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to ũ0 uniformly on Σ, we may localize the problem and assume that M =Ên . More precisely, if the projection of ũ0(z) onto M 4 is (q1, q2, q3, q4)(z), we construct open embeddings Σ ×Ên → Σ × M, (z, q) ↦→ (z, ϕj(z, q)), j = 1, . . . , 4, such that ϕj(z, 0) = qj(z) and D2ϕj(z, 0) is an isometry, for every z ∈ Σ (for instance, by composing an isometric trivialization of q∗ j (TM) by the exponential mapping). The induced open embeddings Σ × T ∗Ên → Σ × T ∗ M, (z, q, p) ↦→ (z, ψj(z, q, p)) := � z, ϕj(z, q), (D2ϕj(z, q) ∗ ) −1 p � , j = 1, . . .,4, are the components of the open embedding Σ × T ∗Ê4n → Σ × T ∗ M 4 , (z, ξ) ↦→ (z, ψ(z, ξ)) := � z, ψ1(z, ξ1), . . . , ψ4(z, ξ4) � . Such an embedding allow us to associate to any ũ : Σ → T ∗ M 4 which is C 0 -close to ũ0 a map w : Σ → T ∗Ê4n =�4n , by setting ũ(z) = ψ(z, w(z)). Then ũ solves (167) if and only if w solves an equation of the form D(w) := ∂sw(z) + J(z, w(z))∂tw(z) + G(z, w(z)) = 0, (172) where J is an almost complex structure on�4n parametrized on Σ and such that J(z, 0) = J0 for any z ∈ Σ, whereas G : Σ ×�4n →�4n is such that G(z, 0) = 0 for any z ∈ Σ. Moreover, ũ solves the asymptotic conditions (171) if and only if w(s, t) tends to 0 for s → ±∞. The maps ψj(z, ·) preserve the Liouville form, so they map conormals into conormals. It easily follows that the boundary condition (168) on ũ is translated into w(s, 1) ∈ N ∗ ∆Ê2n ∀s ∈Ê. (173) Moreover, ũ satisfies the boundary conditon (169) if and only if w satisfies ⎧ ⎨ N w(s, 0) ∈ ⎩ ∗∆Ê2n if s ≤ 0, N ∗∆ΘÊn if 0 ≤ s ≤ 2α, N ∗ (∆Ên × ∆Ên) if s ≥ 2α. Similarly, ũ satisfies the boundary condition (170) if and only if w satisfies � ∗ N ∆Ê2n if s ≤ 0, w(s, 0) ∈ N ∗ (∆Ên × ∆Ên) if s ≥ 0. (174) (175) The element u0 ∈ M Λ Υ (x1, x2; z) corresponds to the solution w0 = 0 of (172)-(175). By using the functional setting introduced in section 5, we can view the nonlinear operator D defined in (172) as a continuously differentiable operator D : X 1,p S ,V ,V ′(Σ,�4n ) → X p S (Σ,�4n ) where S := {0}, V := (∆Ê2n, ∆Ên × ∆Ên), V ′ := (∆Ê2n), and p is some number larger than 2. Since J(z, 0) = J0, the differential of D at w0 = 0 is a linear operator of the kind studied in section 5, and by the transversality assumption it is an isomorphism. Consider the orthogonal decomposition where Ê4n = W1 ⊕ W2 ⊕ W3 ⊕ W4, W1 := ∆ ΘÊn = ∆Ê2n ∩ (∆Ên × ∆Ên), ∆Ê2n = W1 ⊕ W2, ∆Ên × ∆Ên = W1 ⊕ W3, 96
and denote by Pj the the orthogonal projection of�4n onto N ∗ Wj = Wj ⊕ iW ⊥ j . If Tα is the translation operator mapping some w : Σ →�4n into (Tαw)(s, t) := � P1w(s, t), P2w(s, t), P3w(s − 2α, t), P4w(s, t) � , we easily see that w satisfies the boundary condions (175) if and only if Tαw satisfies the boundary conditions (174). Therefore, if we define the operator Dα : X 1,p S ,V ,V ′(Σ,�4n ) → X p S (Σ,�4n ), Dα = D ◦ Tα, we have that w ∈ X 1,p S ,V ,V ′(Σ,�4n ) solves Dα(w) = 0 if and only if Tαw is a solution of (172) satisfying the boundary conditions (173) and (174). The operator [0, +∞[×X 1,p S,V ,V ′(Σ,�4n ) → X p S (Σ,�4n ), (α, w) ↦→ Dα(w), is continuous on the product, it is continuously differentiable with respect to the second variable, and this partial differential is continuous on the product. Moreover, DD0(0) = DD(0) is an isomorphism, so the parametric inverse mapping theorem implies that there are a number α0 > 0 and a neighborhood U of 0 in X 1,p S ,V ,V ′(Σ,�4n ), such that the set of zeroes in [0, α0[×U of the above operator consists of a continuous curve [0, α0[∋ α → (α, wα) starting at w0 = 0. Then α → (α, Tαwα) provides us with the unique curve in M Υ GE (x1, x2; z) converging to (0, u0). This concludes the proof of Proposition 3.13. 