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Action selectors and Maslov class rigidity

In this paper we detect new restrictions on the Maslov class of displaceable Lagrangian submanifolds of symplectic manifolds which are symplectically aspherical. These restrictions are established using action selectors for Hamiltonian flows. In particular, we construct and utilize a new action selector for the flows of a special class of Hamiltonian functions which arises naturally in the study of Hamiltonian paths which minimize the Hofer length functional.


INTRODUCTION
One of the central rigidity phenomena in symplectic topology concerns restrictions on the Maslov class of a Lagrangian submanifold which can not be detected by topological methods alone. This type of rigidity was discovered independently by Viterbo in [Vi] and Polterovich in [Po1], for Lagrangian submanifolds of symplectic vector spaces and certain cotangent bundles. In this paper we study Maslov class rigidity for Lagragian submanifolds of closed symplectic manifolds which are symplectically aspherical. That is, we assume that both the symplectic form and the first Chern class vanish on the second homotopy group. Such symplectic manifolds hold a special place in symplectic topology, in particular for results which concern the contractible periodic orbits of Hamiltonian flows, see for example [CZ,Gi2,Hi,SZ,Schw]. In this setting, Hamiltonian Floer homology becomes a more precise tool for studying periodic orbits, in part, because the actions and Conley-Zehnder indices of these orbits are both well-defined. We will exploit this precision to establish new restrictions on the Maslov class of displaceable Lagrangian submanifolds, i.e., Lagrangian submanifolds which can be displaced by a Hamiltonian diffeomorophism. In the process, we will construct a new action selector for Hamiltonian functions which have properties that are necessary to generate Hamiltonian paths that minimize the (positive) Hofer length, but fail to do so.
Let L be a Lagrangian submanifold of a symplectic manifold (M, ω). Recall that the Maslov class of L is a homomorphism µ L Maslov from the second relative homotopy group π 2 (M, L) to Z, and the minimial Maslov number of L is the largest positive integer N L such that the image of µ L Maslov is contained in N L Z. When (M, ω) is symplectically aspherical, more precisely when the first Chern class of M vanishes on π 2 (M ), the Maslov class descends to a homomorphism µ L Maslov : ker i * → Z, where ker i * is the kernel of the map i * : π 1 (L) → π 1 (M ) induced by the inclusion i : L ֒→ M . In the aspherical setting, the symplectic form also yields a homomorphism ω : ker i * → R defined by integrating ω over spanning discs.
As in [Vi], we will study µ L Maslov by considering Hamiltonian flows which are supported in normal neighborhoods of L, where they can be identified with reparameterizations of geodesic flows on T * L. For displaceable Lagrangian submanifolds, these Hamiltonian flows will eventually fail to minimize the Hofer length functional, and this failure will be shown to imply the existence of special periodic orbits via an action selector which is defined in terms of Hamiltonian Floer theory. The Conley-Zehnder indices of these special periodic orbits will yield bounds on the Maslov indices of the corresponding closed geodesics on L, and these bounds will imply the desired rigidity results.
Let g be a Riemannian metric on L. The closed geodesics of g, with period equal to one, are the critical points of the energy functional E g which is defined on the space of smooth loops C ∞ (R/Z, L) by For a Morse-Bott nondegenerate energy functional E g , we will denote by d g the maximum dimension of a critical submanifold of E g which consists of nonconstant closed geodesics. The nullspace of the Hessian of E g at a closed geodesic q(t) is spanned by the periodic Jacobi fields along q(t). If the metric g has nonpositive sectional, then these Jacobi fields must be constant and so d g ≤ dim L. For example, for the flat metric on a torus one has d g = dim L, and if g is a metric with negative sectional curvature then d g = 1.
The following is our main result.
Theorem 1.1. Let L be a displaceable Lagrangian submanifold of a closed symplectic manifold (M, ω) which has dimension 2n and is symplectically aspherical. Suppose that L admits a metric g of nonpositive sectional curvature whose energy functional E g is Morse-Bott. Then there is a class A in ker i * with ω(A) > 0 and µ L Maslov (A) ∈ [n + 1 − d g , n + 1 (+1)], where the (+1) term does not contribute if L is orientable. In particular, since d g ≤ n, we have N L ∈ [1, n + 1 (+1)].
The bounds of Theorem 1.1 can often be further refined by using the fact that the Maslov class of an orientable Lagrangian submanifold takes values in 2Z. For example, in dimension four we have Theorem 1.2. Let L be a displaceable Lagrangian surface of a closed fourdimensional symplectic manifold (M, ω) which is symplectically aspherical. If L is orientable and has genus at least one, then N L = 2.
