Citation Journal of Symplectic Geometry (201.

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1 Title Symplectic homology of disc Euclidean space cotange Authors) Irie, Kei Citation Journal of Symplectic Geometry 201 Issue Date URL Right by International Press. Type Journal Article Textversion author Kyoto University

2 SYMPLECTIC HOMOLOGY OF DISC COTANGENT BUNDLES OF DOMAINS IN EUCLIDEAN SPACE KEI IRIE Abstract. Let V be a bounded domain with smooth boundary in R n, and D V denote its disc cotangent bundle. We compute symplectic homology of D V, in terms of relative homology of loop spaces on the closure of V. We use this result to show that the Floer- Hofer-Wysocki capacity of D V is between 2rV ) and 2n + 1)rV ), where rv ) denotes the inradius of V. As an application, we study periodic billiard trajectories on V. 1. Introduction 1.1. Main result. Let us consider the symplectic vector space T R n, with coordinates p 1,..., p n, q 1,..., q n and the standard symplectic form ω n := dp 1 dq dp n dq n. For any bounded open set U T R n and real numbers a < b, one can define a Z 2 -module SH [a,b) U), which is called symplectic homology. This invariant was introduced in [7]. Our first goal is to compute SH [a,b) U), when U is a disk cotangent bundle of a domain in R n. First let us fix notations. For any domain i.e. connected open set) V R n, its disc cotangent bundle D V T R n is defined as D V := {q, p) T R n q V, p < 1}. We use the following notations for loop spaces: ΛR n ) := W 1,2 S 1, R n ), where S 1 := R/Z. Λ <a R n ) := {γ ΛR n ) length of γ < a}. For any subset S R n, we set ΛS) := {γ ΛR n ) γs 1 ) S}, Λ <a S) := ΛS) Λ <a R n ). Then, the main result in this note is the following: Theorem 1.1. Let V be a bounded domain with smooth boundary in R n, and V denote its closure in R n. For any a < 0 and b > 0, there exists a natural isomorphism SH [a,b) D V ) = H Λ <b V ), Λ <b V ) \ ΛV ) ). Moreover, for any 0 < b < b + the following diagram commutes: SH [a,b ) D V ) = H Λ <b V ), Λ <b V ) \ ΛV ) ) Date: September 8, SH [a,b+ ) D V ) = H Λ <b + V ), Λ <b+ V ) \ ΛV ) ). 1

3 The left vertical arrow is a natural map in symplectic homology, and the right vertical arrow is induced by inclusion Floer-Hofer-Wysocki capacity and periodic billiard trajectories. By using symplectic homology, one can define the Floer-Hofer-Wysocki capacity, which is denoted as c FHW. The Floer-Hofer-Wysocki capacity was introduced in [8]. We recall its definition in Section 2.4. The Floer-Hofer-Wysocki capacity of a disk cotangent bundle D V is important in the study of periodic billiard trajectories on V for precise definition, see Definition 6.3): Proposition 1.2. Let V be a bounded domain with smooth boundary in R n. Then, there exists a periodic billiard trajectory γ on V with at most n + 1 bounce times such that length of γ = c FHW D V ). Remark 1.3. The idea of using symplectic capacities to study periodic billiard trajectory is due to Viterbo [14]. See also [5], in which a result similar to Proposition 1.2 Theorem 2.13 in [5]) is proved. Proposition 1.2 is essentially the same as Theorem 13 in [11]. However, our formulation of symplectic homology in this note is a bit different from that in [11], in which we used Viterbo s symplectic homology [13]. Hence we include a proof of Proposition 1.2 in Section 6, for the sake of completeness. Given Proposition 1.2, it is natural to ask if one can compute c FHW D V ) by using only elementary i.e. singular) homology theory. The following corollary of our main result gives an answer to this question. For any x V, c x denotes the constant loop at x. Corollary 1.4. Let V be a bounded domain with smooth boundary in R n, and b > 0. Let us define ι b : V, V ) Λ <b V ), Λ <b V ) \ ΛV ) ) by ι b x) := c x. Denote by ι b ) the map on homology induced by ι b. Then, c FHW D V ) = inf{b ι b ) n : H n V, V ) H n Λ <b V ), Λ <b V ) \ ΛV )) vanishes}. To prove Corollary 1.4, we need to combine our main result Theorem 1.1 with results in [10]. Corollary 1.4 is proved in Section Floer-Hofer-Wysocki capacity and inradius. Using Corollary 1.4, one can obtain a quite good estimate of c FHW D V ) by using the inradius of V. First let us define the notion of the inradius: Definition 1.5. Let V be a domain in R n. The inradius of V, which is denoted as rv ), is the supremum of radii of balls in V. In other words, rv ) := sup distx, V ). x V Our estimate of the Floer-Hofer-Wysocki capacity is the following: Theorem 1.6. Let V be a bounded domain with smooth boundary in R n. Then, there holds 2rV ) c FHW D V ) 2n + 1)rV ). Combined with Proposition 1.2, Theorem 1.6 implies the following result: 2

4 Corollary 1.7. Let V be a bounded domain with smooth boundary in R n. There exists a periodic billiard trajectory on V with at most n + 1 bounce times and length between 2rV ) and 2n + 1)rV ). Remark 1.8. Let ξv ) denote the infimum of the lengths of periodic billiard trajectories on V. Corollary 1.7 shows that ξv ) 2n + 1)rV ). When V is convex, this result was already established as Theorem 1.3 in [5]. On the other hand, the main result in [11] is that ξv ) const n rv ) for any domain V with smooth boundary in R n. A weaker result ξv ) const n volv ) 1/n was obtained in [14], [9]. Theorem 1.6 is proved in Section 7. Here we give a short comment on the proof. Actually, the lower bound is immediate from Corollary 1.4, and the issue is to prove the upper bound. By Corollary 1.4, it is enough to show that if b > 2n + 1)rV ), then ι b ) [ V, V )] = 0. We will prove this by constructing a n+1)-chain in Λ <b V ), Λ <b V )\ ΛV ) ) which bounds V, V ). Details are carried out in Section Organization of the paper. In Section 2, we recall the definition and main properties of symplectic homology, following [7]. In Section 3, we recall Morse theory for Lagrangian action functionals on loop spaces, following [1], [3]. The goal in these sections is to fix a setup for the arguments in Sections 4, 5, 6. In Section 4, we prove our main result Theorem 1.1. The proof consists of two steps: Step1: In Theorem 4.2, we prove an isomorphism between Floer homology of a quadratic Hamiltonian on T R n and Morse homology of its fiberwise Legendre transform. Step2: By taking a limit of Hamiltonians, we deduce Theorem 1.1 from Theorem 4.2. Our proof of Theorem 4.2 is based on [2]: we construct an isomorphism by using so called hybrid moduli spaces. However, since we will work on T R n, proofs of various C 0 - estimates for hybrid) Floer trajectories are not automatic. Techniques in [2] in which the authors are working on cotangent bundles of compact manifolds) do not seem to work directly in our setting. To prove C 0 - estimates for Floer trajectories in our setting, we combine techniques in [2] and [7]. Proofs of C 0 - estimates are carried out in Section 5. In Section 6, we discuss the Floer-Hofer-Wysocki capacity and periodic billiard trajectories. The goal of this section is to prove Proposition 1.2 and Corollary 1.4. In Section 7, we prove Theorem 1.6. This section can be read almost independently from the other parts of the paper. 2. Symplectic homology We recall the definition and main properties of symplectic homology. follow [7]. We basically 2.1. Hamiltonian. For H C T R n ), its Hamiltonian vector field X H is defined as ω n X H, ) = dh ). 3

