ⊂ M ⊂ . . . are compact SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 19 convex submanifolds of M . Also assume that [ ω ] | π ( M ) = 0. Examples are ( R n , ω ), cotangentbundles with their usual symplectic form and, more generally, Weinstein manifolds.Given a function H : T × M → R with compactly supported Hamiltonian vector field X H ,choose i so large that the support of X H is contained in the interior of M i . Then A ( H ; M j )is well-defined for j ≥ i . These action selectors are sufficient for proving several results onexhaustions ( M, ω ), like Gromov’s non-squeezing theorem or the Weinstein conjecture fordisplaceable energy surfaces of contact type.Alternatively, one can define one single action selector for (
M, ω ) as follows. While ingeneral it is not clear whether the sequence ( A ( H ; M j )) stabilizes, or whether it is monotone,it is certainly bounded, as it takes values in the spectrum of H . Hence we can define A ( H ) := lim inf j →∞ A ( H ; M j ) ∈ R . One readily checks that A ( H ) is a minimal action selector on the space of functions H : T × M → R with X H of compact support in the sense of Definition 5.2 below.5. Axiomatization and formal consequences
It is useful to define an action selector by a few properties (“axioms”) and to formallyderive other properties from these axioms. In this way, it becomes clearer which properties ofan action selector are fundamental and which other properties are just formal consequences ofthese fundamental ones. The axiomatic approach also makes clear that properties that holdfor some action selectors, but do not follow from the axioms, rely on the specific constructionof the selectors for which they hold. For example, the “triangle inequality” σ ( H H ) ≥ σ ( H ) + σ ( H )and the minimum formula σ ( H + H ) = min { σ ( H ) , σ ( H ) } for functions supported in disjoint incompressible Liouville domains, both hold for the Viterboselector and the PSS selector, but are unknown for general minimal selectors.5.1. Axiomatization.
An attempt to axiomatize action selectors was made in [3], and avery nice and slender set of four axioms was given in [10]. We here give an even smaller listof axioms, that retains the first two axioms in [10], but alters their non-triviality axiom anddiscards the minimum formula axiom.Throughout this section we assume that (
M, ω ) is connected and symplectically aspherical(i.e. [ ω ] | π ( M ) = 0). If M is closed, set H ( M ) = C ∞ ( T × M, R ), and if M is open (i.e. notclosed), let H ( M ) as in Section 4 be the set of functions in C ∞ ( T × M, R ) such that X H hascompact support in the interior of T × M . The spectrum spec ( H ) of H ∈ H ( M ) is againthe set of critical values of the action functional A H . Lemma 5.1.
The spectrum spec ( H ) is a compact subset of R with empty interior.Proof. Since the support S of X H is compact in Int( T × M ) = T × Int( M ), we find a compactsubmanifold with boundary K ⊂ Int( M ) such that S ⊂ T × Int( K ) ⊂ T × K ⊂ T × Int( M ) . It is well known that spec ( H | T × K ) is compact and has empty interior. It therefore sufficesto show that spec ( H ) = spec ( H | T × K ). The inclusion spec ( H ) ⊃ spec ( H | T × K ) is clear. Soassume that x is a 1-periodic orbit of X H that is not contained in K . Since H is locally a function of time on T × ( M \ K ), the orbit x is constant. Since M is connected, there exists y ∈ ∂K such that H ( y, t ) = H ( x, t ) = h ( t ) for all t ∈ T . Hence A H ( y ) = A H ( x ) = R T h ( t ) dt . ✷ Definition 5.2.
An action selector for a connected symplectically aspherical manifold (
M, ω )is a map σ : H ( M ) → R that satisfies the following two axioms. A1 (Spectrality) σ ( H ) ∈ spec ( H ) for all H ∈ H ( M ). A2 ( C ∞ -continuity) σ is continuous with respect to the C ∞ -topology on H ( M ).An action selector is called minimal if, in addition, A3 (Local non-triviality)
There exists a function H ∈ H ( M ) with H ≤ M such that σ ( H ) < Remark 5.3.
Assume that σ : H ( M ) → R satisfies the spectrality axiom A1. Then C ∞ -continuity of σ is equivalent to C -continuity of σ , and continuity of σ implies its monotonicity,see Assertions 5 and 4 of Proposition 5.4 below. On the other hand, it is not clear if mono-tonicity of σ , together with spectrality, implies its continuity, but this is so if σ in addition hasthe shift property σ ( H + c ) = σ ( H ) + c for all H and c ∈ R , cf. the proof of Proposition 3.4.Our selector A on closed or convex symplectically aspherical manifolds is indeed a minimalaction selector, since it is spectral by Proposition 3.2, C ∞ -continuous since even Lipschitzcontinuous with respect to the C -norm by Proposition 3.4, and non-trivial by Proposition 3.9and Remark 3.10. We note that the proof of monotonicity of A can be readily altered nearthe end to show directly that A is C ∞ -continuous. In Proposition 5.4 below we list manyother properties of (minimal) action selectors, some of which we have already verified for A .5.2. Formal consequences.
For H ∈ H ( M ) we abbreviate E + ( H ) = Z T max x ∈ M H ( t, x ) dt, E − ( H ) = Z T min x ∈ M H ( t, x ) dt. The Hofer norm of H is defined as(19) k H k = E + ( H ) − E − ( H ) = Z T (cid:18) max x ∈ M H ( t, x ) − min x ∈ M H ( t, x ) (cid:19) dt. We also recall that the function( H H )( t, x ) := H ( t, x ) + H ( t, ( φ tH ) − ( x ))generates the isotopy φ tH ◦ φ tH .A compact submanifold U of ( M, ω ) is called a
Liouville domain if (
U, ω ) is convex (seeSection 4 for the definition) and if the corresponding Liouville vector field is defined on allof U , not just near the boundary ∂U . Examples are starshaped domains in R n or fiberwisestarshaped neighborhoods of the zero section of a cotangent bundle T ∗ Q . The domain U is incompressible if the map ι ∗ : π ( U ) → π ( M ) induced by inclusion is injective. The aboveexamples are incompressible Liouville domains.Following [30] and [10] we have SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 21
Proposition 5.4.