For the important case of Lagrangian tori we obtain the following result. Theorem 1.3. Let (M, ω) be a closed symplectic manifold of dimension 2n which is symplectically aspherical. If L is a displaceable Lagrangian torus in (M, ω), then N L ∈ [2, n + 1].
This generalizes the main theorem of [Vi], which holds for Lagrangian tori in symplectic vector spaces. By Darboux's theorem, Theorem 1.3 also implies Viterbo's result. On the other hand, one can easily construct examples of displaceable Lagrangian tori which are not contained in any symplectically embedded ball.
Lagrangian intersections. Theorem 1.1 can also be used to prove that certain Lagrangian submanifolds can not be displaced by a Hamiltonian diffeomorphism. Consider the following generalization of Theorem 3.3.3 from Chapter X of [AL]. Let θ 1 , . . . , θ n be coordinates on the n-torus, T n = R/Z × · · · × R/Z, and let G be the group of transformations on T n generated by the map Define K n to be the quotient T n /G. Theorem 1.4. Let (M, ω) be a closed symplectic manifold of dimension 2n ≥ 6 which is symplectically aspherical. If L is a Lagrangian submanifold of (M, ω) which is diffeomorphic to K n , then L can not be displaced by any Hamiltonian diffeomorphism.
Proof. Assume that L is displaceable. For the flat metric on K n we have d g = n and Theorem 1.1 implies that N L is in [1, n + 1 (+1)]. Let U be a neighborhood of L in M which is small enough to be both displaceable and symplectomorphic to a neighborhood of the zero section in T * K n . As described in [AL], pages 296-297, there is a Lagrangian torus L ′ in U such that The Lagrangian torus L ′ is also displaceable and, for n > 3, equation (1) contradicts Theorem 1.3 since N L ′ = (2n − 2)N L > n + 1. For n = 3, K 3 is orientable and so N L ≥ 2. In this case, equation (1) implies that N L ′ ≥ 6 which again contradicts Theorem 1.3. Hence, L is not displaceable.
Split hyperbolic Lagrangian submanifolds. Following [KS], we say that a product manifold L = P 1 × · · · × P k is split hyperbolic if each factor P j admits a metric with negative sectional curvature. By convention, we label the factors of L so that dim P j ≤ dim P j+1 . As noted in [KS], every symplectic manifold with dimension different from two or six, contains Lagrangian submanifolds which are both split hyperbolic and displaceable. For this large class of Lagrangians, Theorem 1.1 yields the following rigidity result.
Theorem 1.5. Let L = P 1 × · · · × P k be a split hyperbolic Lagrangian submanifold of a closed symplectic manifold (M, ω) of dimension 2n which is symplectically aspherical. If L is displaceable, then there is a class A ∈ ker i * such that µ L Maslov (A) ∈ [dim P 1 , n + 1 (+1)]. Proof. For each factor P j fix a metric g j with negative sectional curvature. Each E g j is Morse-Bott, the nonconstant closed geodesics of g j are isolated and noncontractible, and there is a unique closed geodesic of g j in each nontrivial homotopy class of loops in P j . Let g be the metric on L which is the sum of the g j . Then, E g is also Morse-Bott, and the nonconstant 1-periodic geodesics of g occur in Morse-Bott nondegenerate critical submanifolds of dimension no greater than 1+dim P 2 +· · ·+dim P k = 1+n−dim P 1 . Hence, d g = 1 + n − dim P 1 and Theorem 1.1 yields the desired class A ∈ ker i * .
In the aspherical setting, Theorem 1.5 slightly improves the rigidity results for displaceable, split-hyperbolic Lagrangians from [KS], which yield a class in ker i * with Maslov index in the interval [dim P 1 − 1, n + 1 (+1)].
1.1. Maslov class rigidity and the fundamental selector. Under the hypotheses of Theorem 1.1, one can also use the action selector corresponding to the fundamental class in Hamiltonian Floer homology ( [Schw]), to detect a class B ∈ ker i * with µ L Maslov (B) ∈ [n − d g , n + 1 (+1)]. This yields no information for the minimal Maslov number of Lagrangian tori as established in Theorem 1.3 and applied in Theorem 1.4, since the lower bound for µ L Maslov (B) in this case is zero. Moreover, unless one assumes that the Lagrangian submanifold is also monotone, it is not clear whether or not the classes A and B are distinct. Thus, the class B need not yield information which is complimentary to that obtained from the class A.