5 For H C S 1 T R n ) and t S 1, H t C T R n ) is defined as H t q, p) := Ht, q, p). PH) denotes the set of periodic orbits of X Ht ) t S 1, i.e. PH) := {x C S 1, T R n ) ẋt) = X Ht xt))}. x PH) is called nondegenerate if 1 is not an eigenvalue of the Poincaré map associated with x. We introduce the following conditions on H C S 1 T R n ): H0): Every element in PH) is nondegenerate. H1): There exists a 0, ) \ πz such that sup H t Q a C 1 T R n ) t S 1 Q a q, p) := a q 2 + p 2 ). <, where Remark 2.1. The class of Hamiltonians considered in this note is a bit different from that in [7]. To put it more precisely, H1) is more restrictive than conditions 6), 7) in [7]. On the other hand, we do not need condition 8) in [7]. It is easy to see that our definition of symplectic homology is equivalent to that in [7], see Remark 2.6. Lemma 2.2. For any H C S 1 T R n ) which satisfies H1), PH) is C 0 -bounded. In particular, if H also satisfies H0), then PH) is a finite set. Proof. Suppose that there exists H C S 1 T R n ) which satisfies H1) and PH) is not C 0 -bounded. Then there exists a sequence x j ) j=1,2,... in PH) such that R j := max t S 1 x jt) goes to as j. Define v j : S 1 T R n and h j C S 1 T R n ) by v j t) := x j t)/r j, h j t, q, p) := Ht, R j q, R j p)/r 2 j. It is easy to show that v j Ph j ). Moreover, since sup t S 1 dh t dq a C 0 <, 1) lim j sup t S 1 dh j t dq a C 0 = 0. By definition, max v jt) = 1. In particular, v j ) j is C 0 -bounded. Moreover, since t v j = t S 1 X h jv j ), 1) shows that v j ) j is C 1 -bounded. Hence, up to a subsequence, v j ) j converges t in C 0 S 1, T R n ). We denote the limit by v. By the triangle inequality, 1 0 X h jv j t)) X Q avt)) dt t 1 0 X h jv j t)) X Q av j t)) dt + t 1 0 X Q av j t)) X Q avt)) dt. As j, the first term on the RHS goes to 0 by 1), and the second term on the RHS goes to 0 since v j converges to v in C 0. Therefore, for any 0 t 0 1, vt 0 ) v0) = lim j v j t 0 ) v j 0) t0 = lim j 0 X h jv j t)) dt = t 4 t0 0 X Q avt)) dt,

6 hence v PQ a ). On the other hand, it is clear that max vt) = 1. This is a contradiction, since a / πz implies that the only element in PQ a ) is the constant loop at t S1 0,..., 0). H C S 1 T R n ) is called admissible if it satisfies H0) and H1) Truncated Floer homology. Let J = J t ) t S 1 be a time dependent almost complex structure on T R n, such that: J1): For any t S 1, J t is compatible with ω n, i.e. g Jt ξ, η) := ω n ξ, J t η) is a Riemannian metric on T R n. Let H C S 1 T R n ) be an admissible Hamiltonian. For any x, x + PH), we introduce the Floer trajectory space in the usual manner: M H,J x, x + ) := {u : R S 1 T R n s u J t t u X Ht u)) = 0, lim us) = x ±}. s ± We set M H,J x, x + ) := M H,J x, x + )/R, where R acts on M H,J x, x + ) by shift in the s-variable. The standard complex structure J std on T R n is defined as J std pi ) := qi, J std qi ) := pi. Now we state our first C 0 -estimate. It is proved in Section 5. Lemma 2.3. There exists a constant ε > 0 which satisfies the following property: For any admissible Hamiltonian H C S 1 T R n ) and J = J t ) t S 1 which satisfies J1) and sup J t J std C 0 < ε, M H,J x, x + ) is C 0 -bounded t for any x, x + PH). We recall the definition of Floer homology. For any γ C S 1, T R n ), we set ) A H γ) := p i dq i Ht, γt)) dt. S 1 γ i For real numbers a < b, the Floer chain complex CF [a,b) H) is the free Z 2 module generated by {γ PH) A H γ) [a, b)}, indexed by the Conley-Zehnder index ind CZ. For the definition of the Conley-Zehnder index, see Section 1.3 in [7]. Suppose that J = J t ) t S 1 satisfies J1) and each J t is sufficiently close to J std. Lemma 2.3 shows that for generic J, M H,J x, x + ) is a compact 0-dimensional manifold for any x, x + PH) such that ind CZ x ) ind CZ x + ) = 1. We can thus define the Floer differential H,J on CF [a,b) H) as H,J [x ] ) := M H,J x, x + ) [x + ]. ind CZ x + )=ind CZ x ) 1 The usual gluing argument shows that H,J 2 = 0. HF [a,b) H, J) := H CF [a,b) H), H,J ) is called truncated Floer homology. 5

7 2.3. Symplectic homology. Suppose that we are given the following data: Admissible Hamiltonians H, H + C S 1 T R n ) J = J t ) t S 1, J + = J + t ) t S 1, which satisfy J1). Moreover, all J t, J + t are sufficiently close to J std. We assume that HF [a,b) H, J ), HF [a,b) H +, J + ) are well-defined. If H H +, i.e. H t, q, p) H + t, q, p) for any t S 1 and q, p) T R n, one can define monotonicity homomorphism in the following way. HF [a,b) H, J ) HF [a,b) H +, J + ) First we introduce the following conditions on H C R S 1 T R n ): { Hs 0, t, q, p) s s 0 ) HH1): There exists s 0 > 0 such that Hs, t, q, p) = H s 0, t, q, p) s s 0 ). HH2): s Hs, t, q, p) 0 for any s, t, q, p) R S 1 T R n. HH3): There exists as) C R) such that: a s) 0 for any s. as) πz = a s) > 0. Setting s, t, q, p) := Hs, t, q, p) Q as) q, p), there holds sup s,t C 1 T R n ) <, s,t) sup s s,t C 0 T R n ) <. s,t) If H satisfies HH1), HH2), HH3) and H t ± = H ±s0,t, H is called a homotopy from H to H +. For any H and H + such that H H +, there exists a homotopy from H to H +. In fact, take ρ C R) such that s 1 = ρs) = 1, s 0 = ρs) = 0, 0 < s < 1 = ρs) 0, 1), ρ s) > 0. Then Hs, t, q, p) := ρs)h + t, q, p) + 1 ρs))h t, q, p) is a homotopy from H to H +. Next we introduce conditions on J = J s,t ) s,t) R S 1, a family of almost complex structures on T R n parametrized by R S 1 : { J s1,tq, p) s s 1 ) JJ1): There exists s 1 > 0 such that J s,t q, p) = J s1,tq, p) s s 1 ). JJ2): For any s, t) R S 1, J s,t is compatible with ω n. If J satisfies JJ1), JJ2) and J ± t = J ±s1,t, J is called a homotopy from J to J +. Let H C R S 1 T R n ) be a homotopy from H to H +, and J = J s,t ) s,t) R S 1 be a homotopy from J to J +. For any x PH ) and x + PH + ), we define M H,J x, x + ) := {u : R S 1 T R n s u J s,t t u X Hs,t u)) = 0, lim s ± us) = x ±}. Now we state our second C 0 -estimate. It is proved in Section 5. 6

