Displacement energy of coisotropic submanifolds and Hofer's geometry

We prove that the displacement energy of a stable coisotropic submanifold is bounded away from zero if the ambient symplectic manifold is closed, rational and satisfies a mild topological condition.


Introduction and Results
There is positive lower bound for the amount of energy it takes to displace a closed Lagrangian submanifold of a tame symplectic manifold. In particular, every time-dependent function on a symplectic manifold determines a unique Hamiltonian diffeomorphism, and if this diffeomorphism displaces a closed Lagrangian submanifold, then the Hofer norm of the function is bounded away from zero by a constant which depends only on the Lagrangian submanifold and the ambient symplectic manifold. This fundamental fact in symplectic topology was first established for rational Lagrangian submanifolds by Polterovich in [Po], and was later extended to general Lagrangians by Chekanov in [Ch]. Among other things, it implies the nondegeneracy of the Hofer metric on the group of Hamiltonian diffeomorphisms of a tame symplectic manifold.
Recently, Ginzburg proved that there is also a positive lower bound for the amount of energy required to displace certain coisotropic submanifolds. More precisely, in [Gi] it is shown that the displacement energy of a stable coisotropic submanifold of a tame, wide and symplectically aspherical symplectic manifold is bounded away from zero. In the present paper, we extend this coisotropic intersection phenomenon to symplectic manifolds which admit symplectic spheres. The proof utilizes the Floer theoretic methods developed in [Ke], as well as the applications of these methods to the study of Hamiltonian paths which are length minimizing with respect to the Hofer metric.
There is currently no version of Floer theory for the intersection theory of a general coisotropic submanifold and its image under a Hamiltonian diffeomorphism. However, one can study the symplectic properties of a coisotropic submanifold using the Hamiltonian Floer homology of functions which are Date: February 1, 2008. 2000 Mathematics Subject Classification. 53D40, 37J45. This research was partially supported by NSF Grant DMS-0405994 and a grant from the Campus Research Board of the University of Illinois at Urbana-Champaign. supported in (normal) neighborhoods of it. This indirect approach, which goes back to the pioneering work of Viterbo from [Vi], requires a few compromises.
The first compromise involves the submanifolds. To get useful normal neighborhoods, we restrict our attention to stable coisotropic submanifolds. This notion was introduced by Bolle in [Bo1,Bo2] 1 and is defined as follows. Let (M, ω) be a symplectic manifold of dimension 2m and let N be a closed coisotropic submanifold of M with codimension k. Then N is said to be stable if there are one-forms α 1 , . . . , α k on N such that the form α 1 ∧ · · · ∧ α k ∧ (ω| N ) m−k does not vanish on N , and ker dα j ⊃ ker ω| N for j = 1, . . . , k. Examples of stable coisotropic submanifolds include Lagrangian tori and contact hypersurfaces. The stability condition is also closed under products. For more details, the reader is referred to [Bo1,Bo2,Gi].
In using Hamiltonian Floer homology to study the symplectic topology of a coisotropic submanifold, one also needs to recognize nontrivial 1-periodic orbits using only the symplectic action and/or the Conley-Zehnder index. This requires further compromise concerning the ambient symplectic manifolds, (M, ω), we consider. In [Gi], the symplectic manifolds are assumed to be symplectically aspherical. That is, for every class A ∈ π 2 (M ) it is assumed that ω(A) = 0 = c 1 (A), where the notations ω(A) and c 1 (A) refer to the evaluations of the cohomology classes on the elements of H 2 (M ; R) and H 2 (M ; Z) determined by A. With this assumption, the action and index of a periodic orbit are single-valued and any 1-periodic orbit with sufficiently large action (greater than H + as defined below) must be nonconstant.
Here, we allow for the existence of nontrivial symplectic spheres and so the action and index may be multi-valued. To distinguish nonconstant periodic orbits we will assume that the quantity r(M, ω) = inf is positive. 2 A symplectic manifold with r(M, ω) > 0 is said to be rational. We will also assume that (M, ω) satisfies the topological assumption (1) ω(A) = 0 =⇒ c 1 (A) ≥ 0 for all A in π 2 (M ).
Finally, we restrict ourselves, in this work, to the case when M is closed. We expect that the methods developed here are also applicable to symplectic manifolds which are open or have convex boundaries. Before stating the main result, we first recall the definition of the displacement energy. Let C ∞ (S 1 × M ) be the space of smooth time-periodic functions on M , where S 1 = R/Z is the circle parameterized by t ∈ [0, 1].