Remark 1.6. Roughly speaking, the bounds from [KS] correspond to those implied by the class B. However, in the more general setting of [KS], one can not obtain these bounds by using action selectors. This would lead to bounds which only hold modulo the minimal Chern number of the ambient symplectic manifold. The development of techniques which capture the same information as the selector constructed here, and which work for more general classes of symplectic manifolds, will be the subject of future work.
1.2. Maslov class rigidity and holomorphic discs. Starting with the work of Polterovich from [Po1,Po2,Po3], many results on the rigidity of the Maslov class have also been obtained by considering moduli spaces of holomorphic discs with boundary on the Lagrangian submanifold. These restrictions are often as strong or stonger that those obtained here, but hold for the special but important class of monotone displaceable Lagrangian submanifolds. Starting with the work of Oh in [Oh], these results have been obtained primarily through various applications and refinements of Lagrangian Floer homology, [Al, Bi, BCi, BCo, Bu, FOOO]. Since the results presented here make no use of the monotonicity assumption, they can be viewed as complimentary to this body of work.
Recently, Fukaya has established a beautiful link between the moduli spaces of holomorphic discs with boundary on a Lagrangian submanifold and the string topology of the Lagrangian submanifold, [Fu]. Among several other applications, he has used this relationship to prove Audin's conjecture which asserts that every Lagrangian torus in a symplectic vector space has minimal Maslov number equal to two. As described above, our present methods only allow us to reprove Viterbo's results concerning Lagrangian tori in symplectic vector spaces. The coupling of Fukaya's methods with those developed in [KS] and here, will also be the subject of future projects.
1.3. Organization. In the next section we recall some basic definitions from the study of Hofer's geometry and recall a shortening result for certain Hamiltonian paths. In Section 3, we describe some relevant tools from filtered Hamiltonian Floer theory which are then used to define a new action selector in Section 4. In Section 5 we construct special Hamiltonian flows which are supported near a Lagrangian submanifold. Our action selector is then applied to these flows to prove Theorem 1.1 in Section 6.

HAMILTONIAN FLOWS AND HOFER'S GEOMETRY
Henceforth, we will assume that (M, ω) is a closed symplectic manifold of dimension 2n which is symplectically aspherical. A smooth function H on S 1 × M will be referred to as a Hamiltonian on M . Here, we identify the circle S 1 with R/Z and parameterize it with the coordinate t ∈ [0, 1]. Each Hamiltonian H determines a 1-periodic time-dependent Hamiltonian vector field X H via Hamilton's equation where H t (·) = H(t, ·). The time-t flow of X H will be denoted by φ t H . It is defined for all t ∈ [0, 1] (in fact, for all t ∈ R).
For a Hamiltonian H set where Vol(M ) is the volume of M with respect to ω n . Following [Ho], the Hofer length of the Hamiltonian path φ t H is defined to be H = The quantities H + and H − provide different measures of the length of φ t H called the positive and negative Hofer lengths, respectively. We will also consider the quantities For a path of Hamiltonian diffeomorphisms ψ t with ψ 0 = id, let [ψ t ] be the class of Hamiltonian paths which are homotopic to ψ t relative to its endpoints. Denote the set of Hamiltonians which generate the paths in [ψ t ] by The positive and negative Hofer semi-norms of [ψ t ] are defined similarly as , then φ t H fails to minimize the (positive/negative) Hofer length in its homotopy class.
The displacement energy of a subset U ⊂ (M, ω) is the quantity where U denotes the closure of U . The following result relates the negative Hofer semi-norm and the displacement energy. It is a direct application of Sikorav's curve shortening technique and the reader is referred to Lemma 4.2 of [Ke2] for the entirely similar proof of the analogous result for the positive Hofer semi-norm.
In particular, the Hamiltonian path φ t H does not minimize the negative Hofer length in its homotopy class.

FLOER THEORY FOR SYMPLECTICALLY ASPHERICAL MANIFOLDS
For any Hamiltonian H, we will denote the set of contractible 1-periodic M does not have one as an eigenvalue. If every element of P(H) is nondegenerate we will call H a Floer Hamiltonian.