6.4 Proof of Proposition 4.4 Fix some γ1 ∈ P Ω (L1), γ2 ∈ P Ω (L2), and x ∈ P Ω (H1#H2) such that m Ω (γ1; L1) + m Ω (γ2; L2) − µ Ω (x; H1#H2) = 0. By a standard argument in Floer homology, the claim that P K Υ is a chain homotopy between KΩ and ΥΩ ◦ (ΦΩ L1 ⊗ ΦΩ ) is implied by the following statements: L2 (i) For every (u1, u2) ∈ M Ω Φ (γ1, y1) ×M Ω Φ (γ2, y2) and every u ∈ M Ω Υ (y1, y2; x), where (y1, y2) ∈ P Ω (H1) × P Ω (H2) is such that µ Ω (y1; H1) + µ Ω (y2; H2) = µ Ω (x; H1#H2), there exists a unique connected component of M K Υ (γ1, γ2; x) containing a curve (α, uα) such that α → +∞, uα(·, · −1) and uα converge to u1 and u2 in C ∞ loc ([0, +∞[×[0, 1], T ∗ M), while uα(·+σ(α), 2 ·−1) converges to u in C ∞ loc (Ê×[0, 1], T ∗ M), for a suitable function σ diverging at +∞. (ii) For every u ∈ M Ω K (γ1, γ2; x) there exists a unique connected component of M K Υ (γ1, γ2; x) containing a curve α ↦→ (α, uα) which converges to (0, u). Statement (i) can be proved by the standard gluing arguments in Floer theory. Here we prove statement (ii). Given u : [0, +∞[×[−1, 1] → T ∗ M, we define ũ : Σ + → T ∗ M 2 by ũ(s, t) := (−u(s, −t), u(s, t)). If we define ˜x : [0, 1] → T ∗ M 2 and ˜ H ∈ C ∞ ([0, 1] × T ∗ M 2 ) by ˜x(t) := � −x((1 − t)/2), x((1 + t)/2) � , ˜ H(t, x1, x2) := H1(1 − t, x1) + H2(t, x2), 97
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Å�ÜÈÐ�Ò�ÁÒ×Ø�ØÙ
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5.8 Non-local boundary conditions .
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M into Λ(M) as the space of consta
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and converging to Hamiltonian orbit
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1.2 The Chas-Sullivan loop product
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1.1. Remark. The loop product was d
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2.2 Functoriality Let (M1, g1) and
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The orbits of −grad(f1 ⊕f2) are
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this intersection is a finite set o
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(L0) Ω every solution γ ∈ P
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transverse to the stable manifold o
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2.7. Proposition. The homomorphism
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A submanifold L ⊂ T ∗ M is call
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The L 2 -negative gradient equation
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The Riemann surface Σ Ω Υ s = 0
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In the periodic case, we consider H
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3.6. Definition. The Maslov index
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The Riemann surface ΣG is obtained
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3.12. Proposition. For a generic ch
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The following compactness statement
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4.2. Proposition. For a generic cho
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A standard gluing argument shows th
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In the following two sections we wi
Page 48 and 49: We shall apply the above lemma to t
Page 50 and 51: Since the inclusion c : M ֒→ Λ
Page 52 and 53: 4.5 The right-hand square is homoto
Page 54 and 55: commute. Our first aim in this sect
Page 56 and 57: Moreover, �Dαn(βnvn)�0,p;Ê
Page 58 and 59: We identify the product T ∗Ên ×
Page 60 and 61: Up to multiplying u by U, we can re
Page 62 and 63: The symbol C∞ S ,c indicates boun
Page 64 and 65: 5.11. Theorem. (Cauchy-Riemann oper
Page 66 and 67: 5.15. Lemma. Let p > 1 and α ∈Ê
Page 68 and 69: Proof. By Proposition 5.14 the oper
Page 70 and 71: Then ind∂A = ind∂A1 + ind∂A1.
Page 72 and 73: (V0, . . . , Vk), by V ′ the (k
Page 74 and 75: is bounded and Fredholm of index in
Page 76 and 77: so the transversality hypotheses of
Page 78 and 79: 5.9 Coherent orientations As notice
Page 80 and 81: compact subsets of Σ Ω Υ , for
Page 82 and 83: where Ã(z) = CA(z)C ⊕ A(z), C de
Page 84 and 85: By the additivity property of the M
Page 86 and 87: The space M Υ GE . Let x1 ∈ P(H1
Page 88 and 89: The space M Θ Φ . Let γ ∈ PΘ
Page 90 and 91: The embedding Q ֒→ÊN induces an
Page 92 and 93: Given periodic orbits x1 ∈ P(H1),
Page 94 and 95: Since u and ∂u are integrable ove
Page 96 and 97: with non-local boundary conditions
Page 100 and 101: we see that associating ũ to u pro
Page 102 and 103: The energy of a solution u ∈ M K
Page 104 and 105: Since we have applied a conformal r
Page 106 and 107: 6.9. Lemma. There exists a positive
Page 108 and 109: [ChS04] M. Chas and D. Sullivan, Cl
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