8 Lemma 2.4. There exists a constant ε > 0 which satisfies the following property: If J = J s,t ) s,t) R S 1 satisfies sup J s,t J std C 0 s,t C 0 -bounded for any x PH ), x + PH + ). < ε, M H,J x, x + ) is Lemma 2.4 shows that, if J is generic and all J s,t are sufficiently close to J std, M H,J x, x + ) is a compact 0-dimensional manifold for any x PH ), x + PH + ) such that ind CZ x ) = ind CZ x + ). We define Φ : CF [a,b) H, J ) CF [a,b) H +, J + ) by Φ [x ] ) := M H,J x, x + ) [x + ]. ind CZ x + )=ind CZ x ) The usual gluing argument shows that Φ is a chain map. The monotonicity homomorphism Φ : HF [a,b) H, J ) HF [a,b) H +, J + ) is the homomorphism on homology induced by Φ. One can show that Φ does not depend on the choices of H and J, see Section 4.3 in [7]. Remark 2.5. Let H be an admissible Hamiltonian, and J 0, J 1 be S 1 - dependent almost complex structures such that HF [a,b) H, J 0 ), HF [a,b) H, J 1 ) are well-defined. Then, one can show that the monotonicity homomorphism HF [a,b) H, J 0 ) HF [a,b) H, J 1 ) is an i- somorphism. Hence HF [a,b) H, J) does not depend on J, and we denote it by HF [a,b) H). Moreover, for two admissible Hamiltonians H, H + satisfying H H +, the monotonicity homomorphism HF [a,b) H ) HF [a,b) H + ) is well-defined. We define symplectic homology. Let U be a bounded open set in T R n. Let H U denote the set consisting of admissible Hamiltonians H such that H S 1 Ū < 0. H U is a directed set with relation H H + H t, q, p) H + t, q, p) t, q, p) S 1 T R n ). Then, for any < a < b <, we define symplectic homology SH [a,b) U) by SH [a,b) U) := lim HF [a,b) H). H H U If U V, then obviously H V H U. Hence there exists a natural homomorphism SH [a,b) V ) SH [a,b) U). Moreover, for any a ±, b ± R such that a a +, b b +, a < b, a + < b +, there exists a natural homomorphism SH [a,b ) U) SH [a+,b + ) U). Remark 2.6. As noted in Remark 2.1, the class of Hamiltonians considered here is different from that in [7]. However, our definition of symplectic homology given above is equivalent to the definition in [7] see Section 1.6 in [7]). A key fact is that compact perturbations of quadratic Hamiltonians are admissible both in our sense and sense in [7]. 7

9 2.4. Floer-Hofer-Wysocki capacity. Finally, we define the Floer-Hofer-Wysocki capacity, which is originally due to [8]. For any bounded open set U and b > 0, we define SH 0,b) U) := lim ε +0 SH [ε,b) U). When U V, there exists a natural homomorphism SH 0,b) V ) SH 0,b) U). For any p T R n, we define Θ b p) := lim SH 0,b) n+1b 2n p : ε)) ε +0 where B 2n p : ε) denotes the open ball in T R n with center p and radius ε. It is known that Θ b p) = Z 2, see pp in [8]. Let U be a bounded domain hence connected) in T R n. Taking p U arbitrarily, we define the Floer-Hofer-Wysocki capacity of U as c FHW U) := inf{b SH 0,b) n+1u) Θ b p) = Z 2 is onto}. It s known that the above definition does not depend on the choice of p. See pp.604 in [8]. 3. Loop space homology In this section, we recall Morse theory on loop spaces for Lagrangian action functionals. We mainly follow [1], [3] Lagrangian action functional. Recall that we used the notation ΛR n ) := W 1,2 S 1, R n ). Given L C S 1 T R n ), we consider the action functional S L : ΛR n ) R; γ Lt, γt), γt)) dt. S 1 We introduce the following conditions on L: L1): There exists a 0, ) \ πz such that a q 2) sup v 2 C Lt, q, v) t S 1 4a <. 2 T R n ) L2): There exists a constant c > 0 such that 2 vlt, q, v) c for any t, q, v) S 1 T R n. Notice that L1) implies the following estimates: L1) : D 2 Lt, q, v) const, q Lt, q, v) const1 + q ), v Lt, q, v) const1 + v ), Lt, q, v) const1 + q 2 + v 2 ). Lemma 3.1. If L satisfies L1) and L2), the following holds. 1) S L : ΛR n ) R is a Fréchet C 1 function. Its differential ds L is given by ds L γ)ξ) = q Lt, γ, γ)ξt) + v Lt, γ, γ) ξt) dt. S 1 8

10 Moreover, ds L is Gâteaux differentiable. We denote the differential by d 2 S L. 2) γ ΛR n ) satisfies ds L γ) = 0 if and only if γ C S 1, R n ) and q Lt, γ, γ) d dt v Lt, γ, γ) ) = 0. Proof. Using L1) and L2), the proof is the same as Proposition 3.1 in [3]. Let us set PL) := {γ ΛR n ) ds L γ) = 0}. γ PL) is called nondegenerate if d 2 S L γ) is nondegenerate as a symmetric bilinear form on T γ ΛR n ) = W 1,2 S 1, R n ). For each γ ΛR n ), DS L γ) T γ ΛR n ) = W 1,2 S 1, R n ) is defined so that DS L γ), ξ W 1,2 = ds L γ)ξ) ξ W 1,2 S 1, R n ) ). We show that the pair S L, DS L ) satisfies the Palais-Smale PS) condition. First let us recall what the PS condition is: Definition 3.2. Let M be a Hilbert manifold, f : M R be a C 1 function, and X be a continuous vector field on M. A sequence p k ) k on M is called a Palais-Smale PS) sequence, if fp k )) k is bounded, and lim dfxp k )) = 0. The pair f, X) satisfies the PS-condition, if any PS sequence contains a convergent subsequence. Lemma 3.3. Suppose that L C S 1 T R n ) satisfies L1). Let γ k ) k be a sequence on ΛR n ) such that both S L γ k ) and DS L γ k ) W 1,2 are bounded. Then, γ k ) k is C 0 -bounded. Proof. Suppose that there exists a sequence γ k ) k such that both S L γ k ), DS L γ k ) W 1,2 are bounded, and m k := max γ kt) goes to as k. We define δ k ΛR n ) and t S 1 l k C S 1 T R n ) by δ k t) := γ k t)/m k, l k t, q, p) := Lt, m k q, m k p)/m k 2. We show that δ k ) k is W 1,2 -bounded. Since δ k ) k is obviously C 0 -bounded, it is enough to show that δ k ) k is L 2 -bounded. First notice that lim S l k δ k ) = lim S L γ k ) m k 2 = 0. On the other hand, since L satisfies L1), lim S l k δ k ) ) δ k 2 S 4a a δ k 2 dt = 0. 1 Thus δ k ) k is L 2 -bounded. By taking a subsequence of δ k ) k, we may assume that there exists δ ΛR n ) such that lim δ k δ C 0 = 0, and δ k δ k ) weakly in L 2. We prove that ds l δ) = 0, where lt, q, v) := v 2 /4a a q 2. This means that δ C S 1, R n ) and δt) + 4a 2 δt) 0. Since a / πz, this means that δt) 0. However, since max δt) = lim max δ kt) = 1, this is a contradiction. t S1 t S 1 9

11 To prove ds l δ) = 0, first notice that lim DS DS L γ k ) W 1,2 l k δ k ) W 1,2 = lim = 0. m k Hence it is enough to show that for any ξ C S 1, R n ) there holds lim dsl δ) ds l δ k ) ) ξ) = 0, lim dsl δ k ) ds lk δ k ) ) ξ) = 0. To check the first claim, notice the following equation: dsl δ) ds l δ k ) ) δt) δ k t) ξ) = S 2a ξt) 2a δt) δ k t) ) ξt) dt. 1 Then, since δ k converges to δ weakly in L 2, the RHS goes to 0 as k. The second claim follows from lim l l k C 1 = 0. Corollary 3.4. Suppose that L C S 1 T R n ) satisfies L1) and L2). Then, the pair S L, DS L ) satisfies the PS-condition on ΛR n ). Proof. Suppose that γ k ) k is a PS-sequence with respect to S L, DS L ). Then, Lemma 3.3 shows that γ k ) k is C 0 - bounded. Then, Proposition 3.3 in [3] shows that γ k ) k has a convergent subsequence Construction of a downward pseudo-gradient. Suppose that L C S 1 T R n ) satisfies L1) and L2). To define a Morse complex of S L, we need the following condition: L0): Every γ PL) is nondegenerate. The following lemma basically the same as Theorem 4.1 in [3]) constructs a downward pseudo-gradient vector field for S L. For the definitions of the terms Lyapunov function, Morse vector field, Morse-Smale condition, see Section 2 of [3]. Lemma 3.5. If L C S 1 T R n ) satisfies L0), L1), L2), there exists a smooth vector field X on ΛR n ) which satisfies the following conditions: 1) X is complete. 2) S L is a Lyapunov function for X. 3) X is a Morse vector field. Xγ) = 0 if and only if γ PL). Every γ PL) has a finite Morse index, which is denoted by ind Morse γ). 4) The pair S L, X) satisfies the Palais-Smale condition. 5) X satisfies the Morse-Smale condition up to every order. Proof. In the course of this proof, we use the following abbreviation: {a < S L < b} := {γ ΛR n ) a < S L γ) < b}. Moreover, W 1,2 is abbreviated as. Since S L, DS L ) satisfies the PS-condition, and all critical points are nondegenerate, for any a < b there exist only finitely many critical points of S L on {a < S L < b}. We denote them as γ 1,..., γ m. 10