The action of x ∈ P(H) is given by . This is well-defined since ω| π 2 (M ) is trivial. The action spectrum of H is the set Following [CZ], one can also define the Conley-Zehnder index µ CZ (x) of each x in P(H). Here we use the original normalization of this index. In particular, for a local maximum p of an autonomous Morse Hamiltonian such that the Hessian of H at p is arbitrarily small (with respect to a fixed metric on M ), we have µ CZ (p) = −n.
3.1. The Floer complex. Let H be a Floer Hamiltonian on M . For a generic S 1 -family of ω-compatible almost complex structures J t = J, one can define the Floer complex (CF * (H), δ J ) as follows. The cochain group CF * (H) is the vector space over Z 2 which is generated by the elements of P(H) and is graded by the Conley-Zehnder index. The action of an element α = n j x j of CF(H) is defined to be For x and y in P(H), let M(x, y; H, J) be the set of smooth maps w : R × S 1 → M which satisfy Floer's equation and have the following asymptotic behavior with respect to the smooth topology on For a generic choice of J, each M(x, y; H, J) is a smooth manifold of dimension µ CZ (y) − µ CZ (x). Moreover, R acts freely on M(x, y; H, J) by ζ · u(s, t) = u(ζ + s, t). The Floer coboundary map is then defined on generators of CF(H) by where # 2 (M(x, y; H, J)/R) is the number of elements in the zero dimensional manifold M(x, y; H, J)/R, modulo two. Floer's gluing and compactness theorems imply that δ J •δ J = 0, and we denote the resulting cohomology H * (CF(H), δ J ) by HF * (H). Before discussing the construction of Floer continuation maps and their role in identifying HF * (H), we first recall a useful fact concerning the Floer complexes for Hamiltonians H and G which generate the same time one flow, i.e., φ 1 is also well-defined and is canonically isomorphic to (CF * (H), δ J ). In particular, the map ) is a bijection from M(x, y; H, J) to M( x, y; G, J ). We will denote this isomorphism of complexes by . It follows from the work of Seidel in [Se], see also [Schw], that it preserves both the action and Conley-Zehnder index of periodic orbits.
3.2. Floer continuation maps. Suppose that the Floer complexes of two pairs (G, J G ) and (H, J H ) are well-defined. We now recall how Floer continuation maps define a natural isomomorphism between their respective Floer cohomology groups. Consider a smooth homotopy of data (F s , J s ) which equals (G, J G ) for s ≪ 0 and equals (H, J H ) for s ≫ 0. Let M s (x, y; F s , J s ) be the space of maps u : R × S 1 → M which satisfy and have the following asymptotic behavior; For a generic choice of (F s , J s ) each M s (x, y; F s , J s ) is a smooth manifold of dimension µ CZ (x) − µ CZ (y). In this case we will say that the pair (F s , J s ) is regular. The continuation map is then defined on generators by setting This is a chain map and the induced map in cohomology, which we denote by Ψ G H , is independent of the regular homotopy (F s , J s ). For any Floer Hamiltonians G, H and K we have, at the level of cohomology, the factorization identity This can be used to prove that each continuation map is an isomorphism and hence the Floer cohomology HF For small autonomous Hamiltonians we have the following normalization of the Floer cohomology.
Theorem 3.1. [FHS, HS] There is an open and dense set of smooth Morse functions on M such that for any h in this set there is an ω-compatible and t-independent almost complex structure J, and a positive integer m 0 , such that for all m ≥ m 0 the Floer complex , the Morse complex of h/m with respect to the metric g J (·, ·) = ω(·, J·).
Since the homology of a Morse complex is isomorphic to the singular homology of M , this immediately yields the following identification. arguments. Let f and h be two functions in F such that P(f ) = Crit(f ) and P(h) = Crit(h). Suppose that f and h attain their maximum values at points q and p in M , respectively. Then for every Floer Hamiltonian H and regular J equation (3) implies that 3.4. Filtered Floer cohomology. To define the action selectors of the next section we also require tools from the filtered Floer cohomology theory introduced by Floer and Hofer in [FH]. Consider a Floer Hamiltonian H and two real numbers a < b which lie outside of the action spectrum S(H). We define CF (a,b) (H) to be the vector space over Z 2 which is generated by the elements of Hence, the Floer coboundary map δ J decreases actions, and it's restriction to CF (a,b) (H) again satisfies δ 2 J = 0. The filtered Floer cohomology of H for the interval (a, b) is then defined to be HF * (a,b) (H) = H * (CF (a,b) (H), δ J ). For three real numbers a < b < c which lie outside S(H), CF (a,b) (H) is a subcomplex of CF (a,c) (H) and CF (b,c) (H) is naturally isomorphic to the quotient complex. This yields the following long exact sequence We will set HF b (H) = HF (−∞,b) (H), and will denote the inclusion map in the exact sequence above by i b : HF * b (H) → HF * (H). As the notation suggests, HF (a,b) (H) is independent of the choice of the family J but depends on the Hamiltonian H. As we now recall, there are natural morphisms between filtered Floer cohomologies for different Hamiltonians. These maps rely on the useful fact that for a map u in M s (x, y; F s , J s ) we have If G ≥ H one can consider the class of homotopies F s from G to H such that ∂ s F s ≤ 0. These are referred to as monotone homotopies. Inequality (4) implies that a continuation map ψ G H determined by a monotone homotopy induces a chain map from CF * (a,b) (G) → CF * (a,b) (H). The map induced in cohomology, Ψ G H : HF * (a,b) (G) → HF * (a,b) (H), is independent of the choice of monotone homotopy.
