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with non-local boundary conditions ⎧ ⎨ N (u(s, 0), −u(s, 1)) ∈ ⎩ ∗∆M2 if s ≤ 0, N ∗∆Θ M if 0 ≤ s ≤ α, N ∗ (∆M × ∆M) if s ≥ α, and asymptotics (164) lim s→−∞ u(s, t) = � −x1(−t), x2(t) � , lim s→+∞ u(s, t) = � −z((1 − t)/2), z((1 + t)/2) � . (165) Similarly, we can view the space M Λ Υ (x1, x2; z) as the space of maps u : Σ → T ∗M 2 solving the equation (163) with asymptotics (165) and non-local boundary conditions (u(s, 0), −u(s, 1)) ∈ � N ∗ ∆M 2 if s ≤ 0, N ∗ (∆M × ∆M) if s ≥ 0. (166) Compactness. We start with the following compactness results, which also clarifies the sense of the convergence in (ii): 6.6. Lemma. Let (αh, uh) be a sequence in M Υ GE (x1, x2; z) with αh → 0. Then there exists u0 ∈ M Λ Υ (x1, x2; z) such that up to a subsequence uh converges to u0 in C∞ loc (Σ \ {0, i}), in C∞ (Σ ∩ {|Re z| > 1}), and uniformly on Σ. Proof. Since the sequence of maps (uh) has uniformly bounded energy, Proposition 6.2 implies a uniform L ∞ bound. Then, the usual non-bubbling-off analysis for interior points and boundary points away from the jumps in the boundary condition implies that, modulo subsequence, we have uh → u0 in C ∞ loc (Σ \ {0, i}, T ∗ M 2 ), where u0 is a smooth solution of equation (163) on Σ \ {0, i} with bounded energy and satisfying the boundary condions (166), except possibly at 0 and i. By Proposition 6.5, the singularities 0 and i are removable, and u0 satisfies the boundary condition also at 0 and i. By the index formula (162) and transversality, the sequence uh cannot split, so u0 satisfies also the asymptotic conditions (165), and uh → u0 in C ∞ (Σ ∩ {|Rez| > 1}). Therefore, u0 belongs to M Λ Υ (x1, x2; z), and there remain to prove that uh → u uniformly on Σ. We assume by contraposition that (uh) does not converge uniformly on Σ. By Ascoli-Arzelà theorem, there must be some blow-up of the gradient. That is, modulo subsequence, we can find zh ∈ Σ converging either to 0 or to i such that Rh := |∇uh(zh)| = �∇uh�∞ → ∞. For sake of simplicity, we only consider the case where zh = (sh, 0) → 0, 0 < sh < αh → 0. The general case follows along analogous arguments using additional standard bubbling-off arguments. For more details, see e.g. [HZ94, section 6.4]. We now have to make a case distinction concerning the behaviour of the quantity 0 < Rh ·αh < ∞: (a) The case of a diverging subsequence Rhj · αhj → ∞ can be handled by conformal rescaling vj(s, t) := uhj(shj +s/Rhj, t/Rhj) which provides us with a finite energy disk with boundary on a single Lagrangian submanifold of conormal type. This has to be constant due to the vanishing of the Liouville 1-form on conormals, contradicting the convergence of |∇vj(0)| = 1. (b) The case of convergence of a subsequence Rhj · αhj → 0 can be dealt with by rescaling vj(s, t) := uhj(shj + αhj, αhjt). Now vk has to converge uniformly on compact subsets towards a constant map, since �∇vj�∞ = |∇vj(0)| = Rhj · αhj → 0. This in particular implies that uhj(·, 0) |[0,αh j ] converges uniformly to a point contradicting the contraposition assumption. 94
(c) There remains to study the case Rh · αh → c > 0. Again we rescale vh(s, t) = uh(αhs, αht), which now has to converge to a non-constant J-holomorphic map v on the upper half plane. After applying a suitable conformal coordinate change and transforming the non-local boundary conditions into local ones, we can view v as a map on the half strip v: Σ + → T ∗ M 4 , satisfying the boundary conditions v(0, t) ∈ N ∗ ∆ Θ M for t ∈ [0, 1], v(s, 0) ∈ N ∗ (∆M × ∆M) for s ≥ 0, v(s, 1) ∈ N ∗ ∆ M 2 for s ≥ 0. Applying again the removal of singularities for s → ∞ we obtain v as a J-holomorphic triangle with boundary on three conormals. Hence, v would have to be constant, contradicting again the rescaling procedure. This shows the uniform convergence of a subsequence of (uh). Localization. It is convenient to transform the nonlocal boundary conditions (164) and (166) into local boundary conditions, by the usual method of doubling the space: given u : Σ → T ∗ M 2 we define ũ : Σ → T ∗ M 4 as ũ(s, t) := � u(s/2, t/2), −u(s/2, 1 − t/2 � . Then u solves (163) if and only if ũ solves the equation with upper boundary condition where the Hamiltonian ˜ K : [0, 1] × T ∗ M 4 →Êis defined by ∂ J, ˜ K (ũ) = 0, (167) ũ(s, 1) ∈ N ∗ ∆ M 2 ∀s ∈Ê, (168) ˜K(t, x1, x2, x3, x4) := 1 2 K(t/2, x1, x2) + 1 2 K(1 − t/2, −x3, −x4). Moreover, u satisfies (164) if and only if ũ satisfies ⎧ ⎨ N ũ(s, 0) ∈ ⎩ ∗∆M2 if s ≤ 0, N ∗∆Θ M if 0 ≤ s ≤ 2α, N ∗ (∆M × ∆M) if s ≥ 2α, whereas u satisfies (166) if and only if ũ satisfies ũ(s, 0) ∈ � N ∗ ∆M 2 if s ≤ 0, N ∗ (∆M × ∆M), if s ≥ 0. Finally, the asymptotic condition (165) is translated into lim s→−∞ ũ(s, t) = � −x1(−t/2), x2(t/2), x1(t/2 − 1), −x2(1 − t/2) � , lim s→+∞ ũ(s, t) = � −z(1/2 − t/4), z(1/2 + t/4), z(t/4), −z(1 − t/4) � . (169) (170) (171) Let u0 ∈ M Λ Υ (x1, x2; z). We must prove that there exists a unique connected component of M Υ GE (x1, x2; z) containing a curve α ↦→ (α, uα) which converges to (0, u0), in the sense of Lemma 6.6. Let ũ0 be the map from Σ to T ∗ M 4 associated to u0: ũ0 solves (167) with boundary conditions (168), (170), and asymptotic conditions (171). Since we are looking for solutions which converge 95
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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
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 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
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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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