12 For each 1 j m, Lemma 4.1 in [3] shows that there exist U γj, Y γj such that: U γj is a neighborhood of γ j in {a < S L < b}. Y γj is a smooth vector field on U γj. γ j is a critical point of Y γj with a finite Morse index, and there holds ds L Y γj γ)) λγ j ) γ γ j 2 γ U γj ), where λγ j ) is a positive constant. By taking U γj sufficiently small, we may asume that Y γj 1 on U γj. We take a smaller neighborhood V γj such that V γj U γj. Since S L, DS L ) satisfies the PS-condition, there exists ε > 0 such that : for any γ {a < S L < b} \ U γ1 U γm ), DS L γ) ε. For each γ / U γ1 U γm, set Y γ := DS L γ)/ DS L γ). Then, obviously Y γ = 1. Moreover, ds L γ)y γ ) = DS L γ), Y γ = DS L γ) ε. Since S L is C 1 by Lemma 3.1 1), if U γ is a sufficiently small neighborhood of γ, γ U γ = ds L γ )Y γ ) ε/2. We may also assume that U γ is disjoint from V γ1 V γm. Moreover, since ΛR n ) is paracompact, we can define a locally finite open covering {U γ } γ Γ of {a < S L < b} such that γ 1,..., γ m Γ. Let {χ γ } γ Γ be a partition of unity with respect to {U γ } γ Γ. Then we define a vector field Y on {a < S L < b} by Y := γ Γ χ γ Y γ. Since each Y γ satisfies Y γ 1, it is clear that Y 1. Moreover, there exists c > 0 such that γ / V γ1 V γm = ds L γ)y γ)) c. Now we show that S L, Y ) satisfies the PS-condition on {a < S L < b}. Let x k ) k be a sequence on {a < S L < b} such that lim ds L x k )Y x k )) = 0. Then, x k V γ1 V γm for sufficiently large k. By taking a subsequence, we may assume that x k V γ1 for all k. Then, since ds L x k )Y x k )) = ds L x k )Y γ1 x k )) λγ 1 ) x k γ 1 2, there holds lim x k γ 1 = 0. Thus S L, Y ) satisfies the PS condition. We have defined a smooth vector field Y on {a < S L < b}, which satisfies 2), 3), 4) and Y 1. Finally we construct X on ΛR n ). Take a sequence of closed intervals I m ) m Z with the following properties: min I m ) m, max I m ) m are increasing sequences. I m = R. m I m I m if and only if m m 1. For any m Z, I m I m+1 does not contain critical values of S L. 11

13 For every m, there exists a smooth vector field X m on {min I m < S L < max I m } which satisfies 2), 3), 4) and X m 1. Finally, taking a partition of unity ρ m ) m with respect to the open covering {min I m < S L < max I m } m of ΛR n ), we define a vector field X on ΛR n ) by X := ρ m X m. Then, it is easy to check that S L, X) satisfies the m PS condition. Moreover, since X satisfies X 1 everywhere, X is complete. The vector field X defined above satisfies 1)-4) in the statement. Since it is of class C, the Sard-Smale theorem shows that 5) is satisfied by a sufficiently small C perturbation Morse complex. Let X be a downward pseudo-gradient for S L on ΛR n ), which is constructed in Lemma 3.5. Since X is complete, one can define φ X t ) t R, a family of diffeomorphisms on ΛR n ) so that φ X 0 = id ΛR n ), t φ X t ) = Xφ X t ). For each γ PL), its stable and unstable manifolds are defined as W s γ : X) := {p ΛR n ) lim φ X t p) = γ}, t W u γ : X) := {p ΛR n ) lim t φx t p) = γ}. For any γ, γ PL), we set M X γ, γ ) := W u γ : X) W s γ, X). Since M X γ, γ ) consists of flow lines of X, M X γ, γ ) admits a natural R action. We denote the quotient by M X γ, γ ). For any γ, γ PL), W u γ : X) and W s γ : X) are transverse, since X satisfies the Morse-Smale condition. Therefore, M X γ, γ ) is a smooth manifold with dimension ind Morse γ) ind Morse γ ) 1. When ind Morse γ) ind Morse γ ) = 1, M X γ, γ ) consists of finitely many points. For any < a < b <, CM [a,b) L) denotes the free Z 2 -module generated by {γ PL) a S L γ) < b}. We define a differential L,X on CM [a,b) L) by ) L,X [γ] := M X γ, γ ) [γ ]. ind Morse γ )=ind Morse γ) 1 Then CM [a,b) L), L,X ) is a chain complex, and its homology group HM [a,b) L, X) is isomorphic to H {S L < b}, {S L < a}). For details, see [1]. Next we discuss functoriality. Consider L 0, L 1 C S 1 T R n ) which satisfy L0), L1), L2) and L 0 t, q, v) > L 1 t, q, v) for any t, q, v) S 1 T R n. Take vector fields X 0, X 1 on ΛR n ) such that L 0, X 0 ) and L 1, X 1 ) satisfy the conditions in Lemma 3.5. We assume that PL 0 ) PL 1 ) = this can be achieved by slightly perturbing L 0 ). Then, by a C -small perturbation of X 0, one can assume the following: For any γ 0 PL 0 ) and γ 1 PL 1 ), W u γ 0 : X 0 ) is transverse to W s γ 1 : X 1 ). If this assumption is satisfied, M X 0,X 1γ0, γ 1 ) := W u γ 0 : X 0 ) W s γ 1 : X 1 ) is a smooth manifold with dimension ind Morse γ 0 ) ind Morse γ 1 ). 12

14 We define a chain map Φ : CM [a,b) L 0, X 0 ) CM [a,b) L 1, X 1 ) by Φ[γ]) := M X 0,X 1γ, γ ) [γ ]. ind Morse γ )=ind Morse γ) Φ induces a homomorphism on homology, which coincides with the homomorphism induced by the inclusion {S L 0 < b}, {S L 0 < a}) {S L 1 < b}, {S L 1 < a}). 4. Proof of Theorem 1.1 The goal of this section is to prove Theorem 1.1, i.e. to compute SH [a,b) D V ) for a bounded domain V R n with smooth boundary. In Section 4.1, we reduce Theorem 1.1 to Theorem 4.2 and Lemma 4.3. Theorem 4.2 is the main step, and it is proved in Sections 4.2 and 4.3, assuming some C 0 - estimates of Floer trajectories: Lemmas 4.8, 4.9, These C 0 - estimates are proved in Section 5. Lemma 4.3 is a technical lemma on loop space homology, and it is proved in Section Outline. Let us take a m ) m, an increasing sequence of positive numbers such that a m / πz for any m, and lim a m =. We take a sequence k m ) m in C R 0, R) such m that: k1): For every m, t k m t) > 0 and t 2 k m t) 0 for any t 0. k2): For every m, t k m a m on {t k m t) 0}. { k3): k m ) m is strictly increasing. Moreover, sup k m t) = m 0 0 t 1). t > 1) Let us define K m C R n, R) by K m p) := k m p 2 ). Then, k1) implies that K m is strictly convex. Moreover, k2) implies that R n R n ; p vp) := p K m is a diffeomorphism. We denote its inverse by pv), i.e. p K m pv)) = v. Let K m be the Legendre transform of K m, i.e. K mv) := pv) v K m pv)). Then, it is easy to show that K m) m is strictly decreasing, and inf m K mv) = v for any v R n. We take a sequence Q m ) m of smooth functions on R n, such that Q1): There exists a sequence of constants c m ) m such that Q m q) ) a m q 2 + c m is compactly supported. { 0 q Q2): Q m ) m is strictly increasing. Moreover, sup Q m q) = V ) m q / V ). Let H mq, p) := Q m q) + K m p). Then, for every m, H m satisfies H1). Moreover, H m) m is strictly increasing, and { sup H mq, 0 q, p) D V ) p) = m q, p) / D V ). 13