Assume that ( M, ω ) is connected and symplectically aspherical. Then everyaction selector σ on H ( M ) has the following properties.
1. Zero: σ ( H ) = 0 if H ≡ .
2. Shift: σ ( H + r ) = σ ( H ) + R T r ( t ) dt if r : T → R is a function of time.
3. Coordinate change: If ψ is a symplectomorphism of ( M, ω ) that is isotopic to theidentity through symplectomorphisms, then σ ( H ) = σ ( H ◦ ψ ) .
4. Monotonicity: σ ( H ) ≤ σ ( H ) if H ≤ H .
5. Lipschitz continuity: E − ( H − H ) ≤ σ ( H ) − σ ( H ) ≤ E + ( H − H ) . In particular, E − ( H ) ≤ σ ( H ) ≤ E + ( H ) .
6. Energy-Capacity inequality: | σ ( H ) | ≤ k H k if φ H displaces an open set U ⊂ M such that H is supported in T × U .
7. Composition: σ ( H ) + E − ( H ) ≤ σ ( H H ) ≤ σ ( H ) + E + ( H ) .If, in addition, σ is a minimal action selector, then:
8. Non-degeneracy: If H ≤ and H = 0 , then σ ( H ) < .
9. Non-positivity: If H has support in an incompressible Liouville domain, then σ ( H ) ≤ . In particular, σ ( H ) = 0 for all non-negative Hamiltonians which are supported in anincompressible Liouville domain.Outline of the proof. Most properties are proved in [10, § M . Their proof goes through for open M if oneimposes the admissibility condition in Lemma 21 of [10] only on a compact submanifold withboundary K ⊂ Int( M ) that contains the support of all the vector fields X H s . For Property 7we compute, using Lipschitz continuity, σ ( H H ) − σ ( H ) ≤ E + ( H H − H ) = E + ( H ◦ ( φ tH ) − ) = E + ( H )and similarly σ ( H H ) − σ ( H ) ≥ E − ( H H − H ) = E − ( H ) . For the proof of Properties 8 and 9 we need two lemmas. Let U ⊂ M be a Liouville domain(the case U = M is not excluded) and let Y be the corresponding Liouville vector field. Since U is compact and Y points outwards along the boundary, the flow φ tY : U → U of Y exists forall t ≤
0. The property L Y ω = ω of Y integrates to the conformality condition ( φ tY ) ∗ ω = e t ω for t ≤
0. For each τ ≤ U τ = φ τY ( U ), and for a Hamiltonian H : T × M → R with support in U define the Hamiltonian H τ ( t, x ) := (cid:26) e τ H ( t, φ − τY ( x )) if x ∈ U τ , x / ∈ U τ . Then the support of H τ lies in U τ . The following lemma, that goes back to [23, proof of Prop.5.4], is taken from [10, § Lemma 5.5.
Let H : T × M → R be a Hamiltonian with support contained in a disjoint unionof incompressible Liouville domains. Then σ ( H τ ) = e τ σ ( H ) for all τ ≤ . For the proof one shows that spec ( H τ ) = e τ spec ( H ), and so the claim follows from thespectrality and continuity axioms of σ . Lemma 5.6. If G is autonomous with G ≤ and G = 0 , then σ ( G ) < .Proof. Choose a non-empty open set U ⊂ M such that G | U <
0. Let H and B ⊂ M be afunction and a symplectically embedded ball as in Axiom A3, and let 0 ∈ B be the centerof B . Take x ∈ U , and choose a Hamiltonian isotopy ψ of M with ψ (0) = x . Then we find τ < ψ ( B τ ) ⊂ U . Choosing τ smaller if necessary, we have G ≤ H τ ◦ ψ − . UsingProperties 3 and 4 and Lemma 5.5 we obtain σ ( G ) ≤ σ ( H τ ◦ ψ − ) = σ ( H τ ) = e τ σ ( H ) < . (cid:3) Property 8 now readily follows: Given H ≤ H = 0 we find t ∈ (0 ,
1) and x ∈ M with H ( t , x ) <
0. We can thus construct a function of the form α ( t ) G ( x ) with α anon-negative bump function around t and G as in Lemma 5.6 such that H ≤ αG . Then σ ( H ) ≤ σ ( αG ) = σ ( cG ) <
0, where c = R α ( t ) dt >
0. Indeed, the first inequality holds bymonotonicity, and the last inequality by Lemma 5.6. To see the equality σ ( αG ) = σ ( cG ),choose a smooth family of functions α s ( t ), s ∈ [0 , α ( t ) = α ( t ), α ( t ) = c isconstant, and R α s ( t ) dt = c for all s . Then the Hamiltonian functions H s ( t, x ) := α s ( t ) G ( x )all generate the same time-1 map. Lemma 5.7 below combined with the shift property 2 nowyield σ ( αG ) = σ ( H ) = σ ( H ) = σ ( cG ). Let us give a direct and more elementary proof of σ ( H ) = σ ( H ): Since X H s = α s ( t ) X G , the time-1 orbits of H s are reparametrisations of eachother. Let x be a 1-periodic orbit of H , and denote by x s the 1-periodic orbit of H s withthe same trace. Then the area term R D ¯ x ∗ s ω of the action A H s ( x s ) does not depend on s , sincewe can take the same disc for each s , and the same holds for the Hamiltonian term Z T H s ( t, x s ( t )) dt = Z T α s ( t ) G ( x s ( t )) dt = G ( x (0)) Z T α s ( t ) dt = G ( x (0)) c since the autonomous Hamiltonian cG is constant along its orbit x . It follows that spec ( H s )does not depend on s . Since this set has empty interior and σ is continuous, σ ( H s ) neitherdepends on s .We now prove Property 9. Let U ⊂ M be a Liouville domain and choose ε > F : M → R such that F ( x ) = ε on U − , F ( x ) = 0 on M \ U − , such that ε and 0 are the only critical values of F , and such that X F has no non-constant1-periodic orbits. By spectrality, σ ( F ) ∈ { , ε } . Take G as in Lemma 5.6 with support in M \ U − and such that G ≥ − ε . Then F − ε ≤ G and hence σ ( F − ε ) ≤ σ ( G ) <
0. Togetherwith the shift property, σ ( F ) = σ ( F − ε ) + ε < ε , whence σ ( F ) = 0.Given H ∈ H ( U ) we find τ < H τ ≤ F . Then σ ( H τ ) ≤ σ ( F ) = 0 bymonotonicity, and so σ ( H ) = e τ σ ( H τ ) ≤ ✷ Path independence.