More generally, for any two Hamiltonians G and H one can consider homotopies F s for which the quantity R×S 1 max p∈M ∂ s F s (s, t, p) dt ds is bounded from above by some constant C > 0. Following [Gi1], we call such homotopies C-bounded. As in the case of monotone homotopies, inequality (4) implies that each regular homotopy (F s , J s ) for which F s is C-bounded homotopy determines a chain map from CF * (a,b) (G) → CF * (a+C,b+C) (H) and hence a homomorphism (5) Ψ G H : HF * (a,b) (G) → HF * (a+C,b+C) (H). Again, this map is independent on the choice of C-bounded homotopy, [Gi1].
Remark 3.3. To avoid overly cumbersome notation we will use Ψ * * to denote all continuation homomorphisms even when they are defined by special classes of C-bounded homotopies. This latter fact will be implied, when necessary, by the inclusion of the domain and target of the homomorphism, as in equation (5). Hence F s is a (|||H − G||| + )-bounded homotopy which can be used to define a homomorphism (6) Ψ G H : HF * (a,b) (G) → HF * (a+|||H−G||| + ,b+|||H−G||| + ) (H). Henceforth we will denote a chain map defined using a linear homotopy of Hamiltonians from G to H by ψ G H . The corresponding map in (filtered) Floer cohomology will be denoted by Ψ G H . Remark 3.5. To achieve regularity, it might be necessary to perturb the precise linear homotopy defined above away from its endpoints. However, any sufficiently small perturbation yields the same map in cohomology, and so the notation Ψ G H is unambiguous. Since we are working coefficients in Z/2Z, the same is true of the chain level map ψ G H . Let F s be a C-bounded homotopy from G to H, and let F ′ s be a C ′bounded homotopy from H to K. For sufficiently large R > 0 Lemma 3.6. ( [Gi1]) For sufficiently large R > 0, the map s is equivalent to the composition Ψ G H • Ψ H K whose factors are induced by F s and F ′ s . We now define a more subtle fundamental class in HF a (H) for sufficiently large a. This will be a key part of our definition of a new action selector.

Proposition 3.7. For every a > ρ + ([φ t H ])+H av which lies outside of S(H), there is a well-defined and nontrivial class [M ] a in HF −n a (H) with the following properties (i) i a ([M ] a ) = [M ]; (ii) If H is a Floer Hamiltonian and J is regular, then [M ] a is represented by a cycle α in
H ]) such that G av = H av and (7) |||G||| + < min{a, ∆ + H }. Let f be a Morse function in F which satisfies where p is the unique local maximum of f . Since we are working only with Z 2 -coefficients, the class α ′ is uniquely determined by f . Moreover, it follows from Example 3.4 together with (7) and (8) that In fact, for all a we have where J is defined as in (2). Because ψ f G (p) belongs to CF −n a (G), we can therefore identify α ′ with a class α in CF −n a (H) such that If α F is the class in CF −n a (G) which corresponds to ψ f F (p) under the identification then it suffices for us to show that [α F ] = ψ f G (p) . With the Morse function f fixed as above, we will actually prove that . Let x be any element in P(G) with Conley-Zehnder index equal to −n.
In the identification (9), x corresponds to the orbit y(t) = ̺ t (x(t)) in P(F ). To verify (10), we need to prove that (11) # 2 M s (p, x; G s , J G s ) = # 2 M s (p, y; F s , J F s ), where G s and F s are small perturbations of the linear homotopies from f to G and from f to F described in Example 3.4 (see Remark 3.5). Here J G s and J F s are regular families of almost complex structures. The choice of these families does not effect the count, modulo two, of the moduli spaces in (11) which they help define.