15 Let L m be the fiberwise Legendre transform of H m. It is easy to see that L mq, v) = Kmv) Q m q). Then, for every m, { L m satisfies L1) and L2). L m) m is strictly decreasing, and there holds inf v q m L mq, v) = V ) q / V ). Since H m) m is strictly increasing, by sufficiently small perturbations of H m) m, one can obtain a sequence H m ) m on C S 1 T R n ) with the following properties: For every m, H m is admissible. { 0 q, p) D V ) q, p) / D V ). H m ) m is strictly increasing, and sup H m t, q, p) = m For every m, its Legendre transform L m is well-defined, and it satisfies L0), L1), L2). L m ) m is strictly decreasing, and inf m Lm t, q, v) = { v q V ) q / V ). Remark 4.1. For notational reasons, we use superscripts for H m and L m. By the first two properties, SH [a,b) D V ) = lim m HF [a,b) H m ). Now we state the following key result, which is proved in Sections 4.2 and 4.3: Theorem 4.2. For any < a < b < and m, there exists a natural isomorphism HM [a,b) L m ) = HF [a,b) H m ). The following diagram is commutative for every m: HM [a,b) L m ) = HM [a,b) L m+1 ) HF [a,b) H m ) HF [a,b) H m+1 ). = Then we obtain lim m HF [a,b) H m ) = lim m Since L m ) m is strictly decreasing and inf m Lm t, q, v) = Therefore, for any a < 0 and b > 0, HM [a,b) L m ) = H {S L m < b}, ) {S L m < a}. m m inf m S L mγ) < c γs1 ) V or length of γ) < c. { v q V ) q / V, for any c R ) SH [a,b) D V ) = H Λ <b R n ) ΛR n ) \ Λ V )), ΛR n ) \ Λ V ) ) = H Λ <b R n ), Λ <b R n ) \ Λ V ) ) = H Λ <b V ), Λ <b V ) \ ΛV )), where the second isomorphism follows from excision, and the third isomorphism follows from the next Lemma 4.3, which is proved in Section

16 Lemma 4.3. Let V be a bounded domain in R n with smooth boundary. For any 0 < b <, there exists a natural isomorphism H Λ <b R n ), Λ <b R n ) \ Λ V )) = H Λ <b V ), Λ <b V ) \ ΛV )). Finally, we have to check that for any b < b +, the following diagram commutes: SH [a,b ) D V ) = H Λ <b V ), Λ <b V ) \ ΛV ) ) SH [a,b+ ) D V ) = H Λ <b + V ), Λ <b+ V ) \ ΛV ) ). This is clear from the construction, hence omitted Construction of a chain level isomorphism. In this and the next subsection, we prove Theorem 4.2. In this subsection, we define an isomorphism HM [a,b) L m ) HF [a,b) H m ). Following [2], we define this isomorphism by considering so called hybrid moduli spaces. Suppose we are given the following data: J m = Jt m ) t S 1, which is sufficiently close to the standard one, and CF [a,b) H m, J m ) is well-defined. Smooth vector field X m on ΛR n ), such that CM [a,b) L m, X m ) is well-defined. γ PL m ) and x PH m ). We consider the following equation for u W 1,3 S 1 [0, ), T R n ): s u Jt m t u X H m u)) = 0, t πu0)) W u γ : X m ), lim us) = x. s π denotes the natural projection T R n R n ; q, p) q. The moduli space of solutions of this equation is denoted by M X m,h m,jmγ, x). Remark 4.4. In the definition of M X m,h m,j m, we have used a Sobolev space W 1,3 S 1 [0, ), T R n ). One can replace it with W 1,r S 1 [0, ), T R n ) for any 2 < r < 4. The condition 2 < r < 4 is necessary to carry out Fredholm theory and prove C 0 -estimates for Floer trajectories. To define a homomorphism by counting M X m,h m,j mγ, x), we need the following results: Lemma 4.5. For generic J m, M X m,h m,j mγ, x) is a smooth manifold of dimension ind Morse γ) ind CZ x) for any γ PL m ) and x PH m ). Proof. See Section 3.1 in [2]. Lemma 4.6. For any γ PL m ), x PH m ) and u M X m,h m,j mγ, x), there holds S L mγ) S L mπu0))) A H mu0)) A H mx). 15

17 Proof. See pp.299 in [2]. Corollary 4.7. When S L mγ) < A H mx), M X m,h m,j mγ, x) =. When S L mγ) = A H mx), M X m,h m,j mγ, x) if and only if γ = πx). In this case, M X m,h m,jmγ, x) consists of a single element u such that us, t) := xt). We recall that our setup differs from the one of [2] inasmuch as our base manifold is R n, while the authors of [2] work with compact bases. However, their analysis applies to our situation for all aspects except for the C 0 -bounds of Floer moduli spaces. Now, we state our third C 0 -estimate. It is proved in Section 5. Lemma 4.8. There exists ε > 0 such that, if J m satisfies sup Jt m J std C 0 < ε, t M X m,h m,j mγ, x) is C0 -bounded for any γ PL m ) and x PH m ). Suppose that J m satisfies the condition in Lemma 4.5, and it is sufficiently close to J std. By Lemma 4.8, for any γ PL m ) and x PH m ) such that ind Morse γ) ind CZ x) = 0, M X m,h m,jmγ, x) is a compact 0-dimensional manifold. Then, we can define a homomorphism Ψ m : CM [a,b) L m, X m ) CF [a,b) H m, J m ); [γ] M X m,h m,jmγ, x) [x]. ind CZ x)=ind Morse γ) Corollary 4.7 shows that Ψ m is an isomorphism for details, see Section 3.5 in [2]). Gluing arguments show that Ψ m is a chain map for details, see Section 3.5 in [2]). Hence Ψ m induces an isomorphism on homology Chain level commutativity up to homotopy. In the previous subsection, we constructed a chain level isomorphism Ψ m : CM [a,b) L m, X m ) CF [a,b) H m, J m ) for every m. In this subsection, we show that CM [a,b) L m, X m ) Ψ m CF [a,b) H m, J m ) Φ L CM [a,b) L m+1, X m+1 ) CF [a,b) Ψ m+1 H m+1, J m+1 ) commutes up to chain homotopy, where Φ H and Φ L are chain maps constructed in Section 2.3 and Section 3.3, respectively. To prove this, we introduce a chain map Θ : CM [a,b) [γ] Φ H L m, X m ) CF [a,b) H m+1, J m+1 ); ind Morse γ)=ind CZ x) M X m,h m+1,j m+1γ, x) [x]. It is enough to show Φ H Ψ m Θ Ψ m+1 Φ L. means chain homotopic.) 16