Let σ : H ( M ) → R be an action selector. Every function H ∈ H ( M ) generates a Hamiltonian diffeomorphism φ H . Does σ induce a map Ham( M, ω ) → R on the group formed by these diffeomorphisms? SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 23
PSfrag replacements f GH τ ε − ε ∂U − ∂U − ∂U Figure 6.
The functions f , G , and H τ If two functions in H ( M ) differ by a constant, they have the same time-1 map. In thisparagraph we therefore restrict σ to normalized functions: If M is closed, H is normalized if R M H ( t, · ) ω n = 0 for all t ∈ T . If M is open, we fix an end e of Int( M ) and say that H isnormalized if for each t ∈ T the function H ( t, · ) vanishes on e ; notice that this normalizationdepends on the choice of e . Write H ∼ H if H , H are the endpoints of a smooth path H s of normalized functions that all generate the same Hamiltonian diffeomorphism. Lemma 5.7. If H ∼ H , then σ ( H ) = σ ( H ) .Proof. The claim follows from the continuity of σ if one knows that the sets spec ( H s ) areindependent of s . This in turn easily follows for closed M if the flow of H has a contractible1-periodic orbit, see [27, § M if the flow of H has a constant orbit of actionzero, see [4, Cor. 6.2]. The existence of such an orbit for closed M follows from Proposi-tions 2.1 and 2.2, see Remark 2.3, and for open M is obvious (take a point in the end e offthe support of H ). ✷ The lemma implies that σ descends from the set of normalized Hamiltonians to the universalcover ] Ham(
M, ω ), where for open M we denote by Ham( M, ω ) the group of Hamiltoniandiffeomorphisms of M generated by normalized Hamiltonians. Does σ further descend toHam( M, ω )? In other words, is it true that σ ( G ) = σ ( H ) if φ G = φ H for normalized G, H ?This is so if one knows that φ G = φ H for normalized G, H implies that spec ( G ) = spec ( H ),and that σ satisfies the triangle inequality, see [4, proof of Prop. 7.1]. The first requirementalways holds true. Lemma 5.8.
Let ( M, ω ) be a symplectically aspherical manifold. If φ G = φ H for normalizedHamiltonians G, H ∈ H ( M ) , then spec ( G ) = spec ( H ) .Proof. This is again easy to verify for M open [4, Cor. 6.2]. For M closed the proof is moredifficult. Under the additional assumption that also the first Chern class of ( M, ω ) vanisheson π ( M ), the proof is given by Schwarz [27, Theorem 1.1]. One can dispense with thisassumption thanks to results of McDuff [14]. We give a rough outline of the argument.Let γ be the loop in Ham( M, ω ) obtained by first going along φ tG and then along φ − tH , for t ∈ [0 , γ one associates a bundle E with fiber M and base S by gluing two copiesof the trivial bundle M × D over the closed disk along their boundaries via the loop γ . Thetotal space E comes with a closed 2-form ω E , the so-called coupling form, that restricts to ω on each fiber. The assertion of the lemma will follow if we can show that(20) Z S s ∗ ω E = 0for one and hence any section s : S → E , see [27, Lemma 4.6].Let J be an almost complex structure on E that is ω -compatible on each fibre and suchthat the projection E → S is J - i -holomorphic, where i is the usual complex structureon S . Since [ ω ] vanishes on π ( M ), for a generic choice of J the space M ( J ) of holomorphicsections s : S → E is a closed manifold, and its dimension is 2 n . Further, for given z ∈ S the evaluation map M ( J ) → M, ev z ( u ) = u ( z )has non-vanishing degree, see [14, p. 117]. Now (20) follows exactly as in the proof of Corol-lary 4.14 in [27]. ✷ On the other hand, we do not know whether the triangle inequality holds for our actionselector A . At least for Liouville domains, one can go around the triangle inequality andprove the following result. Proposition 5.9.