Since both F and G belong to C ∞ ([φ t H ]) we can choose a family of functions
Consider the family of contractible Hamiltonian loops and let . For each value of s, ̺ s,t is a loop based at the identity, and From ̺ s,t we obtain two families of normalized Hamiltonians, A s and B s , defined by ∂ s (̺ s,t (p)) = X As (̺ s,t (p)) and ∂ t (̺ s,t (p)) = X Bs (̺ s,t (p)).

The standard composition formula for Hamiltonian flows implies that
where (G s • ̺ −1 s,t )(t, p) = G s (t, ̺ −1 s,t (p)). To prove (11), we first note that the image of M s (p, x; G s , J G s ) under the map is the space M s (p, y; F s , A s , J s ) of smooth maps v : R × S 1 → M which satisfy the equation To see this, note that the desired limiting behavior of v = ̺ s,t (u) ∈ M s (p, y; F s , A s , J s ) is determined by that of u. The fact that v satisfies equation (12) then follows from the following simple computation For λ ∈ R, let ( F λ s , A λ s , J λ s ) be a triple of smooth families of functions and ω-compatible almost complex structures such that For a generic choice of the family J λ s , the space W(p, y; F λ s , A λ s , J λ s ) is a smooth cobordism between M s (p, y; F s , A s , J s ) and M s (p, y; F s , 0, J F s ) = M s (p, y; F s , J F s ). Hence, # 2 M s (p, x; G s , J G s ) = # 2 M s (p, y; F s , A s , J s ) = # 2 M s (p, y; F s , J F s ), as required.
Remark 3.8. For a > |||H||| + , we can choose G = H in the construction of [M ] a . In this case, for any function f ∈ F with 2 f < a − |||H||| + .

AN ACTION SELECTOR FOR PINNED HAMILTONIANS
We now define a new action selector σ for a special, but useful, class of Hamiltonians. Pinned Hamiltonians arise naturally in the study of the length minimizing properties of Hamiltonian paths with respect to Hofer lengths. In particular, slightly weaker versions of these two properties are necessary for a Hamiltonian to generate a positive length minimizing path in its homotopy class, [BP, LMcD, Us].
In what follows, the point Q at which a pinned Hamiltonian attains its maximum values will be specified. In this case, we say that H is pinned at Q.
Proof. The bounds on the action of β follow from inequality (4) and the fact that our homotopy from f H to H is monotone. If one presumes that the linear homotopy F s = (1 − b(s))f H + b(s)H is part of a regular set of continuation data (F s , J s ), then by (4) the only element of M s (Q, Q; F s , J s ) is the constant map u(s, t) = Q and we are done. Without this presumption, one must prove that the number of elements in M s (Q, Q; F ′ s , J s ) is equal to one modulo two, where F ′ s is a generic monotone perturbation of F s for which (F ′ s , J s ) is regular. This can be established as in Lemma 2.5 of [Gi2].
We can now define another version of the fundamental class for the filtered Floer cohomology of pinned Hamiltonians.
. If a is sufficiently large, for example if a is greater than max{A H (x) | x ∈ P(H)}, then we have [α] = [γ] ∈ HF a (H). However, for values of a just slightly larger than |||H||| + we have To see this, note that the action inequalities (13) and (14) imply that Q appears with coefficient one in γ − α = Q + β − α. Since H is a Floer This change in behavior as a approaches |||H||| + from infinity, leads one to the following definition. For Floer Hamiltonian H in H(Q) set Remark 4.5. By the discussion above, we have σ(H) > |||H||| + . Hence, σ(H) is not equal to any of the action selectors defined by Schwarz in [Schw] for closed, symplectically aspherical symplectic manifolds.
Lemma 4.6. If H ∈ H(Q) is a Floer Hamiltonian, then there is an element x in P(H) such that µ CZ ( x) = −n − 1 and A H ( x) = σ(H).
Proof. As described above, for every a > σ(H) the class [Q + β − α] is trivial in HF a (H). Hence, there is a class η in CF a (H) such that A H (η) ≥ σ(H) and δ J (η) = Q + β − α. In particular, for all a > σ(H) there is an orbit in P(H) whose Conley-Zehnder index is −n − 1 and whose action is in the interval [ σ(H), a). Since there are only finitely many elements of P(H), the result follows.
