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The embedding Q ֒→ÊN induces an embedding T ∗ Q ֒→ T ∗ÊN ∼ =Ê2N ∼ =�N . Since the embedding Q ֒→ÊN is isometric, the standard complex structure J0 ofÊ2N restricts to the metric almost complex structure J on T ∗ Q. If z = (q, p) is an element of T ∗ Q, we denote by z the element (q, −p). This notation is justified by the fact that in the embedding T ∗ Q ⊂Ê2N ∼ =�N the map z ↦→ z is the complex conjugacy. Let H ∈ C ∞ ([0, 1] × T ∗ Q) be a Hamiltonian satisfying (H1) and (H2). Fix real numbers −∞ = s0 < s1 < · · · < sk < sk+1 = +∞, and let u :Ê×[0, 1] → T ∗ Q be a solution of the Floer equation satisfying the non-local boundary conditions for every j = 0, . . .,k. The map u satisfies the energy identity � b � 1 a 0 ∂su + J(u)(∂tu − XH(t, u)) = 0, (146) (u(s, 0), u(s, 1)) ∈ N ∗ Qj ∀s ∈ [sj−1, sj], (147) |∂su(s, t)| 2 � dt ds =�H(u(a, ·)) −�H(u(b, ·)) + (u(·, 1) [a,b] ∗ η − u(·, 0) ∗ η) =�H(u(a, ·)) −�H(u(b, ·)), (148) for every a < b, where the integral over [a, b] vanishes because η ⊕ (−η) vanishes on N ∗ Qj. The following result is proven in [AS06b, Lemma 1.12] (in that lemma different boundary conditions are considered, but the proof makes use only of the energy identity (148) coming from those boundary conditions). 6.1. Lemma. For every a > 0 there exists c > 0 such that for every solution u :Ê×[0, 1] → T ∗Q of (146), (147), and � �Ê×]0,1[ |∂su(s, t)| 2 ds dt ≤ a, we have the following estimates: for every interval I. �u� L 2 (I×]0,1[) ≤ c|I| 1/2 , �∇u� L 2 (I×]0,1[) ≤ c(1 + |I| 1/2 ), The proof of the following result follows the argument of [AS06b, Theorem 1.14], using the above lemma together with the elliptic estimates of Proposition 5.10. 6.2. Proposition. For every a > 0 there is c > 0 such that for every solution u :Ê×[0, 1] → T ∗Q of (146), (147), with energy bound � �Ê×]0,1[ |∂su(s, t)| 2 ds dt ≤ a, we have the following uniform estimate: �u� L ∞ (Ê×]0,1[) ≤ c. Proof. By using the above embedding, the Floer equation (146) can be rewritten as ∂u = J0XH(t, u). (149) 88
We can pass to local boundary conditions by considering the map v :Ê×[0, 1] → T ∗ (Q × Q) ⊂�2N , v(z) := (u(z/2), u(z/2 + i)). The map v satisfies the boundary conditions Moreover, v(s, 0) ∈ N ∗ Qj ⊂ N ∗ Vj, if s ∈ [2sj−1, 2sj], v(s, 1) ∈ N ∗ ∆Q ⊂ N ∗ ∆ÊN, ∀s ∈Ê. (150) ∂v(z) = 1 � � ∂u(z/2), ∂(z/2 + i) , 2 so by (149) and by the fact that XH(t, q, p) has quadratic growth in |p| by (29), there is a constant c such that |∂v(z)| ≤ c(1 + |v(z)| 2 ). (151) Let χ be a smooth function such that χ(s) = 1 for s ∈ [0, 1], χ(s) = 0 outside [−1, 2], and 0 ≤ χ ≤ 1. Given h ∈�set w(s, t) := χ(s − h)v(s, t). Fix some p > 2, and consider the norm � · �X p introduced in section 5.3, with S = {2s1, . . .,2sk}. The map w has compact support and satisfies the boundary conditions (150), so by Proposition 5.10 we have the elliptic estimate Since we obtain, together with (151), �∇w�Xp ≤ c0�w�X p + c1�∂w�X p. ∂w = χ ′ (s − h)v + χ(s − h)∂v = χ′ w + χ(s − h)∂v, χ �∇w�X p ≤ (c0 + c1�χ ′ /χ�∞) �w�X p + c1�χ(· − h)∂v�X p ≤ (c0 + c1�χ ′ /χ�∞) �w�X p + c1c�χ(· − h)(1 + |v| 2 )�X p. Therefore, we have an estimate of the form �∇w�Xp ≤ a�w�Xp + b�χ(· − h)(1 + |v|2 )�Xp. (152) Since w has support in the set [h − 1, h + 2] × [0, 1], we can estimate its Xp norm in terms of its X1,2 norm, by Proposition 5.13. The X1,2 norm is equivalent to the W 1,2 norm, and the latter norm is bounded by Lemma 6.1. We conclude that �w�Xp is uniformly bounded. Similarly, the Xp norm of χ(· − h)(1 + |v| 2 ) is controlled by its W 1,2 norm, which is also bounded because of Lemma 6.1. Therefore, (152) implies that w is uniformly bounded in X1,p . Since p > 2, we deduce that w is uniformly bounded in L∞ . The integer h was arbitrary, hence we conclude that v is uniformly bounded in L∞ , and so is u. Let us explain how all the solution spaces M considered in this paper can be viewed in terms of the above general setting. We describe explicitly the reduction in the case of the spaces M Λ Υ associated to the pair-of-pants product as described in sections 3.2 and 3.3, the argument being analogous for all the other solution spaces. The pair-of-pants Riemann surface Σ Λ Υ is described as the quotient of the disjoint union of two stripsÊ∪[−1, 0] andÊ×[0, 1] with respect to the identifications (s, −1) ∼ (s, 0−), (s, 0+) ∼ (s, 1) ∀s ≤ 0, (153) (s, −1) ∼ (s, 1), (s, 0−) ∼ (s, 0+) ∀s ≥ 0. (154) 89
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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
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 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 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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