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compact subsets of Σ Ω Υ , for which there exists s0 > 0 such that u(s, t − 1) = exp x1(t) ζ1(s, t), ∀(s, t) ∈] − ∞, −s0[×[0, 1], u(s, t) = exp x2(t) ζ2(s, t), ∀(s, t) ∈] − ∞, −s0[×[0, 1], u(s, 2t − 1) = exp y(t) ζ(s, t), ∀(s, t) ∈]s0, +∞[×[0, 1], for suitable W 1,p sections ζ1, ζ2, ζ of the vector bundles x ∗ 1 (TT∗ M) →] − ∞, −s0[×[0, 1], x ∗ 2 (TT∗ M) →] − ∞, −s0[×[0, 1], y ∗ (TT ∗ M) →]s0, +∞[×[0, 1]. Here “exp” is the exponential map given by some metric on T ∗M, but the space W Ω Υ does not depend on the choice of this metric. Notice also that when we say “of Sobolev class W 1,p on compact subsets of ΣΩ Υ ”, we consider ΣΩΥ endowed with its smooth structure (and not with the structure endowed by the singular coordinate z = s + it). Since p > 2, the space W Ω Υ is an infinite dimensional manifold modeled on the real Banach space W 1,p iÊn(Σ Ω Υ ,�n ) = W 1,p 0 (Σ Ω Υ ,Ên ) ⊕ W 1,p (Σ Ω Υ , iÊn ). Notice that by our definition of the smooth structure of ΣΩ 1,p Υ , a Banach norm of WiÊn(ΣΩ Υ ,�n ) is �v� p 1 := � (|v(z)| |Im z|1 p + |Dv(z)| p � � p |v(z)| )ds dt + |z|
where A is a smooth map taking value into L(Ê2n ,Ê2n ). Since u(s, t − 1) converges to x1(t) for s → −∞, u(s, t) converges to x2(t) for s → −∞, and u(s, 2t − 1) converges to y(t) for s → +∞, for any t ∈ [0, 1], the L(Ê2n ,Ê2n )-valued function A has the following asymptotics: A(s + (t − 1)i) → A − 1 (t), A(s + ti) → A− 2 (t) for s → −∞, A(s + (2t − 1)i) → A+ (t), for s → +∞, for any t ∈ [0, 1], where A − 1 (t), A− 2 (t), and A+ (t) are symmetric matrices such that the solutions of the linear Hamiltonian systems d dt Ψ− 1 (t) = iA− 1 (t)Ψ− 1 (t), d dt Ψ− 2 (t) = iA− 2 (t)Ψ− 2 (t), d dt Ψ+ (t) = 2iA + (t)Ψ + (t), Ψ − 1 (0) = Ψ− 2 (0) = Ψ+ (0) = I, are conjugated to the differential of the Hamiltonian flows along x1, x2, and y: Ψ − 1 (t) ∼ Dxφ H1 (1, x1(0)), Ψ − 2 (t) ∼ Dxφ H2 (1, x2(0)), Ψ + (t) ∼ Dxφ H1#H2 (1, y(0)). Then, by the definition of the Maslov index µ Ω in terms of the relative Maslov index µ, we have µ Ω (x1) = µ(Ψ − 1 iÊn , iÊn ) − n 2 , µΩ (x2) = µ(Ψ − 2 iÊn , iÊn ) − n 2 , µΩ (y) = µ(Ψ + iÊn , iÊn ) − n 2 . We claim that the linear operator is Fredholm of index D + G : W 1,p iÊn(Σ Ω Υ,�n ) → Ω 0,1 L p(ΣΩ Υ,�n ). ind (D + G) = µ Ω (x1) + µ Ω (x2) − µ Ω (y). (128) In order to deduce this claim from Theorem 5.9, we show that the operator D + G is conjugated to a linear perturbed Cauchy-Riemann operator on a strip with jumping Lagrangian boundary conditions, in the sense of section 5.3. Indeed, given v : Σ Ω Υ →�n let us consider the�2n -valued map ˜v on Σ = {0 ≤ Im z ≤ 1} defined as The map v ↦→ ˜v gives us an isomorphism where ˜v(z) := (v(z), v(z)). W 1,p iÊn(Σ Ω Υ,�n ) ∼ = X 1,p S ,V ,V ′(Σ,�2n ), S = {0}, V = ((0), ∆Ên), V ′ = (0), ∆Ên being the diagonal subspace ofÊn ×Ên , and (0) being the zero subspace ofÊ2n . This follows from comparing the norm (126) to the X 1,p S the other hand, by comparing the norm (127) to the X p S w ↦→ 2 � w[∂s] gives us an isomorphism norm by means of (87), (88), (90) and (91). On norm by (89), we see that the map Ω 0,1 L p(ΣΩ Υ ,�n ) ∼ = X p S (Σ,�2n ). It is easily seen that composing the operator D + G by these two isomorphisms produces the operator ∂ Ã : X1,p S ,V ,V ′(Σ,�2n ) → X p S (Σ,�2n ), u ↦→ ∂u + Ãu, 79
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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
Page 30 and 31: The Riemann surface Σ Ω Υ s = 0
Page 32 and 33: In the periodic case, we consider H
Page 34 and 35: 3.6. Definition. The Maslov index
Page 36 and 37: The Riemann surface ΣG is obtained
Page 38 and 39: 3.12. Proposition. For a generic ch
Page 40 and 41: The following compactness statement
Page 42 and 43: 4.2. Proposition. For a generic cho
Page 44 and 45: A standard gluing argument shows th
Page 46 and 47: 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 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 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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