18 First we show that Ψ m+1 Φ L Θ. For any γ PL m ) and x PH m+1 ), N 0 γ, x) denotes the set of α, u, v), where α [0, ), u : [0, α] ΛR n ), v W 1,3 [0, ) S 1, T R n ) which satisfy the following conditions: u0) W u γ : X m ), us) = φ Xm+1 s u0)) 0 s α), s v Jt m+1 t v X H m+1v)) = 0, πv0)) = uα), lim vs) = x. t s We state our fourth C 0 -estimate. It is proved in Section 5. Lemma 4.9. There exists ε > 0 which satisfies the following property: If J m+1 satisfies sup Jt m+1 J std C 0 < ε, v C 0 is uniformly bounded for t any α, u, v) N 0 γ, x), where γ PL m ) and x PH m+1 ). Suppose that J m is generic and sufficiently close to J std. Then, due to Lemma 4.9 and gluing arguments, the following holds: When ind Morse γ) ind CZ x) = 1, N 0 γ, x) is a compact 0-dimensional manifold. Every α, u, v) N 0 γ, x) satisfies α > 0. When ind Morse γ) ind CZ x) = 0, N 0 γ, x) is a 1-dimensional manifold with boundary. Its boundary is {α = 0}, and its end is compactified by the following moduli spaces we set k := ind Morse γ) = ind CZ x)): M X mγ, γ ) N 0 γ, x) γ PL m ), ind Morse γ ) = k 1), M X m,x m+1γ, γ ) M X m+1,h m+1,j m+1γ, x) γ PL m+1 ), ind Morse γ ) = k), N 0 γ, x ) M H m+1,j m+1x, x) x PH m+1 ), ind CZ x ) = k + 1). Let us define K 0 : CM <a L m, X m ) CF <a +1H m+1, J m+1 ) by K 0 [γ] := N 0 γ, x) [x]. ind CZ x)=ind Morse γ)+1 Then, the above results show that H m+1,j m+1 K0 + K 0 L m,x m = Ψm+1 Φ L + Θ. Next we show that Φ H Ψ m Θ. Let H C R S 1 T R n ) be a homotopy from H m to H m+1, and J = J s,t ) s,t) R S 1 be a homotopy from J m to J m+1. By HH1) and JJ1), there exists s 0 > 0 such that { Ht m, Jt m ) s s 0 ) H s,t, J s,t ) = Ht m+1, Jt m+1. ) s s 0 ) For any γ PL m ) and x PH m+1 ), N 1 γ, x) denotes the set of β, w), where β, s 0 ], w W 1,3 [β, ) S 1, T R n ) 17

19 which satisfy the following properties: πwβ)) W u γ : X m ), s w J s,t t w X Hs,t w)) = 0, lim ws) = x. s Now we state our fifth C 0 - estimate. It is proved in Section 5. Lemma There exists ε > 0 which satisfies the following property: If J satisfies sup J s,t J std C 0 < ε, w C 0 is uniformly bounded for any s,t β, w) N 1 γ, x), where γ PL m ), x PH m+1 ). Suppose that J is generic and sufficiently close to J std. gluing arguments, the following holds: Then, by Lemma 4.10 and When ind Morse γ) ind CZ x) = 1, N 1 γ, x) is a compact 0-dimensional manifold. Every β, w) N 1 γ, x) satisfies β < s 0. When ind Morse γ) ind CZ x) = 0, N 1 γ, x) is a 1-dimensional manifold with boundary. Its boundary is {β = s 0 }, and its ends are compactified by the following moduli spaces we set k := ind Morse γ) = ind CZ x)): M X mγ, γ ) N 1 γ, x) γ PL m ), ind Morse γ ) = k 1), M X m,h m,j mγ, x ) M H,J x, x) x PH m ), ind CZ x ) = k), N 1 γ, x ) M H m+1,j m+1x, x) x PH m+1 ), ind CZ x ) = k + 1). Let us define K 1 : CM [a,b) L m, X m ) CF [a,b) +1 H m+1, J m+1 ) by K 1 [γ] := N 1 γ, x) [x]. ind CZ x)=ind Morse γ)+1 Then, the above results show that H m+1,j m+1 K1 + K 1 L m,x m = Θ + ΦH Ψ m Proof of Lemma 4.3. Finally, we prove Lemma 4.3. Through this section, V denotes a bounded domain in R n with smooth boundary. First we need the following lemma: Lemma For any open neighborhood W of V and b > 0, the natural homomorphism is an isomorphism. H Λ <b W ), Λ <b W ) \ Λ V )) H Λ <b W ), Λ <b W ) \ ΛV )) Proof. This is equivalent to showing that H Λ <b W ) \ ΛV ), Λ <b W ) \ Λ V )) = 0. Let us take a k-dimensional singular chain α = i c i α i C k Λ <b W ) \ ΛV )) c i Z 2, α i : k Λ <b W ) \ ΛV ) are continuous maps) such that α C k 1 Λ <b W ) \ Λ V )). Since k is compact, there exists b < b such that α i k ) Λ <b W ) for all i. Let us take a compactly supported smooth vector field Z on W, which points outwards on V. Let φ Z t ) t R be the isotopy on W generated by Z, i.e. φ Z 0 = id W, t φ Z t = Zφ Z t ). 18

20 Take δ > 0 and define α t i : k ΛW ) by α t ip) := φ Z δt α i p) p k ). When δ > 0 is sufficiently small, α t i k ) Λ <b W ) for any i and 0 t 1. It is easy to see that α t i satisfies the following properties for any i and 0 t 1: α 0 i = α i. α t := i c i α t i satisfies α t C k Λ <b W ) \ ΛV )) and α t C k 1 Λ <b W ) \ Λ V )) for any 0 t 1. α 1 C k Λ <b W ) \ Λ V )). Then we obtain [α] = [α 0 ] = [α 1 ] = 0 in H k Λ <b W ) \ ΛV ), Λ <b W ) \ Λ V )). Corollary For any open neighborhood W of V, the natural homomorphism H Λ <b W ), Λ <b W ) \ ΛV )) H Λ <b R n ), Λ <b R n ) \ ΛV )) is an isomorphism. Proof. Consider the following commutative diagram: H Λ <b W ), Λ <b W ) \ Λ V )) H Λ <b R n ), Λ <b R n ) \ Λ V )) H Λ <b W ), Λ <b W ) \ ΛV )) H Λ <b R n ), Λ <b R n ) \ ΛV )) Then, vertical arrows are isomorphism by Lemma 4.11, and the top arrow is an isomorphism by excision. Therefore the bottom arrow is an isomorphism. Applying Lemma 4.11 with W = R n, H Λ <b R n ), Λ <b R n ) \ Λ V )) H Λ <b R n ), Λ <b R n ) \ ΛV )) is an isomorphism. Hence, to prove Lemma 4.3 it is enough to show that the natural homomorphism H Λ <b V ), Λ <b V ) \ ΛV )) H Λ <b R n ), Λ <b R n ) \ ΛV )) is an isomorphism. To show this, we need the following trick: take a sequence g l ) l of Riemannian metrics on R n, with the following properties: g-1): For any tangent vector ξ on R n, ξ g l is decreasing in l: ξ g 1 > ξ g 2 >. g-2): For any tangent vector ξ on R n, lim ξ g l = ξ, where is the standard metric. l g-3): For any l 1, there exists an embedding τ l : V ε l, ε l ) R n with the following properties: τ l x, 0) = x for any x V. τ 1 l V ) = V ε l, 0). τl g l is a product metric of g l V and the standard metric on ε l, ε l ). We set W l := V Im τ l. For each l we define Λ <b l R n ) := { γ ΛR n ) } γt) g l dt < b S 1 19, Λ <b l V ) := Λ <b l R n ) Λ V ).