The Hofer norm of a function H in
where H t (·) = H(t, ·). One can also associate to H its Hamiltonian vector X H via the equation The time-t flow of this vector field, also referred to as the Hamiltonian flow of H, is denoted by φ t H and is defined for all t ∈ [0, 1]. The group of Hamiltonian diffeomorphisms consists of all the time-1 maps, φ 1 H , obtained in this way.
The displacement energy of a subset U of M is defined as the minimum variation of a function which generates a Hamiltonian diffeomorphism that moves U off of itself. Our main result is the following: Theorem 1.1. Let N be a stable coisotropic submanifold of a closed and rational symplectic manifold satisfying (1). There is a positive constant ∆ > 0 such that e(N ) ≥ ∆.
Of course, one starts with the assumption that N can be displaced by some Hamiltonian diffeomorphism, i.e., e(N ) < ∞. This has deep implications for the Hamiltonian flows supported near N . In turn, these flows can be used to probe the geometry of N . It is this interaction between the displacability of N and its geometry, which leads to the proof of Theorem 1.1.
The primary difference between the proof of Theorem 1.1 and the proof of the main result in [Gi] is the contribution coming from Floer theory. In [Gi], both the action filtration and action selector are used to prove the existence of a Floer trajectory whose energy yields the crucial estimate for the displacement energy, (Proposition 5.1 of [Gi]). For a rational symplectic manifold, the action filtration and selector can not be used in the same manner. Instead we use the Floer theoretic techniques which were developed in [Ke] to study the length minimizing properties of Hamiltonian paths. These tools allow us to detect a perturbed holomorphic cylinder in Proposition 2.5 whose energy recovers the crucial estimate.
Remark 1.2. Another approach to studying coisotropic intersections is to consider general leaf-wise intersections under Hamiltonian diffeomorphisms, [Mo, EH, Ho]. The most recent work in this direction is [Dr], where Dragnev establishes the existence of leaf-wise intersections for a stable coisotropic submanifold N of R 2n , and its image under any Hamiltonian diffeomorphism with energy less than the Floer-Hofer capacity of N .
1.1. Organization. The proof of Theorem 1.1 is described in the next section, assuming the contribution from Floer theory, Proposition 2.5. In the third section, we recall the required Floer theory methods and applications from [Ke]. The proof of Proposition 2.5 is then contained in the final section of the paper.
1.2. Acknowledgments. The author would like to thank Peter Albers and Viktor Ginzburg for their helpful comments.
Before presenting the proof of Theorem 1.1 in §2.4, we discuss some preliminary notions and results.
2.1. Properties of stable coisotropic submanifolds. We begin by recalling some useful implications of the stability assumption. The proofs of these results can be found in [Bo1,Bo2,Gi].
Let N be a stable coisotropic submanifold of codimension k in a symplectic manifold (M, ω) of dimension 2m. We then have the following normal neighborhood result.
Here, |p| denotes the standard norm of p = (p 1 , . . . , p k ) ∈ R k , and π : U r → N is the obvious projection.
Recall that the characteristic foliation F of N is determined by the integrable distribution ker ω| N . The normal form above implies that for each manifold N p = N × p with |p| < r we have ω| Np = ω| N . Hence, each of the N p in the tubular neighborhood U r is a coisotropic submanifold with the same characteristic foliation.
The relevant Hamiltonian dynamical system is the following leaf-wise geodesic flow on the tubular neighborhood U r of N . Bo1,Bo2,Gi]). The Hamiltonian flow of the function 1 2 |p| 2 on the normal neighborhood U r is the leaf-wise geodesic flow of the leaf-wise metric k j=1 (α j ) 2 on F. Moreover, this metric is leaf-wise flat.
This implies that a nonconstant periodic orbit x of the flow of 1 2 |p| 2 corresponds to a closed geodesic γ contained on a leaf of F in N . The fact that the leaf-wise metric is flat implies that this geodesic is noncontractible within its leaf.
For any closed curve γ contained in a leaf of F, set Bo2,Gi]). There is a constant δ N > 0 such that for every nontrivial closed geodesic γ of the leaf-wise metric k j=1 (α j ) 2 .

