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Then ind∂A = ind∂A1 + ind∂A1. The proof is analogous to the proof of Theorem 3.2.12 in [Sch95], and we omit it. When there are no jumps, that is S = ∅ and V = (V ), V ′ = (V ′ ), Theorem 5.11 shows that the index of the operator is ∂A : X 1,p ∅,(V ),(V ′ ) (Σ,�n ) = W 1,p N ∗ V,N ∗ V ′(Σ,�n ) → L p (Σ,�n ) = X p ∅ (Σ,�n ) In the general case, Proposition 5.19 shows that ind∂A = µ(Φ − N ∗ V, N ∗ V ′ ) − µ(Φ + N ∗ V, N ∗ V ′ ). = µ(Φ − N ∗ V0, N ∗ V ′ 0 ) − µ(Φ+ N ∗ Vk, N ∗ V ′ where the correction term c satisfies the additivity formula ind (∂A : X 1,p S ,V ,V ′(Σ,�n ) → X p S (Σ,�n )) k ′) + c(V0, . . .,Vk; V ′ 0 , . . .,V ′ k ′), c(V0, . . . , Vk+h; V ′ 0, . . .,V ′ k ′ +h ′) = c(V0, . . . , Vk; V ′ 0, . . . , V ′ k ′) + c(Vk, . . .,Vk+h; V ′ ′ k ′, . . .,V k ′ +h ′). (112) (113) Since the Maslov index is in general a half-integer, and since we have not proved that the cokernel of ∂A is finite dimensional, the correction term c takes values in (1/2)�∪{−∞}. Actually, the analysis of this section shows that c is always finite, proving that ∂A is Fredholm. Clearly, we have the following direct sum formula c(V0 ⊕ W0, . . . , Vk ⊕ Wk; V ′ 0 ⊕ W ′ ′ 0 , . . .,V k ′ ⊕ W ′ k ′) = c(V0, . . . , Vk; V ′ 0, . . .,V ′ k ′) + c(W0, . . .,Wk; W ′ 0, . . . , W ′ k ′). Note also that the index formula of Theorem 5.11 produces a correction term of the form (114) c(λ; ν) = µ(λ, ν), (115) where λ and ν are asymptotically constant paths of Lagrangian subspaces on�n . The Liouville type results of the previous section imply that Indeed, by Proposition 5.17 the operator c((0),Ên ;Ên ) = − n 2 = c(Ên , (0);Ên ). (116) ∂αI : X 1,p {0},((0),Ên ),(Ên ) (Σ,�n ) → X p {0} (Σ,�n ) is an isomorphism if 0 < α < π/2. By (80), the Maslov index of the path e iαtÊn , t ∈ [0, 1], with respect toÊn is −n/2. On the other hand, the Maslov index of the path e iαt iÊn , t ∈ [0, 1], with respect toÊn is 0 because the intersection is (0) for every t ∈ [0, 1]. Inserting the information about the Fredholm and the Maslov index in (112), we find 0 = ind∂αI = n 2 + c((0),Ên ; (0)), which implies the first identity in (116). The second one is proved in the same way by using Proposition 5.18. 68
5.20. Lemma. Let (V0, V1, . . . , Vk) be a (k + 1)-uple of linear subspaces ofÊn , with Vj−1 and Vj partially orthogonal for every j = 1, . . .,k, and let W be a linear subspace ofÊn . Then c(V0, . . . , Vk; W) = c(W; V0, . . . , Vk) = − 1 2 k� (dimVj−1 + dimVj − 2 dimVj−1 ∩ Vj). Proof. Let us start by considering the case W =Ên . By the additivity formula (113), c(V0, . . . , Vk;Ên ) = j=1 k� j=1 c(Vj−1, Vj;Ên ). j Since Vj−1 and Vj are partially orthogonal,Ên has an orthogonal splittingÊn = X1 ⊕Xj 2 ⊕Xj 3 ⊕Xj 4 where Vj−1 = X j 1 ⊕Xj 2 and Vj = X j . By the direct sum formula (114) and by formula (116), Since c(Vj−1, Vj;Ên ) = c(X j 1 , Xj 1 ; Xj 1 1 ⊕Xj 3 ) + c(Xj 2 , (0); Xj 2 ) + c((0), Xj 3 ; Xj 3 ) + c((0), (0); Xj 4 ) = 0 − 1 2 dimXj 1 2 − 2 dimXj 3 + 0 = −1 2 dimXj 2 ⊕ Xj 3 dimX j 2 ⊕ Xj 3 = dimVj−1 + dimVj − 2 dimVj−1 ∩ Vj, the formula for c(V0, . . .,Vk;Ên ) follows. Now let λ :Ê→L (n) be a continuous path of Lagrangian subspaces such that λ(s) =Ên for s ≤ −1 and λ(s) = N ∗ W for s ≥ 1. By an easy generalization of the additivity formula (113) to the case of non-constant Lagrangian boundary conditions, c(N ∗ V0; λ) + c(V0, . . .,Vk; W) = c(V0, . . . , Vk;Ên ) + c(N ∗ Vk; λ). (117) By (115), c(N ∗ V0; λ) = −µ(λ, N ∗ V0) and c(N ∗ Vk; λ) = −µ(λ, N ∗ Vk), so (117) leads to c(V0, . . .,V k ; W) = c(V0, . . . , V k ;Ên ) − (µ(λ, N ∗ Vk) − µ(λ, N ∗ V0)) = c(V0, . . . , V k ;Ên ) − h(N ∗ V0, N ∗ Vk;Ên , N ∗ W), where h is the Hörmander index. By Lemma 5.2, the above Hörmander index vanishes, so we get the desired formula for c(V0, . . .,Vk; W). The formula for c(W; V0, . . . , Vk) follows by considering the change of variable v(s, t) = u(s, 1 − t). The additivity formula (113) leads to c(V0, . . . , Vk; V ′ 0, . . . , V ′ k ′) = c(V0, . . . , Vk; V ′ 0) + c(Vk; V ′ 0, . . . , V ′ k ′), and the index formula in the general case follows from (112) and the above lemma. This concludes the proof of Theorem 5.9. 5.7 Half-strips with jumping conormal boundary conditions This section is devoted to the analogue of Theorem 5.9 on the half-strips Σ + := {z ∈�|0≤Im z ≤ 1, Re z ≥ 0} and Σ − := {z ∈�|0≤Im z ≤ 1, Rez ≤ 0}. In the first case, we fix the following data. Let k, k ′ ≥ 0 be integers, let 0 = s0 < s1 < · · · < sk < sk+1 = +∞, 0 = s ′ 0 < s ′ 1 < · · · < s ′ k ′ < s′ k ′ +1 = +∞, be real numbers, and let W, V0, . . . , Vk, V ′ 0, . . .,V ′ k ′ be linear subspaces ofÊn such that Vj−1 and Vj, V ′ ′ j−1 and V j , W and V0, W and V ′ 0 , are partially orthogonal. We denote by V the (k +1)-uple 69
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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
Page 20 and 21: (L0) Ω every solution γ ∈ P
Page 22 and 23: transverse to the stable manifold o
Page 24 and 25: 2.7. Proposition. The homomorphism
Page 26 and 27: A submanifold L ⊂ T ∗ M is call
Page 28 and 29: 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 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 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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