21 By g-1), Λ <b l R n )) l, Λ <b By g-2), Λ <b l R n ) = Λ <b R n ), l l l V )) l are increasing sequences of open sets in Λ <b R n ), Λ <b V ). V ) = Λ <b V ). Thus there holds Λ <b l H Λ <b V ), Λ <b V ) \ ΛV )) = lim H Λ <b l V ), Λ <b l V ) \ ΛV )), l H Λ <b R n ), Λ <b R n ) \ ΛV )) = lim H Λ <b l R n ), Λ <b l R n ) \ ΛV )). l Therefore Lemma 4.3 is reduced to the following lemma: Lemma For any l 1, the natural homomorphism H Λ <b l V ), Λ <b l V ) \ ΛV )) H Λ <b l R n ), Λ <b l R n ) \ ΛV )) is an isomorphism. Proof. Let us take W l V as in g-3). Since Corollary 4.12 is valid also for g l, H Λ <b l W l ), Λ <b l W l ) \ ΛV )) H Λ <b l R n ), Λ <b l R n ) \ ΛV )) is an isomorphism. Hence it is enough to show that I : H Λ <b l V ), Λ <b l V ) \ ΛV )) H Λ <b l W l ), Λ <b l W l ) \ ΛV )) is an isomorphism. We check surjectivity and injectivity. We prove surjectivity of I. Take α = i c i α i C k Λ <b l W l )) such that α C k 1 Λ <b l W l )\ ΛV )). Since k is compact, there exists b < b such that length of α i p) with respect to g l < b i, p k ). Let us take ρ C ε l, ε l )) with the following properties: ρs) 0 on [0, ε l ). 0 ρ s) b/b, ε l < ρs) 0 on ε l, 0). ρs) s near ε l. Then we define a smooth map φ : W l [0, 1] W l ; x, t) φ t x) such that: If x / Im τ l, φ t x) = x. If x = τ l y, s), φ t x) = τ l y, 1 t)s + tρs)). It is easy to check the following properties of φ: φ 0 = id Wl, φ 1 W l ) = V. For any 0 t 1, φ t W l \ V ) W l \ V, φ t V ) = V. For any tangent vector ξ on W l and 0 t 1, dφ t ξ) g l b/b ) ξ g l. We define αi t : k ΛW l ) by αip) t := φ t α i p) p k ). By the last property of φ, αi t k ) Λ <b l W l ). Moreover, α t := c i αi t satisfies the following properties: i α 0 = α. 20

22 α t C k Λ <b l W l )), α t C k 1 Λ <b l W l ) \ ΛV )) for any t [0, 1]. α 1 C k Λ <b l V )), α 1 C k 1 Λ <b l V ) \ ΛV )). Thus we obtain [α] = [α 0 ] = [α 1 ] Im I. Hence we have proved surjectivity of I. We prove injectivity of I. Let α = i c i α i C k Λ <b l V )) such that α C k 1 Λ <b l V )\ ΛV )). We show that if I[α]) = 0 then [α] = 0. By I[α]) = 0, there exists β = j d j β j C k+1 Λ <b l W l )) such that β α C k Λ <b l W l ) \ ΛV )). Since k, k+1 are compact, there exists b < b such that length of α i p), β j q) with respect to g l < b i, j, p k, q k+1 ). Taking φ : W l [0, 1] W l as before, we set α t i := φ t α i, α t := i Then, it is easy to confirm the following claims: c i α t i, β t j := φ t β j, β t := j For any 0 t 1, α t C k Λ <b l V )), α t C k 1 Λ <b l V ) \ ΛV )). β 1 C k+1 Λ <b l V )). β 1 α 1 C k Λ <b l V ) \ ΛV )). d j β t j. Thus we obtain [α] = [α 1 ] = [ β 1 ] = 0 in H k Λ <b l V ), Λ <b l V ) \ ΛV )). Hence we have proved injectivity of I. 5. C 0 -estimates The goal of this section is to prove Lemmas on C 0 -estimates for Floer trajectories: Lemma 2.3, 2.4, 4.8, 4.9, Our arguments in this section are based on techniques in [2] and [7] W 1,2 -estimate. The goal of this subsection is to prove the following W 1,2 - estimate. In the following statement, an expression c 0 H, M) means that c 0 is a constant which depends on H and M. Proposition 5.1. For any H C R S 1 T R n ) satisfying HH2), HH3) and M > 0, there exists a constant c 0 H, M) > 0 which satisfies the following property: Let I R be a closed interval of length 3, and J s,t ) s,t) I S 1 be a I S 1 -family of almost complex structures on T R n, such that every J s,t is compatible with ω n. Suppose that there holds ξ 2 2 ω nξ, J s,t ξ)) 2 ξ 2 for any s I, t S 1 and tangent vector ξ on T R n. Then, for any W 1,3 -map u : I S 1 T R n which satisfies s u J s,t t u X Hs,t u)) = 0, AHs us)) M, 21 sup s I

23 there holds u W 1,2 I S 1 ) c 0. Remark 5.2. H s C S 1 T R n ) is defined as H s t, q, p) := Hs, t, q, p). A crucial step is the following lemma. Lemma 5.3. Let H and I be as in Proposition 5.1. Then, there exists a constant c 1 H) > 0 such that: for any x C S 1, T R n ) and s I, there holds ) x 2 L + tx 2 2 L c t x X Hs,t xt)) 2 + s H s,t xt)) dt. S 1 Proof. Let us take c 2 H) so that c 2 > sup s s,t C 0 recall that s,t was defined in s,t HH3)). Then we show that there exists a constant c 3 H) > 0 such that there holds ) 2) x 2 L c 2 3 c 2 + t x X Hs,t xt)) 2 + s H s,t xt)) dt S 1 for any x C S 1, T R n ) and s I. Suppose that this does not hold. Then, there exists a sequence x k ) k and s k ) k such that 3) x k 2 L 2 c 2 + S 1 t x k X Hsk,tx k t)) 2 + s H sk,tx k t)) dt Since c 2 + s H s,t q, p) > 0 for any s, t, q, p), there also holds k ). t x k X Hsk x k ) L 2 x k L 2 0 k ). Let us set m k := x k L 2, and v k := x k /m k. Then, obviously v k L 2 = 1. We show that v k ) k is W 1,2 -bounded, i.e. t v k ) k is L 2 -bounded. To show this, we set h k t, q, p) := H sk,tm k q, m k p)/m 2 k, and consider the inequality t v k X h kv k ) L 2 t v k L 2 t v k X h kv k ) L 2 + X h kv k ) L 2. is bounded in k, since 4) t v k X h kv k ) L 2 = tx k X Hsk x k ) L 2 m k 0 k ). To bound X h kv k ) L 2, we use the inequality 5) X Q as k )v k ) X h kv k ) L 2 X Q as k )v k ) X h kv k ) C 0 sup t S 1 s k,t C 1 m k. Then, it is easy to see that there exists c 4 H) > 0 such that X h kv k ) L 2 c 4 1+ v k L 2). Thus we have proved that v k ) k is W 1,2 -bounded. By taking a subsequence of v k ) k, we may assume that there exists v W 1,2 S 1, T R n ) such that lim v v k C 0 = 0, and t v k converges to t v weakly in L 2. Moreover, we may assume that s k ) k converges to s I. We show that lim X Q as)v) X h kv k ) L 2 = 0. By the triangle inequality, X Q as)v) X h kv k ) L 2 X Q as)v) X Q as)v k ) L 2 + X Q as)v k ) X h kv k ) L 2. 22

24 Then, lim X Q as)v) X Q as)v k ) L 2 5) shows that lim X Q as)v k ) X h kv k ) L 2 = 0. = 0 since lim v v k L 2 Now we show that t v X Q as)v) = 0 in L 2 S 1, T R n ), i.e. t v X Q as)v), ξ L 2 = 0 for any ξ C S 1, T R n ). This follows from t v X Q as)v), ξ L 2 = lim t v k X h kv k ), ξ L 2 = 0. The first equality holds since in L 2 S 1, T R n ) t v k converges to t v weakly), The second equality follows from 4). = 0. On the other hand, X h kv k ) converges to X Q as)v) in norm). Now we have shown that t v X Q as)v) = 0 in L 2 S 1, T R n ). Therefore, by a boot strapping argument, we conclude that v C S 1, T R n ). This implies that as) πz, hence a s) > 0 by HH3). Hence we obtain m 2 k c 2 + S 1 s H sk,tx k t)) dt m 2 k Q S 1 a s k) x k t)) dt 1 = Q S 1 a s k) v k t)) dt 1 k ). a s) v 2 L 2 However, this contradicts the assumption that x k ) k satisfies 3). Hence we have proved 2). Setting c 5 := max{c 2 c 3, c 3 }, there holds ) 6) x 2 L c t x X Hs,t xt)) 2 + s H s,t xt)) dt S 1 for any x C S 1, T R n ) and s I. Now, it is enough to show that there exists c 6 H) > 0 such that ) 7) t x 2 L c t x X Hs,t xt)) 2 + s H s,t xt)) dt S 1. By using t x L 2 t x X Hs x) L 2 + X Hs x) L 2 t x X Hs x) L 2 + 2as) x L 2 + sup s,t C 1, s,t 7) follows easily from 6). Now we can prove Proposition 5.1. Proof of Proposition 5.1. Suppose that u W 1,3 I S 1, T R n ) satisfies s u J s,t t u X Hs,t u)) = 0, AHs us)) M. By elliptic regularity, u is C on inti S 1. By the assumption on J s,t, it is easy to see that J s,t s u 2 4 s u 2, s u 2 2ω n s u, J s,t s u). 23 sup s I