Hofer's length functional.
Following [Ho], this time-dependent generating function is used to define the Hofer length of the path ψ t by The quantities H + and H − provide different measures of ψ t called the positive and negative Hofer lengths, respectively. The positive Hofer length will play an important role in the proof of Theorem 1.1.
2.3. Right asymptotic spanning discs. A spanning disc for a loop y : S 1 → M is a smooth map w from the unit disc in C to M such that w(e 2πit ) = y(t). A right asymptotic spanning disc for the loop y is a smooth map v : • there is a sequence s + j → +∞ for which v(s + j , t) converges to a constant map t → p for some point p ∈ M. Here, convergence is with respect to the smooth topology on C ∞ (S 1 , M ).
We will detect right asymptotic spanning discs for 1-periodic orbits of the Hamiltonian flow of a function H in C ∞ (S 1 × M ). They will be constructed using a smooth (R × S 1 )-family of ω-compatible almost complex structures, J s , which is independent of s ∈ R for |s| sufficiently large. This last auxiliary structure is used to define the energy of v by For each integer j, we will also consider the quantities 2.4. The proof of Theorem 1.1. We may assume that (2) 3e(N ) < r(M, ω), otherwise we are done. We will prove the following result which clearly implies Theorem 1.1.
Theorem 2.4. For sufficiently small values of r > 0, there is a positive constant ∆ > 0, independent of r, such that e(U r ) > ∆.
By (2), for all sufficiently small values of r > 0 we have Fix an R > 0 for which this inequality holds. Henceforth, we will consider only neighborhoods U r for 0 < r < R/2. In order to relate the assumption that N can be displaced by a Hamiltonian diffeomorphism to the properties of the flow from Proposition 2.2, we reparameterize this flow so that it extends to a global flow on M which is supported in U r .
Here, A and B are positive constants. We then define the function The Hamiltonian flow of H r is trivial way from U r , and inside of U r it is a reparameterization of the geodesic flow from Proposition 2.2. Clearly, H r + = A and H r − = B. We choose the constant A so that 2e(U r ) < A < 3e(U r ).
We then choose a constant B satisfying so that H r is normalized. Further restricting R, if necessary, we may also assume that The following technical result is proved in the final section of the paper using the methods developed in [Ke]. The existence of the map v described below is implied by the fact that the Hamiltonian path generated by H r does not minimize the positive Hofer length in its homotopy class (see §4.2).
Proposition 2.5. For the function H r above, there is an ǫ > 0, a family of almost complex structures J s as in §2.3, and a nonconstant 1-periodic orbit y of H r with a right asymptotic spanning disc v such that for a smooth nondecreasing function η(s) which equals zero for s ≤ −1 and equals one for s ≥ 1.
The following inequality for the energy of the map v detected in Proposition 2.5, is easily derived from the work of Bolle and Ginzburg. We include a proof for the sake of completeness.
Lemma 2.6 ( [Bo1,Bo2,Gi]). There is a constant c R > 0 such that for the periodic orbit y and the asymptotic right spanning disc v of Proposition 2.5, we have E(v) > c R · δ(π(y)).
Proof. Letf : [0, R) → R be a smooth nonincreasing function which is equal to one on [0, R/2) and is equal to zero near R. Let f be the function which equalsf (|p|) in U R and vanishes outside of U R . For the one-forms σ i = f π * α i , we have In particular, away from U r we have X Hr = 0. Within U r , σ i = π * α i and X Hr is a reparameterization of the leaf-wise geodesic flow from Proposition 2.2. Hence, i X Hr dσ i = i X Hr π * dα i = 0 since the forms dα i vanish on the leaves of F. It follows from (7) that i X g Hs dσ i = 0 for both of the possible functions H s from Proposition 2.5. The s-norm of a tangent vector X ∈ T p M is defined to be X s = ω(X, J s X). Since J s does not depend on s when |s| is large, we can find constants c i > 0 such that for any pair of tangent vectors X, Y ∈ T p M and every s ∈ R.
For the asymptotic right spanning disc v detected in Proposition 2.5, we then have Since y is nonconstant, it is contained in U r where σ i = π * α i . The inequality above then implies that We can now complete the proof of Theorem 1.1. By Proposition 2.5, we have Together with inequality (4) and Lemmas 2.6 and 2.4, this implies that Setting ∆ = c R · δ N /3, the proof of Theorem 2.4, and hence Theorem 1.1, will be complete once we prove Proposition 2.5.

Floer caps and chain isomorphisms in Morse homology
Throughout this section H will be a normalized function in C ∞ 0 (S 1 × M ) whose contractible periodic orbits with period equal to one are nondegenerate. This (finite) set of 1-periodic orbits will be denoted by P(H).
Let J (M, ω) be the space of smooth almost complex structures on M which are compatible with ω, and let J S 1 (M, ω) denote the space of smooth S 1 -families of elements in J (M, ω). Fixing a J in J S 1 (M, ω), we refer to (H, J) as our Hamiltonian data.