Assume that ( M, ω ) is a Liouville domain. Then for any action selector σ on H ( M ) it holds that σ ( G ) = σ ( H ) whenever G, H are normalized Hamiltonians with φ G = φ H .Proof. The claim is shown for ( R n , ω ) in [9, Proposition 11 in § L be a normalized Hamiltonian such that φ L = id, that is, φ tL , t ∈ [0 , M, ω ). The ends of Int( M ) are in bijection with the components N , . . . , N k of theboundary of M . By assumption, L ( x, t ) = h j ( t ) for x near N j , and one of these functionsvanishes. Denote the Liouville vector field on M again by Y , and for τ ≤ M τ = φ τY ( M ).Then the function L τ ( t, x ) = e τ L ( t, φ − τY ( x )) if x ∈ M τ ,e τ h j ( t ) if x ∈ [ τ 0. Hence σ ( H ) ≤ σ ( G ). In the same way, σ ( G ) ≤ σ ( H ). ✷ Corollary 5.10. The conclusion of Proposition 5.9 also holds for all compact 2-dimensionalsymplectic manifolds ( M, ω ) with M not diffeomorphic to the 2-sphere. SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 25 Proof. If ( M, ω ) is closed and different from the sphere, then the fundamental group ofHam( M, ω ) is trivial [22, § M is not closed, then ( M, ω )is a Liouville domain, see [16, Exercise 3.5.30], and so the claim follows from Lemma 5.9. ✷ Three applications of the existence of a minimal action selector In this section we illustrate by three examples how the existence of a minimal action selectorprovides short and elementary proofs of theorems in symplectic geometry and Hamiltoniandynamics. Our examples are Gromov’s non-squeezing theorem, the existence of periodic orbitsnear displaceable energy surfaces, and the unboundedness of Hofer’s metric.6.1. Gromov’s non-squeezing theorem. In the standard symplectic vector space ( R n , ω )with n ≥ ω = P nj =1 dx j ∧ dy j we consider the open ball B n ( r ) of radius r and thecylinder Z n ( R ) = B ( R ) × R n − = { x + y < R } . For any r > B n ( r ) embedsinto Z n ( R ) by a volume preserving embedding; just take a suitable diagonal linear map ofdeterminant one. Every symplectic embedding ϕ : B n ( r ) → Z n ( R ) is volume preserving,since ϕ ∗ ( ω n ) = ( ϕ ∗ ω ) n = ω n , but there is no such embedding if r > R . This celebrated theorem of Gromov [6] shows thatsymplectic mappings are much more rigid than volume preserving mappings. Theorem 6.1. If r > R there exists no symplectic embedding of the ball B n ( r ) into thecylinder Z n ( R ) .Proof. Let ϕ : B n ( r ) → Z n ( R ) be a symplectic embedding. Fix ε ∈ (0 , r ), and choose asymplectic ellipsoid E ( R , . . . , R n ) = ( ( x , y , . . . , x n , y n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 x j + y j R j < ) such that(21) ϕ (cid:0) B n ( r − ε ) (cid:1) ⊂ E ( R , . . . , R n ) ⊂ Z n ( R ) . By the elementary Extension after restriction principle from [1], see also [26, Appendix A],there exists a compactly supported Hamiltonian function G on T × R n such that φ G = ϕ on B n ( r − ε ). Choose ρ so large that the support of G is contained in B n ( ρ ). Let σ be aminimal action selector on the convex symplectic manifold ( M, ω ) = ( B n ( ρ ) , ω ). For everyopen subset U of the interior of M we set c σ ( U ) = sup {| σ ( H ) | | H has compact support in T × U } . We will prove that π ( r − ε ) (cid:13) ≤ c σ ( B n ( r − ε )) (cid:13) = c σ ( φ G ( B n ( r − ε ))) (cid:13) ≤ c σ ( E ( R , . . . , R n )) (cid:13) ≤ π ( R + ε ) (cid:13) ≤ π ( R + ε ) . Theorem 6.1 then follows since ε > f : [0 , + ∞ ) → R with support in [0 , π ( r − ε ) ) such that f (0) = min f < , f ′ ( s ) ∈ [0 , ∀ s ∈ [0 , + ∞ ) , f ( s ) = 0 if f ′ ( s ) = 0 . While all orbits of the Hamiltonian flow of the function H f : R n → R , H f ( z ) = f ( π | z | ),are closed, only those on the sphere of radius p s/π with f ′ ( s ) ∈ Z have period one. Hencethe spectrum of H f contains only 0 and min H f . By the non-degeneracy property 8 inProposition 5.4, σ ( H f ) < σ ( H f ) = min H f by the spectrality axiom. Since we canchoose f such that min H f is as close to − π ( r − ε ) as we like, inequality 1 (cid:13) follows.Equality 2 (cid:13) follows from the coordinate change property 3 in Proposition 5.4.Inequality 3 (cid:13) follows from the first inclusion in (21): We can use more Hamiltonian func-tions in E ( R , . . . , R n ) than in φ G ( B n ( r − ε )) ⊂ ϕ ( B n ( r − ε )) ⊂ E ( R , . . . , R n ).It is easy to construct a compactly supported Hamiltonian function K on R such that k K k ≤ π ( R + ε ) and such that φ K displaces B ( R ), see [9, p. 171]. Let K be a compactlysupported cut-off of the function ( x , y , . . . , x n , y n ) K ( x , y ) such that φ K displaces E ( R , . . . , R n ) and k K k = k K k . Choosing ρ larger if necessary, we can assume that thesupport of K is contained in B n ( ρ ). The energy-capacity inequality 6 in Proposition 5.4 nowimplies that c σ ( E ( R , . . . , R n )) ≤ π ( R + ε ) .Finally, the second inclusion in (21) shows that π ( R + ε ) ≤ π ( R + ε ) . ✷ Existence of periodic orbits near a given energy surface. The search for periodicorbits of prescribed energy is a traditional topic of celestial mechanics and therefore also ofHamiltonian dynamics. We consider an autonomous Hamiltonian H : M → R on a symplecticmanifold ( M, ω ), and assume that c is a regular value of H with compact energy surface S c = H − ( c ). By preservation of energy, S c is invariant under the Hamiltonian flow of H .Examples by Ginzburg [5] and Herman [8] show that S c may carry no periodic orbit. Wetherefore look