25 By Lemma 5.3, the following inequality holds for any s inti: ) us) 2 L + tus) 2 2 L c s us, t) 2 + s H s,t us, t)) dt. S 1 The RHS is bounded by 4 s us, t) 2 + s H s,t us, t)) dt S 1 8ω n s u, J s,t s u) + s H s,t us, t)) dt S 1 By similar arguments, it is easy to show that Therefore I I Thus we get 8 s AHs us)) ). S 1 s us, t) 2 dt 2 s AHs us)) ). us) 2 L + tus) 2 2 L ds c s A Hs us))) ds c M), I s us) 2 L ds 2 2 s A Hs us))) ds 4M. I I S 1 us, t) 2 + t us, t) 2 + s us, t) 2 dsdt 3c c 1 + 4)M. This concludes the proof of Proposition Proof of Lemma 2.3, 2.4. First notice that Lemma 2.3 is a special case of Lemma 2.4. Hence it is enough to prove Lemma 2.4. First we need the following lemma: Lemma 5.4. Suppose that H C R S 1 T R n ) is a homotopy from H to H +. Then, there exists M > 0 which depends only on H such that AHs us)) M for any s R and u M H,J x, x + ), where x PH ), x + PH + ). Proof. Since PH ) and PH + ) are finite sets, there exists M > 0 such that A H x), A H +y) [ M, M] x PH ), y PH + ) ). Since A Hs us)) is decreasing on s, A Hs us)) [ M, M] for any u M H,J x, x + ). Now we prove Lemma 2.4. In the course of the proof, constants which we do not need to be specified are denoted as const. Proof of Lemma 2.4. To estimate u C 0, it is enough to bound u [j,j+1] S 1 C 0 each integer j. Take a cut-off function χ so that suppχ 1, 2), χ [0,1] 1, 0 χ 1, 2 χ 2. Setting v j s, t) := χs j)us, t), it is enough to bound v j C 0 in the following, we omit the subscript j). First notice that v C 0 const v W 1,3 const v L 3, 24 for

26 where the first inequality is a Sobolev estimate, and the second one is Poincaré inequality. By the Calderon-Zygmund inequality, there exists c > 0 such that v L 3 c s J std t )v L 3 + v L 3). We claim that ε := 1/2c satisfies the requirement in Lemma 2.4. Suppose that sup J std s,t J s,t C 0 1/2c. Then c s J std t )v L 3 c ) J std J s,t C 0 t v L 3 + s J s,t t )v L 3 Hence we obtain v L 3/2 + c s J s,t t )v L 3. v L 3 2c v L 3 + s J s,t t )v L 3). Since vs, t) = χs j)us, t), it is clear that v L 3 u L 3 [j 1,j+2] S 1 ). On the other hand, since s J s,t t )vs, t) = χ s j)us, t) + χs j)j s,t u)x Hs,t u), and H satisfies HH3), it is easy to see Then we conclude that s J s,t t )v L 3 const1 + u L 3 [j 1,j+2] S 1 )). v L 3 const1 + u L 3 [j 1,j+2] S 1 )) const1 + u W 1,2 [j 1,j+2] S 1 )). Then, Lemma 5.4 and Proposition 5.1 shows that the RHS is bounded Proof of Lemma 4.8, 4.9, These lemmas are consequences of the following proposition: Proposition 5.5. There exists a constant ε > 0 which satisfies the following property: Suppose we are given the following data: H C R S 1 T R n ) which satisfies HH2), HH3). J = J s,t ) s,t) R S 1 which satisfies JJ2) and sup J s,t J std C 0 < ε. s,t) Constants M 0, M 1 > 0. Then, there exists a constant ch, M 0, M 1 ) > 0 such that, for any σ R and u W 1,3 [σ, ) S 1, T R n ) satisfying s u J s,t t u X Hs,t u)) = 0, πuσ)) W 2/3,3 S 1,R n ) M 1, there holds u C 0 ch, M 0, M 1 ). sup s σ AHs us)) M0, In this subsection, we deduce Lemmas 4.8, 4.9, 4.10 from Proposition 5.5. First notice that Lemma 4.8 is a special case of Lemma Hence it is enough to prove Lemma 4.9 and Lemma Proof of Lemma 4.9. Since PL m ) and PH m+1 ) are finite sets, there exists M > 0 such that S L mγ), A H m+1x) [ M, M] 25

27 for any γ PL m ), x PH m+1 ). For any α, u, v) N 0 γ, x), there holds S L mγ) S L mu0)) S L m+1u0)) S L m+1uα)) A H m+1v0)) A H m+1x). In particular, S L m+1uα)) is bounded from below. Now we use the following lemma: Lemma 5.6. For any γ PL m ) and d R, φ Xm+1 [0, ) W u γ : X m )) {S L m+1 precompact in ΛR n ). d} is Proof. This lemma is an immediate consequence of Proposition 2.2, Corollary 2.3 in [1]. Let γ k, t k ) k 1 be a sequence, where γ k W u γ : X m ) and t k 0, such that, with γ k := φ Xm+1 t k γ k ), S L m+1γ k) d. Since S L mγ k ) S L m+1γ k) d, Corollary 2.3 in [1] shows that γ k ) k has a convergent subsequence. Then, Proposition 2.2 2) in [1] implies the conclusion. Since S L m+1uα)) is bounded from below for any α, u, v) N 0 γ, x), Lemma 5.6 shows that uα) W 1,2 is bounded for any α, u, v). Therefore, πv0)) W 2/3,3 const πv0)) W 1,2 = const uα) W 1,2 is bounded from above the first inequality is a Sobolev estimate). On the other hand AH m+1vs)) M. Hence Proposition 5.5 shows that v C 0 is bounded. sup s 0 Proof of Lemma Suppose that β, w) N 1 γ, x). Then, there holds S L mγ) S L mπwβ))) A H mwβ)) A Hβ wβ)) A H m+1x). Then, sup AHs ws)) is bounded. Moreover, since SL mπwβ))) is bounded from below, s β πwβ)) W 1,2 is bounded. Hence w C 0 is bounded Proof of Proposition 5.5. Finally we prove Proposition 5.5. It is enough to bound sup us, t) for each integer j 0. The proof for s σ,t) [j,j+1] S 1 j 1 is as the proof of Lemma 2.4. Hence we only consider the case j = 0. We denote the q-component and p-component of u by u q, u p, i.e. us, t) = u q s, t), u p s, t) ). By the theory of Sobolev traces, there exists ũ q s, t) W 1,3 [σ, ) S 1 : R n ) such that ũ q σ, t) = u q σ, t) for any t S 1, and there holds ũ q W 1,3 [σ, ) S 1 :R n ) const u q σ) W 2/3,3 S 1 :R n ). Take a cut-off function χ C [0, )) such that suppχ [0, 2), χ [0,1] 1, 0 χ 1, 2 χ 0. We set ws, t) := χs σ)u q s, t) ũ q s, t), u p s, t)). Since u C 0 [σ,σ+1] S 1 ) w C 0 + ũ q C 0 [σ,σ+1] S 1 ) w C 0 + const ũ q W 1,3, it is enough to bound w C 0. It is easy to see that w C 0 const w W 1,3 const w L 3 26