Homotopy triples and Floer caps.
Any such λ will be referred to as a horizon of the compact homotopy.
A homotopy triple for the pair (H, J) is a collection where H s is a compact homotopy from a constant function c to H, K s is a compact homotopy from the zero function to itself, and J s is a compact homotopy in J S 1 (M, ω) from some J − to J. Although the constant c is an important part of the homotopy triple, it will be suppressed from the notation for simplicity. For a homotopy triple H = (H s , K s , J s ), we consider smooth maps u from the infinite cylinder R × S 1 to M which satisfy the following equation The energy of a solution u of (9) is defined as If the energy of u is finite, then it follows from standard arguments that for some 1-periodic orbit x ∈ P(H). The assumption of finite energy also implies that It is clear from the asymptotic behavior described above, that each left Floer cap u in L(x; H) determines a unique homotopy class of spanning discs for x and hence a well-defined Conley-Zehnder index µ CZ (x, u). This index is normalized here so that if x(t) = p is a constant 1-periodic orbit of a C 2 -small Morse function and u(z) = p is the constant spanning disc, then Given any map of the form F (s, ·) for s ∈ R, we set For a homotopy triple H = (H s , K s , J s ) we will also consider maps v : R × S 1 → M which satisfy the equation The energy of such a map is defined by If a map v satisfies (10) and has finite energy, then v(+∞) is a point in M and v(−∞) is a 1-periodic orbit of H. In particular, such a v is a right asymptotic spanning disc, as defined in §2.3. The space of right Floer caps for x ∈ P(H) is defined by and define the Conley-Zehnder index µ CZ (x, ← − v ) and the action Here we use the convention {H, K} = ω(X K , X H }. The curvature relates the energy and action of solutions of equations (9) and (10) as follows. Given a solution u of (9), set We then derive the following identity from (9) For a solution v of (10), the corresponding map Remark 3.1. In this notation, we have for the quantities appearing in Proposition 2.5. Any function G which generates a path that is homotopic to φ t H , relative its endpoints, can be used to construct a useful homotopy triple for H. The following basic result in this direction is a simple consequence of Propositions 2.6 and 2.7 from [Ke]. Here, u#v denotes the obvious concatenation of the maps, and [u#v] is the element of π 2 (M ) determined by u#v. We will refer to (u, v) above as a central pair of Floer caps for x.
For a period orbit x in P(H) and Floer caps u ∈ L(x; H L ) and v ∈ R(x; H R ), equations (11) and (13) where c L and c R are the constants for H L and H R , respectively. If (u, v) is a central pair of Floer caps for x with respect to H, then (15) and (16) imply that This constant is positive and greater than or equal to r(M, ω). We now describe a chain isomorphism in Morse homology which can be constructed using cap data H that satisfies This chain map will be used in §4.3 to find central periodic orbits whose right Floer caps will, in turn, be used to detect the right asymptotic spanning disc of Proposition 2.5 in §4.4.

