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we see that associating ũ to u produces a one-to-one correspondence between M Ω K (γ1, γ2; x) and the space � M Ω K (γ1, γ2; x) consisting of the maps ũ : [0, ∞[×[0, 1] → T ∗ M 2 solving ∂ J, ˜ H (ũ) = 0 with boundary conditions ũ(s, 0) ∈ N ∗ ∆M ∀s ≥ 0, (176) ũ(s, 1) ∈ T ∗ q0M × T ∗ q0M ∀s ≥ 0, (177) π ◦ ũ1(0, 1 − ·) ∈ W u (γ1), π ◦ ũ2(0, ·) ∈ W u (γ2), (178) lim ũ(s, ·) = ˜x. (179) s→+∞ Similarly, we have a one-to-one correspondence between M K Υ (γ1, γ2; x) and the space � M K Υ (γ1, γ2; x) consisting of pairs (α, ũ) where α is a positive number and ũ : Σ + → T ∗ M 2 is a solution of the problem above, with (176) replaced by ũ(s, 0) ∈ T ∗ q0 M × T ∗ q0 M ∀s ∈ [0, α], ũ(s, 0) ∈ N ∗ ∆M ∀s ≥ α. (180) Fix some ũ 0 ∈ � M Ω K (γ1, γ2; x). Since we are looking for solutions near ũ 0 , we can localize the problem as follows. Let k = m Ω (γ1; L1) + m Ω (γ2; L2). Let q :Êk × Σ + → M 2 be a map such that q(0, s, t) = π ◦ ũ 0 (s, t) ∀(s, t) ∈ Σ + , q(λ, s, ·) → π ◦ ˜x for s → +∞, ∀λ ∈Êk , and such that the mapÊk ∋ λ ↦→ (q1(λ, 0, 1 − ·), q2(λ, 0, ·)) ∈ Ω 1 (M 2 ) is a diffeomorphism onto a neighborhood of π ◦ (˜x1(−·), ˜x2) in W u (γ1) × W u (γ2). By means of a suitable trivialization of q ∗ (TM 2 ) and using the usual W 1,p Sobolev setting with p > 2, we can transform the problem of finding mas ũ solving ∂ J, ˜ H (ũ) = 0 together with (177), (178) and (179) and being close to ũ 0 , into the problem of finding pairs (λ, u) ∈Êk ×W 1,p (Σ + , T ∗Ê2n ), solving an equation of the form with boundary conditions ∂u(z) + f(λ, z, u(z)) = 0 ∀z ∈ Σ + , (181) u(0, t) ∈ N ∗ (0) ∀t ∈ [0, 1], u(s, 1) ∈ N ∗ (0) ∀s ≥ 0. (182) Then the boundary conditon (176) is translated into u(s, 0) ∈ N ∗ ∆Ên ∀s ≥ 0, (183) and the solution ũ 0 corresponds to the solution λ = 0 and u ≡ 0 of (181), (182), and (183). On the other hand, the problem � M K Υ (γ1, γ2; x) of finding (α, ũ α ) solving ∂ J, ˜ H (ũ α ) = 0 together with (177), (178), (179) and (180) corresponds to the problem of finding (λ, u) ∈Êk ×W 1,p (Σ + , T ∗Ê2n ) solving (181) with boundary conditions (182) and u(s, 0) ∈ N ∗ (0) ∀s ∈ [0, α], u(s, 0) ∈ N ∗ ∆Ên ∀s ≥ α. (184) In order to find a common functional setting, it is convenient to turn the boundary condition (184) into (183) by means of a suitable conformal change of variables on the half-strip Σ + . The holomorphic function z ↦→ cosz maps the half strip {0 < Re z < π, Im z > 0} biholomorphically onto the upper half-planeÀ={Im z > 0}. It is also a homeomorphism between the closure of these domains. We denote by arccos the determination of the arc-cosine which is the inverse of this function. Then the function z ↦→ (1 + cos(iπz))/2 is a biholomorphism from the interior of Σ + toÀ, mapping 0 into 1 and i into 0. Let ǫ > 0. If we conjugate the linear automorphism z ↦→ (1 + ǫ)z ofÀby the latter biholomorphism, we obtain the following map: ϕǫ(z) = i π arccos� (1 + ǫ)cos(iπz) + ǫ � . 98