We now establish a continuity property for the action selector σ.
Proof. We first consider the case of two Hamiltonians H and G with H av = G av . Let f be a dominating function for both H and G and fix an a > σ(H).

A SPECIAL HAMILTONIAN FLOW NEAR L
In this section we describe the construction of a Hamiltonian H L whose flow is supported near L and does not minimize the negative Hofer length in its homotopy class. The nonconstant contractible periodic orbits of this Hamiltonian are related directly to closed geodesics of a metric g on L. One aspect of this relation is in terms of familiar indices; the Conley-Zehnder index of a 1-periodic orbit of H L is related to the Morse index of the corresponding closed geodesic and the Maslov index of the class in ker i * that it represents. This index relation is recalled at the end of the section as Proposition 5.2. 5.1. Geodesic flows. Let g be a Riemannian metric on a closed manifold L. As noted in the introduction, the energy functional of g on the space of smooth loops C ∞ (S 1 , L), is defined by and the critical points of E g are the closed geodesics of g with period equal to one. The closed geodesics of g with any positive period T > 0 correspond to the 1-periodic orbits of the metric 1 T g, and are thus the critical points of the functional E 1 T g . Denote the Hessian of E g at a critical point q(t), by Hess(E g ) q . The dimension of the space on which Hess(E g ) q is negative definite is finite. This dimension is the Morse index of q and will be denoted by I Morse (q). The Hessian of E g at a critical point also has a finite dimensional kernel which is always nontrivial (unless L is a point).
A submanifold D ⊂ C ∞ (S 1 , L) which consists of critical points of E g is said to be Morse-Bott nondegenerate if the dimension of the kernel of Hess(E g ) q is equal to the dimension of D for every q ∈ D. The energy functional E g is said to be Morse-Bott if all the 1-periodic geodesics are contained in Morse-Bott nondegenerate critical submanifolds of E g . Note that if E g is Morse-Bott, then so is E 1 T g for any T > 0.

5.2.
Geodesic flows supported near a Lagrangian submanifold. Let L be a displaceable Lagrangian submanifold of a symplectic manifold (M, ω). Any Riemannian metric g on L determines a cometric on T * L which we may use to define the neighborhood of the zero section Weinstein's neighborhood theorem implies that, for sufficiently small r > 0, there is a neighborhood of L in (M, ω) which is symplectomorphic to U r where T * M is equipped with the symplectic form dθ where θ is the Liouville one-form. We fix a value of r for which this holds, and will henceforth identify U r with a neighborhood of L in (M, ω). Decreasing this value of r, if necessary, we may also assume that U r has finite displacement energy. For a subinterval I ⊂ [0, r), we will use the notation For a numbers ǫ ∈ (0, r/8) and C > 0, let ν = ν ǫ,C : [0, +∞) → R be a smooth function with the following properties . For future reference, we note that these properties imply that the function s → −ν(s) + ν ′ (s)s is nondecreasing, and takes values in the interval [0, ǫ] for s ∈ [0, r − 2ǫ].
Define the Hamiltonian K ν on M by The Hamiltonian flow of K ν is trivial on U ǫ and the complement of U r−ǫ . Elsewhere, where K g = 1 2 |p| 2 is the kinetic energy Hamiltonian which generates the cogeodesic flow of g on (T * L, dθ). In particular, if x(t) is a nonconstant 1-periodic orbit of K ν , then x(t) can be expressed in standard (q, p) coordinates on T * L as x(t) = (q(t), p(t)), where q(t) is a closed geodesic of g on L with period |p(0)|/ν ′ (|p(0)|) and length Hence, for every nonconstant orbit x(t) = (q(t), p(t)) in P(K ν ) we have the uniform bound (20) |q(t)| ≤ C.
Moreover, if C is not the length of a closed geodesic of g on L, then the nonconstant orbits of K ν occur on the level sets contained in U (ǫ,2ǫ) or U (r−2ǫ,r−ǫ) , where ν is convex or concave, respectively. Note that .
By construction, A > Cr 2 , so for C > E we have In this case, Proposition 2.1 implies that the Hamiltonian flow of K ν does not minimize the negative Hofer length. We will henceforth assume that C > E.

5.3.
A perturbation of K ν . The following result describes a perturbation of K ν which is a Floer Hamiltonian whose constant periodic orbits capture the Morse homology of M and whose nonconstant periodic orbits retain many of the properties of those of K ν . It follows from the proof of Proposition 3.5 in [KS].