is a chain map which is chain homotopic to the identity.
This result is strongly motivated by the work of Chekanov in [Ch]. The proof of Proposition 3.4 is contained in [Ke] where it appears as Proposition 2.4. While it is assumed there that c L = c R = 0, the proof from [Ke] extends easily to the present setting. The genericity assumption of this result concerns the almost complex structure J as well as the families of almost complex structures appearing in the cap data H. As usual, this assumption is included to ensure that the moduli spaces used to construct the maps are regular. These almost complex structures should also by chosen to lie in specific open sets of J (M, ω) so that inequality (19) can be used to avoid bubbling. These technical details, which are discussed in detail in [Ke], can be safely ignored in the present discussion.
Since the maps Φ R , Φ L and Φ H play important roles in the proof of Proposition 2.5, we will recall the relevant aspects of their constructions. We begin with the map Φ L .
A left or right Floer cap will be called short if its energy is less than . The subset of short elements in L(x; H L ) will be denoted by L ′ (x; H L ). Consider the space of left-half gradient trajectories; For a critical point p of f and an orbit x in P(H), set Here, R ′ (x; H R ) is the set of short right Floer caps of x. For generic data, each R(x, q; H R , f ) is a smooth manifold, and the dimension of the component containing (v, β) is µ CZ (x, ← − v ) − ind(q) + m. Let R 0 (x, q; H L , f ) be the set of zero-dimensional components in R(x, q; H L , f ). The map Φ H is then defined by setting the coefficient of q in Φ H (p) to be the integer The map Φ R : CF(H) → CM(f ) is determined by Φ L and Φ H as follows. Let V L be the submodule of CF(H) generated by the orbits in P(H) which appear in an element in the image of Φ L with a nonzero coefficient. The maps Φ L and Φ H uniquely determine the restriction of Φ R to V L . Setting Φ R = 0 on the complement of V L we obtain the full map. In particular, the coefficient of q in Φ R (x) is the signed count of elements (v, β) ∈ R 0 (x, q; H R , f )) for which there is an element (u, α) in some L 0 (p, x; H R , f )) such that [u#v] = 0.
Step 1: approximating H r by generic functions. The results of the previous section can not be applied directly to H r because the elements of P(H r ) are degenerate. To overcome this, we now approximate H r by a sequence of functions H k whose 1-periodic orbits are nondegenerate. These functions are constructed explicitly in order to retain suitable control over their periodic orbits. Let For a sufficiently small choice of ǫ N > 0, F is a Morse function whose critical points away from U r/4 agree with those of F 0 and whose critical points in U r/4 are precisely the critical points of f N on N ⊂ M . Now let Each H 0 k is also a Morse function with Crit(H 0 k ) = Crit(F ). As well, Q is the only critical point of H 0 k with Morse index equal to 2m. For an interval I ⊂ [0, R/2], we introduce the notation U I = {(q, p) ∈ U R | |p| ∈ I}. When k is sufficiently large, the 1-periodic orbits of H 0 k are either critical points or nonconstant orbits contained in U (r/3,2r/3) . In fact, these nonconstant orbits lie in U [r/3+δ,2r/3−δ] for some δ > 0. This follows from the fact that dH 0 k converges to zero in the C ∞ -topology along the boundary of U (r/3,2r/3) .
Perturbing each H 0 k away from Crit(F ), one obtains a sequence of functions H k with the following properties • The orbits in P(H k ) are nondegenerate and are of two types: constant orbits which coincide with the critical points of F , and nonconstant orbits in U [r/3+δ,2r/3−δ] for some δ > 0. • The constant periodic orbits equipped with their constant spanning discs have Conley-Zehnder indices less than m except for the constant orbit at the point Q ∈ N , which has Conley-Zehnder index equal to m. The final detail to account for is the normalization condition. If we add the function − 1 0 H k (t, ·) ω m to H k , the resulting function is normalized and retains the properties described above. In particular, it determines the same Hamiltonian vector field. Abusing notation, this new normalized function will still be denoted by H k .
The following lemma provides a simple criteria for detecting nonconstant periodic orbits of H k .
Proof. Arguing by contradiction, we assume that x(t) = P for some point P in M . The spanning disk w then represents an element [w] in π 2 (M ), and we have Moreover, the point P corresponds to a critical point of F and H k is C 2small near P , so our normalization of the Conley-Zehnder index yields If ω([w]) = 0, then assumption (1) implies that c 1 ([w]) ≥ 0. 3 It then follows from (21) that the Morse index of P must be 2m. This implies that P = Q, since Q is the unique fixed local maximum of H k . However, the action of Q with respect to a spanning disc w with ω([w]) = 0 is equal to H k + . This is outside the assumed action range and hence a contradiction. We must therefore have ω([w]) = 0 and thus |ω([w])| ≥ r(M, ω) > H k .

For the case ω([w]) < 0, this implies that
which is a contradiction, as above. If ω([w]) > 0, then Both of these conclusions again contradict our hypotheses. Therefore x(t) must be nonconstant.

4.2.
Step 2: curve shortening. We now prove that the Hamiltonian path φ t Hr does not minimize the positive Hofer length in its homotopy class. We also show that the same is true of the paths φ t H k when k is sufficiently large. For a Hamiltonian path ψ t , let [ψ t ] be the class of Hamiltonian paths which are homotopic to ψ t relative to its endpoints. Denote the set of normalized functions which generate the paths in [ψ t ] by

The Hofer semi-norm of [ψ t ] is then defined by
The positive and negative Hofer semi-norms of [ψ t ] are defined similarly as In these terms, the displacement energy of a subset U ⊂ M is equal to The following result is an easy application of Sikorav's curve shortening procedure. The proof follows very closely the proof of Proposition 2.1 in [Sc].