The map ϕǫ is a homeomorphism of Σ + onto itself, it is biholomophic in the interior, it preserves the upper part of the boundary i +Ê+ , while it slides the left part i[0, 1] and the lower partÊ+ by moving the corner point 0 into the real positive number α(ǫ) := − i arccos(1 + 2ǫ). π The function ǫ ↦→ α(ǫ) is invertible, and we denote by α ↦→ ǫ(α) its inverse. Moreover, ϕǫ converges to the identity uniformly on compact subsets of Σ + for ǫ → 0. An explicit computation shows that If u : Σ + → T ∗Ê2n and α > 0, we define ϕ ′ ǫ − 1 → 0 in Lp (Σ + ), if 1 < p < 4. (185) v(z) := u(ϕ ǫ(α)(z)). Since ϕǫ is holomorphic, ∂(u ◦ ϕǫ) = ϕ ′ ǫ · ∂u ◦ ϕǫ. Therefore, u solves the equation (181) if and only if v solves the equation Given 2 < p < 4, we set ∂v(z) + ϕ ′ ǫ(α) (z)f(λ, ϕ ǫ(α)(z), v(z)) = 0. (186) W 1,p ∗ (Σ + , T ∗Ê2n ) = � v ∈ W 1,p (Σ + , T ∗Ê2n ) | v(s, 0) ∈ N ∗ ∆Ên ∀s ≥ 0, and we consider the operator v(s, 1) ∈ N ∗ (0) ∀s ≥ 0, v(0, t) ∈ N ∗ (0) ∀t ∈ [0, 1] � , F : [0, +∞[×Êk × W 1,p ∗ (Σ + , T ∗Ê2n ) → L p (Σ + , T ∗Ê2n ), F(α, λ, v) = ∂v + ϕ ′ ǫ(α) f(λ, ϕ ǫ(α)(·), v), where ϕ0 = id. The problem of finding (α, ũ) in � M K Υ (γ1, γ2; x) with ũ close to ũ 0 is equivalent to finding zeroes of the operator F of the form (α, λ, v) with α > 0. By (185), the operator F is continuous, and its differential D (λ,v)F with respect to the variables (λ, v) is continuous. The transversality assumption that ũ 0 is a non-degenerate solution of problem � M Ω K (γ1, γ2; x) is translated into the fact that D (λ,v)F(0, 0, 0) is an isomorphism. Then the parametric inverse maping theorem implies that there is a unique curve α ↦→ (λ(α), v(α)), 0 < α < α0, converging to (0, 0) for α → 0, and such that (λ(α), v(α)) is the unique zero of F(α, ·, ·) in a neighborhood of (0, 0). This concludes the proof of statement (ii). 6.5 Proof of Proposition 4.7 The setting. We recall the setting of section 4.4. Let γ1 ∈ P(L1), γ2 ∈ P(L2), and x ∈ P Θ (H1 ⊕H2). If α ≥ 0, M K α (γ1, γ2; x) is the space of solutions u : [0, +∞[×[0, 1] → T ∗ M 2 of the equation satisfying the boundary conditions ∂H1⊕H2,J(u) = 0, (187) π ◦ u(0, ·) ∈ W u (γ1; −grad g1ËΛ L1 ) × W u (γ2; −grad g2ËΛ L2 ), (188) (u(s, 0), −u(s, 1)) ∈ N ∗ ∆M×M if 0 ≤ s < α, (189) (u(s, 0), −u(s, 1)) ∈ N ∗ ∆ Θ M if s ≥ α, (190) 99 lim u(s, ·) = x. (191) s→+∞
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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
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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 98 and 99: to ũ0 uniformly on Σ, we may loca
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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