for periodic orbits on nearby energy surfaces S c ′ = H − ( c ′ ). Theorem 6.2. Let ( M, ω ) be a compact symplectically aspherical symplectic manifold, whichis either closed or convex, and assume that the compact and regular energy surface S c = H − ( c ) is disjoint from ∂M and displaceable, namely there exists a smooth function K : T × M → R with support in T × ( M \ ∂M ) such that φ K ( S c ) ∩ S c = ∅ . Then there exists a sequence c j → c of regular values of H such that every energy surface S c j carries a periodic orbit ofthe flow of H . By applying this result to sufficiently large balls or disc bundles we obtain the existence ofnearby periodic orbits for compact regular hypersurfaces in R n and (under the displaceabilityassumption) in cotangent bundles. Proof of Thereom 6.2. Since c is a regular value and S c is disjoint from ∂M , we find anopen interval I = ( c − ε, c + ε ) of regular values of H such that the union U = ` c ′ ∈ I S c ′ ofdiffeomorphic hypersurfaces forms an open neighbourhood of S c in M \ ∂M . Choose a smoothfunction K : T × M → R with support in T × ( M \ ∂M ) such that φ K ( S c ) ∩ S c = ∅ . Then φ K displaces a whole neighbourhood of S c . We can therefore choose ε smaller if necessary, suchthat φ K ( U ) ∩ U = ∅ . Let f ε : R → R be a smooth non-positive function with support in I whose only critical values are 0 and −k K k − 1, see Figure 7.The function H ε := f ε ◦ H has support in U . Let σ be a minimal action selector for M (it existsby our construction in Sections 3 and 4 and by Remark 4.3). By the non-degeneracy property 8in Proposition 5.4, σ ( H ε ) < 0. Further, the energy-capacity inequality 6 in Proposition 5.4shows that | σ ( H ε ) | ≤ k K k . Hence −k K k − < σ ( H ε ) < . SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 27 PSfrag replacements R c − ε c + ε −k K k − f ε Figure 7. The function f ε Since the only critical values of H ε are 0 and −k K k − 1, and since σ ( H ε ) belongs to thespectrum of H ε , it follows that H ε has a non-constant 1-periodic orbit in U . A constantreparametrization of this orbit is a periodic orbit γ of H , and H ( γ ) ∈ I . Since ε > ✷ Theorem 6.2 can be improved in two directions: First, an elementary additional argumentshows that the set of energies c ′ ∈ ( c − ε, c + ε ) at which the flow of H has a periodic orbitactually forms a set of full Lebesgue measure. Secondly, assume in addition that near S c one can find a Liouville vector field Y transverse to S c . In this case, S c is called of contacttype. Using the local flow φ tY of Y we define another foliation ` c ′ ∈ I e S c ′ =: e U with centralleaf e S c = S c by e S c ′ := φ c ′ − cY S c . We now look at the “tautological” function e H : e U → R given by e H ( x ) = c ′ if x ∈ e S c ′ . Therestrictions of the Hamiltonian flow of e H to S c and e S c ′ are conjugate under φ c ′ − cY up to aconstant time-change. By Theorem 6.2 we find c ′ such that the flow of e H has a periodic orbiton e S c ′ . Hence the flow of e H also has a periodic orbit on S c . A reparametrisation of this orbitis a periodic orbit of the flow of our original function H . This result proves a special caseof the Weinstein conjecture on the existence of a periodic Reeb orbit on any closed contactmanifold. We refer to Sections 4.2 and 4.3 of [9] for detailed proofs of these improvements.6.3. Unboundedness of Hofer’s metric. By Darboux’s theorem, every symplectic mani-fold ( M, ω ) locally looks like the standard symplectic vector space of the same dimension, andso there are no local geometric invariants of ( M, ω ). However, on the group Ham( M, ω ) ofHamiltonian diffeomorphisms there is a bi-invariant Finsler metric, the so-called Hofer metric,which is defined by d ( φ, id) = inf H k H k , where H varies over those H ∈ H ( M ) with φ H = ϕ and where k H k is the Hofer normdefined by (19). The only difficult point in verifying that d is indeed a metric is its non-degeneracy. For closed or convex symplectically aspherical manifolds, this can be done byusing any minimal action selector. We leave this nice exercise to the reader. Note that in the above infimum we can ask the Hamiltonian H to be normalized as inSection 5.3, which for M closed means that Z M H ( t, · ) ω n = 0 ∀ t ∈ T . Indeed, any Hamiltonian can be normalized by adding a suitable function of t , and this opera-tion neither affects the Hamiltonian vector field nor the Hofer norm. Symplectic geometers usetheir metric intuition to prove results on the metric space (Ham( M, ω ) , d ), which in turn helpunderstanding the dynamics and the symplectic topology of the underlying manifold ( M, ω ),see for instance [22]. A first question one can ask on a metric space is whether it has boundeddiameter. The following result was proven by Ostrover [20] under the assumption that alsothe first Chern class vanishes on π ( M ), and by McDuff [14] without this assumption. Theyboth used the PSS selector. Theorem 6.3. Let ( M, ω ) be a closed symplectically aspherical manifold. Then the Hofermetric on Ham( M, ω ) is unbounded.Proof. Let B ⊂ M be a symplectically embedded ball in M , so small that there exists aHamiltonian diffeomorphism h of M with h ( B ) ∩ B = ∅ . We can assume that h is the time-1map of an autonomous and normalized Hamiltonian H . Let f : M → R be a function suchthat f = 1 on M \ B and R M f ω n = 0.PSfrag replacements f h B Figure 8. The function f and the map h For s ∈ R consider the