Lemma 4.2. Let H be an autonomous normalized Hamiltonian that is constant and equal to its minimal value on the complement of an open set
In other words, φ t H does not minimize the positive Hofer semi-norm in its homotopy class.
Proof. Let φ t and ψ t be Hamiltonian paths and let ϕ be a symplectomorphism. The following properties of the positive and negative Hofer seminorms are easily checked.
. Now choose a Hamiltonian path ψ t starting at the identity such that The path φ t H can then be factored as follows. Hence, For the first summand on the right, we have To see this, consider the Hamiltonian path φ t H k • (φ t Hr ) −1 which is generated by the function These functions clearly converge to zero in the C ∞ -topology. The path is homotopic to φ t H k and is generated by the function For large enough k we then have

4.3.
Step 3: nontrivial linear right Floer caps. Fix a family of almost complex structures J k for each H k such that the J k converge to J in J S 1 (M, ω). As in Example 3.2, set where the (R × S 1 )-families of almost complex structures J k,s converge to a compact homotopy J s from some J − to J. The linear homotopy triples H k then converge to the linear homotopy triple H r = (H s , 0, J s ) for (H r , J). and Proof. As shown above, for large enough k there is a function G k such that the Hamiltonian path φ t G k is homotopic to φ t H k , relative endpoints, and G k + ≤ H k + + ǫ. Applying Proposition 3.3 to G k , we get a J k in J S 1 (M, ω) and a homotopy triple H G k for (H k , J k ) such that We now consider the following cap data for (H k , J k ), By inequalities (23), (26), and the curvature norm estimates for linear homotopy triples derived in Example 3.2, we have By construction (see inequalities (3) and (4)) we also have H r < r(M, ω).
Hence, for sufficiently large k, (27) implies that From this point on, we will assume that k is large enough for this inequality to hold. Since r(M, ω) ≤ , inequality (28) allows us to apply Proposition 3.4 to the homotopy data H k . In particular, for any Morse-Smale pair (f, g) on M we can construct two Z-module homomorphisms which is chain homotopic to the identity.
For simplicity, we choose the Morse-Smale pair (f, g) so that the function f has a unique local (and hence global) maximum at a point q ∈ M . Standard arguments imply that q is the unique nonexact cycle of degree 2m in the Morse complex (CM(f ), ∂ g ), and so Φ H k (q) = q.
Let V L,k be the submodule of CF(H k ) generated by 1-periodic orbits of H k which appear in an element of the image of Φ L,k with a nonzero coefficient. Let K R,k be the submodule of CF(H k ) generated by periodic orbits which lie in the kernel of Φ R,k and let p k : V L,k → V L,k /K R,k be the projection map. We then have Φ H k = Φ R,k • p k • Φ L,k . It follows from the definitions of these maps that any periodic orbit which appears in the image of p k • Φ L,k is central with respect to H k . Let By the construction of Φ H k , X k is a finite sum of the form where the n j k are nonzero integers and the x j k are central 1-periodic orbits of H k .
Since X k gets mapped to q under Φ L,k , the moduli space which determines the image Φ R,k (X k ), must be nonempty. Choose a (v k , σ k ) in R 0 (X k , q; H k , f ) for each k. The caps v k belongs to R(x k ; H k ) for some x k in P(H k ) which appears in X k with a nonzero coefficient. Moreover, v k is part of a central pair for x k with respect to H k , and so by (17), (23) and Inequality (18) together with (28) yields the desired uniform energy bound It only remains to show that the orbits x k are nonconstant. Each x k appears in p k • Φ L,k (q) with a nonzero coefficient. Hence, there is a pair of maps (α k , u k ) in L [0] (q, x k ; f, H G k ) such that u k is part of a central pair for x k with respect to H k . The existence of the regular pair (α k , u k ) together with the dimension formula for Since u k is part of a central pair for x k the action A H k (x k , u k ) satisfies the same bounds, (25), as Lemma 4.1 then implies that the orbits x k are nonconstant and the proof of Proposition 4.3 is complete.