Proposition 5.1. Let L be a Lagrangian submanifold of (M, ω) and let g be a metric on L whose energy functional is Morse-Bott. Let ν = ν ǫ,C be a function as above, with C > E. There is an ǫ 0 > 0 such that for every 0 < ǫ < ǫ 0 there is a Floer Hamiltonian H L = H ǫ,C L which has the following properties: In particular, we may assume that the C 0 -distance between them is small enough so that  The construction of H L from [KS] also implies that each nonconstant contractible 1-periodic orbit x(t) of H L projects to a nondegenerate critical point q(t) of a perturbed energy functional of the form Here, T > 0 and V : S 1 ×L → R is smooth and arbitrarily small with respect to the C ∞ -metric. Thus, the bound (23) follows immediately from (20). The perturbed geodesic q can be associated to a unique critical submanifold D of E 1 T g and the Morse index of q belongs to [I Morse (D), I Morse (D) + dim D]. The following index relation follows from the work of Duistermaat [Du], Weber [We], and Viterbo [Vi]. For a proof the reader is referred to Proposition 4.4 of [KS].
Proposition 5.2. Let L be a Lagrangian submanifold of (M, ω) and let g be a metric on L which has nonpositive sectional curvature and whose energy functional E g is Morse-Bott. Denote by d g the maximum dimension of a critical submanifold of E g which consists of nonconstant closed geodesics. Let x(t) = (q(t), p(t)) be a contractible 1-periodic orbit of a Hamiltonian H L as in Proposition 5.1. If x is contained in U (ǫ,2ǫ) , then Moreover, the map A e H φ t e H (x(0)) = −A H (x). Let ν ǫ,C be a family of smooth functions defined as in Section 5.2, which depend smoothly on the parameters ǫ and C. Fix a corresponding smooth family of Hamiltonians H ǫ,C L , as in Proposition 5.1, for (ǫ, C) ∈ (0, ǫ 0 ) × (E, ∞). Set is well-defined on (0, ǫ 0 )×(E, ∞). By Proposition 4.7, this function is continuous. The following result relates σ(ǫ, C) to 1-periodic orbits of the Hamiltonians H ǫ,C L .
Proof. Lemma 4.6 implies the existence of an orbit x ǫ with properties (25), and (26). By condition (H2), every constant 1-periodic orbit of H ǫ,C L has Conley-Zehnder index no less than −n, and so (26) implies that x ǫ must be nonconstant.
Proof. We begin by showing that the limit as ǫ → 0 of σ(ǫ, C) exists for each C. For any sequence ǫ j in (0, ǫ 0 ), Proposition 4.7 and (H1) imply that Hence, if ǫ j → 0, then the sequence σ(ǫ j , C) is Cauchy and converges. It suffices to show that the limit of this sequence is independent of the choice of ǫ j → 0.
For the orbits x ǫ = ( q ǫ , p ǫ ), it follows from (23) that ˙ q ǫ (t) < 2C for all t ∈ [0, 1] and all ǫ ∈ (0, ǫ 0 ). By the Arzela-Ascoli theorem, for every sequence ǫ j → 0 there is a subsequence ǫ j k such that the q ǫ j k converge to a closed geodesic q 0,C of g. The corresponding lifts x ǫ j k = ( q ǫ j k , p ǫ j k ) converge to some ( q 0,C , p 0,C ) and hence For small ǫ > 0, the fact that H ǫ,C L is within ǫ of K ν ǫ,C in the C ∞ -topology, implies that As well, the defining properties of the ν ǫ,C imply that each function Now, the union of the quantities ℓ( q 0,C ) and D 2 q * 0,C ω, over all closed geodesics q 0,C , is a countable set. By (27), this implies that lim j→∞ σ(ǫ j , C) is independent of the sequence ǫ j → 0. Hence, lim ǫ→0 σ(ǫ, C) exists for each C ∈ (E, ∞). We denote this limit by σ(0, C).
It remains to show that σ(0, C) is continuous in C. This is an immediate consequence of the following inequalities If C is not the length of a closed geodesic of g on L, then property (H3) implies that the nonconstant orbits x ǫ lie in either U (ǫ,2ǫ) or U (r−2ǫ,r−ǫ) . Proposition 6.3. If C is sufficiently large and is not the length of a closed geodesic of g on L, then for some ǫ ∈ (0, ǫ 0 ) the orbit x ǫ lies in U (ǫ,2ǫ) .