Hamiltonian diffeomorphism φ s = h ◦ φ sf = h ◦ φ sf . Let G s be any normalized Hamiltonian generating φ s . We shall prove that(22) spec ( G s ) = spec ( H ) + s. Now let σ be any (not necessarily minimal) action selector on H ( M ). Since spec H hasempty interior and σ is continuous, (22) implies that σ ( G s ) = s + s for some s ∈ R andevery s ∈ R . Further, since G s is normalized, E − ( G s ) ≤ ≤ E + ( G s ) for every s . Property 5in Proposition 5.4 thus implies that k G s k = E + ( G s ) − E − ( G s ) ≥ E + ( G s ) ≥ σ ( G s ) = s + s. This holds for all normalized Hamiltonians generating φ s , and so d ( φ s , id) ≥ s + s → + ∞ as s → + ∞ . SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 29 In order to prove (22) we use that by Lemma 5.8 the set spec ( G s ) does not depend onthe specific choice of the normalized Hamiltonian G s generating φ s . It therefore suffices toprove (22) for the “natural” Hamiltonian generating h ◦ φ sf given by b G s ( t, x ) = ( α ( t + ) s f ( x ) if t ∈ (cid:2) , (cid:3) ,α ( t ) H ( x ) if t ∈ (cid:2) , (cid:3) , that first generates the map φ sf in time and then generates the map h in time , yielding h ◦ φ sf in time 1. Here, α : R → R is a smooth non-negative function with support in ( , R R α ( t ) dt = 1. At first reading one should take α ≡ 2, but this would result in aHamiltonian b G s not smooth at t = .Since h ( B ) ∩ B = ∅ , the contractible 1-periodic orbits of φ t b G s are exactly the contractible1-periodic orbits of φ tαH : Such an orbit γ must start outside the ball B and does not movefor t ∈ (cid:2) , (cid:3) , hence f ≡ γ . The autonomous Hamiltonian H is also constant along γ .After reparametrization, γ corresponds to a 1-periodic γ H of φ tH .Given such an orbit γ and a disc γ that restricts to γ along its boundary, we compute theactions A b G s ( γ ) = Z γ ω + s Z α ( t + ) f ( γ ( t )) dt + Z α ( t ) H ( γ ( t )) dt = Z γ ω + s + Z H ( γ H ( t )) dt = A H ( γ H ) + s. Claim (22) follows. ✷ Further directions and open problems In this section we describe a few modifications of the construction in Section 3. For theproofs of the claims made we refer to [7]. To fix the ideas we assume that the symplecticallyaspherical manifold ( M, ω ) is closed.7.1. Smaller deformation spaces. Our definition of an action selector admits several vari-ations.7.1.1. Smaller classes of functions deforming H . The set K ( H ) is a large class of deforma-tions of H , and it might be useful to consider smaller classes. Definition 7.1. A subset K ′ ( M ) = S H K ′ ( H ) of K ( M ) is admissible if the followingholds: For any pair H ≥ H and for any K ∈ K ′ ( H ) with supp ∂ s K ⊂ [ s − , s + ], every K ∈ K ( H ) with ∂ s K ≤ s ≤ s − and K = K for s ≥ s − belongs to K ′ ( H ).For every admissible set K ′ ( M ) ⊂ K ( M ) and D ′ ( H ) := K ′ ( H ) × J ω ( M ), A ′ ( H ) = sup ( K,J ) ∈ D ′ ( H ) min U ( K,J ) a − H defines a minimal action selector.Examples of admissible sets are given by the monotone decreasing deformations ( ∂ s K ≤ c by the set K c ( M ) = { K ∈ K ( M ) | K + = c } . Of course, A ′ ≤ A for every admissible subset K ′ ( M ) of K ( M ). For the classes K c ( M ) equality holds, see [7,Prop. 4.5.2]: Proposition 7.2. For every c ∈ R we have A c ( H ) = A ( H ) for all H ∈ C ∞ ( T × M, R ) . For functions K ∈ K ( H ), the removal of singularity theorem shows that the elementsof U ( K, J ) are actually open disks which are J + -holomorphic near the origin and satisfy theFloer equation on a collar of the boundary equipped with cylindrical coordinates. These areexactly the objects which are used in the PSS isomorphism from [21], see Section 7.3 below.7.1.2. Smaller classes of almost complex structures. Given an ω -compatible almost complexstructure J on M that does not depend on s , define A J ( H ) = sup K ∈ K ( H ) min U ( K,J ) a − H . While we do not know if A J ( H ) depends on J , the number sup J A J ( H ) is of course indepen-dent of J . All of the functions A J ( H ) and sup J A J ( H ) on C ∞ ( T × M ) are minimal actionselectors, by the same (and sometimes easier) arguments as for A ( H ). We have chosen togive the construction for A ( H ) since this is more natural given our deformation approach.Clearly, A ( H ) ≥ sup J A J ( H ) ≥ A J ( H ) for every J and H .Are these inequalities all equalities? A class of Hamiltonian functions for which A ( H ) = A J ( H ) for every J is given in Remark 7.6. A somewhat different class is given by theintersection of the Hamiltonians in Proposition 3.9 and the proposition below.For the selectors A J and hence also for sup J A J we have the following variant of Proposi-tion 3.9, which is Proposition 4.4.2 in [7]. Its proof appeals to the transversality and gluinganalysis from Floer theory. Proposition 7.3. Let H ∈ C ∞ ( M ) be an autonomous Hamiltonian such that X H has nonon-constant contractible closed orbits of period T ∈ (0 , . Then for every ω -compatible J , A J ( H ) = min M H. Action selectors associated to other cohomology classes. By using the resultstated in Remark 2.4, one can construct spectral values A ( ξ, H ) ∈ spec ( H ) for every non-zero cohomology class ξ ∈ H ∗ ( M ; Z ). In the case ξ = 1 ∈ H ( M ; Z ), the value A (1 , H )agrees with A ( H ). These spectral values are monotone and continuous in H and hence areaction selectors, but for ξ = 1 they are in general larger than the action selector A and notminimal. For instance, for the generator [ M ] of H n ( M ; Z ) and for C -small autonomousHamiltonians with exactly two critical values we have A ([ M ] , H ) = max M H. We refer to [7, § 5] for the proofs and for further properties of these action selectors.7.3. Comparison with the PSS selector. Recall that in [27] and [4] the PSS selector wasconstructed on closed and convex symplectically aspherical manifolds with the help of Floerhomology. While our selector A already has many applications to Hamiltonian dynamics andsymplectic geometry, some of the applications of the PSS selector rely on additional properties,that we were not able to verify for the selector A . One such property is the triangle inequality σ PSS ( G H ) ≥ σ PSS ( G ) + σ PSS ( H ) , SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 31 that is stronger than the composition property 7 in Proposition 5.4. Proving the triangleinequality requires the compatibility of the selector with the pair of paints product. Thetriangle inequality can be used, for instance, to define a bi-invariant metric on the groupHam( M, ω ) that in general is different from the Hofer metric, and to construct partial sym-plectic quasi-states [2, 24]. Another property of the PSS selector is the minimum formulafrom [10]: Given H and H with support in disjoint incompressible Liouville domains, σ PSS ( H + H ) = min { σ PSS ( H ) , σ PSS ( H ) } . It is shown in [10] that for any minimal action selector σ satisfying this formula there is analgorithm for computing σ on autonomous Hamiltonians on surfaces different from the sphere.All properties of the PSS selector would of course hold for our selector A if we couldshow that they agree. The selectors A and σ PSS both select “essential” critical values, but ina rather different way: While A ( H ) is the highest critical value of A H such that all strictlylower critical points can be “shaken off”, σ PSS ( H ) is the A H -action of the lowest homologicallyvisible generator of the Floer homology of H . Assuming that the reader is familiar with Floerhomology, we describe σ PSS ( H ) in a way relevant for its comparison with A ( H ).By the C -continuity of both selectors, we can assume that all contractible 1-periodic orbitsof H are non-degenerate, in the sense that for every such orbit x , 1 is not in the spectrum ofthe linearized return map dφ H ( x (0)). There are then finitely many 1-periodic orbits of φ tH .Fix K ∈ K ( H ) such that K + = 0 and J with J − generic. Recall that for such functions K ,for every element u ∈ U ( K, J ) the limit ev( u ) := lim s → + ∞ u ( s, t ) ∈ M exists. Choose aMorse function f on M with only one minimum m , and let W s ( m ) be the stable manifoldof m with respect to the gradient flow −∇ f of a generic Riemannian metric on M . Then σ PSS ( H ) is the smallest action A H ( x ) of a contractible 1-periodic orbit x with the followingproperties: x is a generator of HF ( H, J − ; Z ) (namely x is in the kernel of the Floer boundaryoperator ∂ J − but not in its image, and x has Conley–Zehnder index 0), and the number ofthose elements u ∈ U ( K, J ) that start at x and satisfy ev( u ) ∈ W s ( m ) is odd. Then clearly σ PSS ( H ) ≥ min U ( K,J ) a − H . Since σ PSS ( H ) does not depend on the choice of K ∈ K ( H ) noron J , we conclude that σ PSS ( H ) ≥ A ( H ). Together with Proposition 7.2 we obtain thefollowing result, which is Proposition 9.1.1 in [7]. Proposition 7.4. σ PSS ( H ) ≥ A ( H ) for all H ∈ C ∞ ( T × M, R ) . Open Problem 7.5. Is it true that A ( H ) = σ PSS ( H ) for all H ∈ C ∞ ( T × M, R ) ? The following remark was made by the referee. Remark 7.6. The equality A ( H ) = σ PSS ( H ) holds for C -small autonomous Hamiltonians H ,and more generally for those H with the following properties. (H1) There exists m ∈ M such that H t ( m ) = min x ∈ M H t ( x ) for every t ∈ T . (H2) dφ tH ( m ) for all t ∈ (0 , . (H3) The flow of X H has no non-constant contractible closed orbits of period T ∈ (0 , .Indeed, for such Hamiltonians we have (23) A ( H ) = σ PSS ( H ) = Z T H t ( m ) dt. For the proof we shall show that Z T H t ( m ) dt = σ PSS ( H ) ≥ A ( H ) ≥ Z T H t ( m ) dt. The first inequality follows from Proposition 7.4. Assumption (H3) in particular impliesthat the only contractible 1-periodic orbits of the flow of X H are the rest points. Togetherwith (H1) we obtain R T H t ( m ) dt = min spec ( H ), whence the second inequality follows inview of the spectrality of A . For the equality(24) Z T H t ( m ) dt = σ PSS ( H )we first notice that the assumptions (H1) and (H3) imply that the constant orbit m is acritical point of the action functional A sH for every s ∈ [0 , 1] and that for any other criticalpoint y of A sH ,(25) A sH ( y ) ≥ A sH ( m ) = s Z T H t ( m ) dt, s ∈ [0 , . By the continuity of σ PSS it suffices to prove (24) for a C -close Hamiltonian. In view of thenon-degeneracy assumption (H2) we find a C -small perturbation such that the contractible1-periodic orbits of the new H are non-degenerate and such that (H1), (H2), and (25) stillhold for the same point m . (There now may be non-constant contractible 1-periodic orbits y .)In the above description of the PSS selector we then choose the Morse function f suchthat m is the unique minimum and the deformation K of the form K = β ( s ) H with a cut-offfunction β . One can now show using (25) that for a generic choice of the path J s the criticalpoint m is indeed selected by σ PSS , see the proof of Theorem 5.3 