4.4.
Step 4: A nonconstant limit of linear right Floer caps. Let C be the closed subset of C ∞ (R × S 1 , M ) consisting of maps v : R × S 1 → M such that v(0, t) is a contractible loop in M . We consider this space as being equipped with the C ∞ loc -topology. By Proposition 4.3 we have a sequence of nonconstant periodic orbits x k ∈ P(H k ) and a sequence of right Floer caps v k ∈ R(x k , H k ) which satisfy (24) and (25). The linear homotopy triples H k were chosen so that they converge to H r = (H s , 0, J s ). Together with uniform energy bound (24), this implies that there is a subsequence of the v k which converges in C to a mapṽ. This limiting mapṽ is a solution of the equation It also satisfies The mapṽ may or not be constant. To find the the periodic orbit and the right asymptotic spanning disc of Proposition 2.5, we need to consider both possibilities. 4.4.1. Case 1: a nonconstant limit. We assume here that the subsequence, which we still denote by v k , converges to a nonconstant solutionṽ of (29). In this case, the mapṽ will be the asymptotic right spanning disc of Proposition 2.5 and we will writeṽ = v.
The energy bound (30) implies that the limit v(+∞) = lim s→+∞ v(s, t) is a point in M . It also implies that there is a sequence s − j → −∞ such that v(s − j , t) converges to some y(t) in P(H r ). For simplicity we assume that the sequence s − j is monotone decreasing and that s − 1 < −1. It remains for us to show that the limiting periodic orbit y is nonconstant and that there is an ǫ > 0 such that (5) holds for all j, i.e., We begin by proving that (5) holds for ǫ = 1 2 ( H r + − G + ). By Remark Since the v k converge to v in the C ∞ loc -topology, and H k converges to H r in the C ∞ -topology, it suffices to show that for large enough k we have The first of these inequalities follows immediately from equation (13). In particular, this identity implies that To prove the second inequality in (31), we first note that for i > j inequality (14) yields Hence, for each k, the sequence Thus, (31) holds and we have established inequality (5).
Finally we must show that the periodic orbit y is nonconstant. This is easily derived from (5) as follows. Set can be extended and reparameterized to form spanning discs for y ∈ P(H) in a fixed homotopy class. These extensions can be made arbitrarily small for sufficiently large j. Hence, by inequality (5) and the assumption that v is nonconstant, we can choose such a spanning disc w for y such that (32) − B < A Hr (y, w) < A.
Assume now that y is a constant periodic orbit, i.e., y(t) = P for some critical point P of H r . Then w represents a class [w] ∈ π 2 (M ) and If ω([w]) = 0, then (32) and (33)  However, this implies that A Hr (y, w) fails to lie in the interval (−B, A), which contradicts (32). The orbit y must therefore be nonconstant. 4.4.2. Case 2: a constant limit. We now assume that the maps v k converge in C to a constant mapṽ(s, t) = P . In this case, we will adapt a topological argument from [Gi] to prove that there is a sequence τ k → −∞, such that v k (s + τ k , t) converges to a nonconstant solution v of the equation This will be the right asymptotic spanning disc of Proposition 2.5.
To detect this map, we first pass to a subsequence of the v k whose negative asymptotic limits converge to a nonconstant element of P(H r ). Recall that x k = v k (−∞) is a nonconstant 1-periodic of H k . Since the x k are nonconstant, they are contained in the region U [r/3+δ,2r/3−δ] . By Arzela-Ascoli, there is a convergent subsequence of the x k that converges to some x ∈ P(H r ). Since it is contained in U [r/3+δ,2r/3−δ] , the orbit x is also nonconstant. From this point on, we restrict our attention to a subsequence of the v k for which the x k converge to x. For simplicity, this subsequence will still be denoted by v k .
There is a natural action of R on C defined by τ · v(s, t) = v(s + τ, t). We set Γ(v k ) = {τ · v k | τ ∈ R}, and define Σ to be the set of limits of all convergent sequences of the form There are two continuous maps on Σ which will be useful in what follows. The first is the evaluation map ev : Σ → M defined by ev(v) = v(0, 0).

The second map is the function
and it suffices to show that for sufficiently large k we have The proof of these inequalities is entirely similar to the proof of (31). In particular, (13) implies that On the other hand, (14) yields and by (25), we then have Lemma 4.5. Every element of Σ is a solution of (34) with energy less that r(M, ω).

Proof.
Let v = lim k→∞ τ k · v k . By (24), the energy of each v k is less than r(M, ω). Since E(v k ) = E(τ k · v k ), the energy of v is also less than r(M, ω). It only remains to show that v is a solution of (34).
Recall that, ← − H s is a compact homotopy from H r to −B. If τ k → −∞, then v is clearly a solution of (34). If the sequence of shifts τ k is bounded, then v is equal to the constant mapṽ(s, t) = P . Sinceṽ is also a solution of (29), we must have X← − H s ( P ) = 0 for all s ∈ R. In other words, P is a critical point of H r and hence must lie in U r/3 ∪ U [2r/3,+∞) , where U [2r/3,+∞) denotes the complement of U 2r/3 in M . Lemma 4.4 implies that Hence, P belongs to U [2r/3,+∞) . On this set ← − H s = H r = −B, and so v is a trivial solution of (34).
Finally, when the shifts τ k → ∞, the limit v is a solution of with energy less than r(M, ω). Any such map can be uniquely extended to a holomorphic sphere with the same energy. Since r(M, ω) < , the almost complex structure J − can be chosen, at the outset, to satisfy (J − ) > H r .
The map v must therefore be constant. Lemma 4.4 implies that the constant maps in Σ all lie in U [2r/3,+∞) . Hence, v is again a trivial solution of (34).
Here, the elements of P(H r ) are identified with elements of C that do not depend on s.
Proof. For τ ′ > τ , a simple computation yields Since the integrand is nonnegative the function for τ ′ > τ , then (36) implies that ∂ s v = 0 for s ∈ (τ, τ ′ ). By Lemma 4.5, v is a solution of (34), and hence v(s, t) = v(t) is a 1-periodic orbit of H r for s ∈ (τ, τ ′ ). The Unique Continuation Theorem of [FHS] then implies that v(s, t) = v(t) for all values of s.
Following [Gi] we now prove: Lemma 4.7. The set Σ has the following properties.
(i) the point P and the nonconstant 1-periodic orbit x(t) belong to Σ; (ii) the subsets Γ(v k ) ⊂ C converge to Σ in the Hausdorff topology; (iii) the set Σ is connected, compact and preserved by the R-action on C; (iv) The action of R on Σ is nontrivial.
Proof. The first two properties follow almost immediately from the definition of Σ. The same is true of the fact that Σ is invariant under the R-action. The compactness of Σ follows from Lemma 4.5 and the fact that Σ is closed. In particular, the subset of C consisting of solutions of (34) with energy less than is itself compact by the usual Floer compactness theorem.
To prove that Σ is connected, consider any two disjoint open sets in C, U 1 and U 2 , which cover Σ. Let Σ e P be the component of Σ which contains P and suppose that Σ e P ⊂ U 1 . By (ii), the Γ(v k ) are contained in U 1 ∪ U 2 for all sufficiently large k. Since the Γ(v k ) are connected and P is a limit point of the Γ(v k ), they must be contained in U 1 for large k. Thus, Σ ∩ U 2 = ∅ and it follows that Σ must be connected.
We now consider the set The properties of Σ established above imply that Σ min is comprised of elements in P(H r ). In particular, for v ∈ Σ min choose a τ < 0. Lemmas 4.4, 4.6, and 4.7 yield The second statement of Lemma 4.6 then implies that v belongs P(H r ).
Note that the constant elements of Σ min take values in the set U [2r/3,+∞) . Identifying U [2r/3,+∞) with the space of constant maps in C which take values in U [2r/3,+∞) , we define The set C is a compact subset of Σ. By property (iv) of Lemma 4.7, C is also a proper subset of Σ. Most importantly, C is nonempty because it contains P .
Lemma 4.8. The set C is a union of connected components of Σ min .
Proof. If one assumes the contrary, then there is a sequence of nonconstant periodic orbits x − k ∈ Σ min C which converges to a point of C. This is a contradiction since the nonconstant orbits are contained in the closure of U 2r/3−δ .  Proof. Assume the contrary. Then there is neighborhood V ⊃ C 0 and a sequence c i → 0 + such that V c i is not contained in V. Let v i be an element in V c i V. Since Σ is compact, there is a subsequence of the v i which converges to an element of i V c i V. On the other hand, i V c i is a connected subset of Σ min which contains C 0 . This contradicts the fact that C 0 is a connected component of Σ min .
We can now complete the proof of Proposition 2.5 in the present case. By Lemma 4.9, we can find a constant c > 0 such that V c ⊂ ev −1 (U (2r/3−δ,+∞) ). Either V c ∩ C = C 0 or V c ∩ C is disconnected. In both cases, the fact that Σ is connected and contains the nonconstant orbit x(t) implies that Let v be any map in V c C. We will show that v is a right asymptotic spanning disc with the desired properties.
Properties (5) and (6) are easily verified. By Lemma 4.5, v is a solution of (34) and hence (6) for H s = H r . By the definition of Σ, v = lim k→∞ τ k · v k and so A j