in [4].We also remark that the inequality A ( H ) ≥ A J ( H ) and the spectrality of A J imply that onthe above class of Hamiltonian functions, every action selector A J is also equal to the threequantities in (23). Appendix A.In this appendix we prove the following existence result for zeroes of a section of a Banachbundle, which is used in the proof of Proposition 2.2. Results of this kind are well-knownand widely used in nonlinear analysis. The proof uses standard ideas from degree theory forproper Fredholm maps. Theorem A.1. Let π : E → B be a smooth Banach bundle over the Banach manifold B andlet S : [0 , × B → E be a C map such that S ( t, · ) is a section of E for every t ∈ [0 , . Assume that S satisfiesthe following conditions: (i) The inverse image S − (0 E ) of the zero section E is compact. (ii) For every ( t, x ) ∈ S − (0 E ) the fiberwise differential of the section S ( t, · ) at x is a Fred-holm operator of index . (iii) There exists a unique x ∈ B such that S (0 , x ) ∈ E . (iv) The fiberwise differential of S (0 , · ) at x is an isomorphism.Then the restriction of the projection [0 , × B → [0 , to S − (0 E ) is surjective. In particular,there exists at least one x ∈ B such that S (1 , x ) ∈ E .Proof. In order to simplify the notation, we assume that the Banach bundle E has a globaltrivialization E ∼ = B × Y , where the Banach space Y is the typical fiber of E . The bundle towhich we applied the theorem in the proof of Proposition 2.2 has a global trivialization, since SIMPLE CONSTRUCTION OF AN ACTION SELECTOR 33 its typical fiber is an L p -space and the general linear group of L p -spaces is contractible by aversion of Kuiper’s theorem, [17]. By using such a trivialization, we write S ( t, x ) = ( x, F ( t, x ))for a suitable C -map F : [0 , × B → Y that has the following properties:(i’) The inverse image F − (0) of 0 ∈ Y is compact.(ii’) For every ( t, x ) ∈ F − (0) the differential of the map F ( t, · ) at x is a Fredholm operatorof index 0.(iii’) There exists a unique x ∈ B such that F (0 , x ) = 0.(iv’) The differential of F (0 , · ) at x is an isomorphism.We wish to show that the restriction of the projection [0 , × B → [0 , 1] to F − (0) issurjective. By (ii’) the differential of F at each ( t, x ) ∈ F − (0) is a Fredholm operator ofindex 1. Since Fredholm operators of a given index form an open set, there exists an openneighborhood U ⊂ [0 , × B of F − (0) on which F is a Fredholm map of index 1. Fix p ∈ F − (0). Since Fredholm maps are locally proper [28, p. 862, (1.6)], we find an openneighborhood V ( p ) such that V ( p ) ⊂ U and F | V ( p ) is a proper map. Since F | V ( p ) : V ( p ) → Y is a proper C Fredholm map of index 1, the Sard–Smale theorem [28] implies that theset R ( p ) of regular values of F | V ( p ) : V ( p ) → Y is residual in Y (actually, in [28] the Sard–Smale theorem is stated under the assumption that the domain of the map – in our case[0 , × B – is second countable, but the proof consists in showing the above local statementfor proper Fredholm maps, see also [25, p. 1106]). Here, if V ( p ) intersects { , } × B we view V ( p ) as a manifold with boundary V ( p ) ∩ ( { , } × B ), and a boundary point is consideredto be regular if it is regular for the restriction of F to the boundary. By (i’) we find finitelymany points p j ∈ F − (0) such that F − (0) ⊂ S j V ( p j ) =: V . The set R := T j R ( p j ) ⊂ Y isalso residual, and F | V is a proper map.By (iii’) and (iv’), the point (0 , x ) belongs to F − (0) and F (0 , · ) is a local C -diffeomorphismat x . Since x is the unique zero of F (0 , · ), up to reducing V we may assume that the re-striction of F to V ∩ ( { } × B ) is a diffeomorphism onto an open subset of Y containing aball of radius r centered in 0.Denote by ∂V the topological boundary of V in [0 , × B . In Figure 9 this set is indicatedby the dashed curves. Proper maps between metric spaces are closed, so F ( ∂V ) is a closed setand, since it does not contain 0, there exists a positive number r such that all the elementsof F ( ∂V ) have norm at least r .Altogether, for every natural number n we can find a regular value y n ∈ Y with(26) k y n k < min (cid:8) − n , r , r (cid:9) . The set F − ( { y n } ) ∩ V is a one-dimensional submanifold of V , and its boundary is precisely F − ( { y n } ) ∩ V ∩ ( { , } × B ). Indeed, the fact that k y n k < r implies that F − ( { y n } ) doesnot intersect the topological boundary ∂V of V in [0 , × B . Together with the fact that F | V is proper, this implies that the set F − ( { y n } ) ∩ V is compact. Therefore, F − ( { y n } ) ∩ V is afinite union of C -embedded images of S and [0 , , 1] have boundarypoints on V ∩ ( { , } × B ). The fact that k y n k < r and the property of the restriction of F to V ∩ ( { } × B ) stated above imply that F − ( { y n } ) ∩ V has exactly one point on { } × B ,that we denote by (0 , z n ). Denote by Γ n the connected component of F − ( { y n } ) ∩ V thatcontains (0 , z n ). By what we have said above, Γ n is an embedded image of [0 , 1] with one PSfrag replacements 0 1 txBx z n V Γ n Figure 9. 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Alberto Abbondandolo, Fakult¨at f¨ur Mathematik, Ruhr-Universit¨at Bochum E-mail address : [email protected] Carsten Haug, Institut de Math´ematiques, Universit´e de Neuchˆatel E-mail address : [email protected] Felix Schlenk, Institut de Math´ematiques, Universit´e de Neuchˆatel E-mail address ::