NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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CALDERÓN <strong>IN</strong>VERSE PROBLEM ON RIEMANN SURFACES 93 real-valued function with χ(z) = 1 in {|z|
94 GUILLARMOU and TZOU At a point (p, u) such that b(p, u) = 0, a simple computation yields that the differential D(p,u)b : TpƔ ′ 0 × HƔ ′ 0 → T(p,∂νu(p))(N ∗ Ɣ ′ 0 ) is given by (w, u′ ) ↦→ (w, ∂τ ∂νu(p)w + ∂νu ′ (p)). This observation combined with Lemma 2.3.4 shows that, for all (p, u) ∈ Ɣ ′ 0 × HƔ ′ 0 such that b(p, u) = (p, 0), b is transverse to N ∗ 0 Ɣ′ 0 at (p, 0). Now we can apply Theorem 2.3.3; with X = Ɣ ′ 0 , W = N ∗Ɣ ′ 0 ,andW ′ = N ∗ 0 Ɣ′ 0 , we see that the set {u ∈ HƔ ′ 0 ; bu is transverse to N ∗ 0 Ɣ′ 0 } is residual in HƔ ′ . In view of Lemma 2.3.5, we 0 deduce the following. LEMMA 2.3.7 The set of functions u ∈ HƔ ′ 0 such that u has no degenerate critical point on Ɣ′ 0 is residual in HƔ ′ 0 . Observing the general fact that finite intersection of residual sets remains residual, the combination of Lemmas 2.3.7 and 2.3.2 yields this corollary. COROLLARY 2.3.8 The set of functions u ∈ HƔ ′ 0 points on Ɣ ′ 0 is residual in HƔ ′ 0 with respect to the Ck ( dense. which are Morse in O and have no degenerate critical Ō) topology. In particular, it is We are now in a position to give a proof of the main proposition of this section. Proof of Proposition 2.1 As explained above, choose O in such a way that Ō is a smooth surface with boundary, containing M0, sothatƔ0⊂∂O andsothatOcontains ∂M0\Ɣ0. LetƔ ′ 0 be an open and such subset of the boundary of Ō such that the closure of Ɣ0 is contained in Ɣ ′ 0 that ∂Ō\Ɣ′ 0 =∅.Letpbe an interior point of M0. By Lemma 2.2.4, there exists a holomorphic function f = u + iv on Ō such that f is purely real on Ɣ′ 0 , v(p) = 1, and df (p) = 0 (thus v ∈ HƔ ′ 0 ). By Corollary 2.3.8, there exists a sequence (vj)j of Morse functions vj ∈ HƔ ′ 0 such that vj → v in C k (M0) for any fixed k large. By the Cauchy integral formula, there exist harmonic conjugates uj of vj such that fj := uj + ivj → f in C k (M0). Let ɛ>0 be small, and let U ⊂ O be a neighborhood containing p andnoother critical points of f , and with boundary a smooth circle of radius ɛ. In complex local coordinates near p, we can identify ∂f and ∂fj to holomorphic functions on an open set of C. Then by Rouche’s theorem, it is clear that ∂fj has precisely one zero in U and that vj never vanishes in U if j is large enough. Fix to be one of the fj for j large enough. By construction, is Morse in O and has no degenerate critical points on Ɣ0 ⊂ Ɣ ′ 0 . We notice that, since the imaginary part of vanishes on all of Ɣ′ 0 ,itis clear from the reflection principle applied after using the Riemann mapping theorem
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NEAR OPTIMAL BOUNDS IN FREIMAN’S
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SUR LA NON-DENSITÉ DES POINTS ENTI
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30 CASIM ABBAS Recall that the Reeb
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32 CASIM ABBAS that Here the energy
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34 CASIM ABBAS • uτ ( ˙S) ∩ u
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36 CASIM ABBAS punctured surfaces w
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38 CASIM ABBAS that is, the central
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40 CASIM ABBAS even have A(r) ≡ 0
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Page 43 and 44: 44 CASIM ABBAS the standard complex
Page 45 and 46: 46 CASIM ABBAS and Jδε2 = xγ1(r)
Page 47 and 48: 48 CASIM ABBAS (λ0,J0) with vanish
Page 49 and 50: 50 CASIM ABBAS defined as H 1 j (S)
Page 51 and 52: 52 CASIM ABBAS and denoting the cor
Page 53 and 54: 54 CASIM ABBAS formula (2.2) for th
Page 55 and 56: 56 CASIM ABBAS Restricting any of t
Page 57 and 58: 58 CASIM ABBAS where fτ (∞) = li
Page 59 and 60: 60 CASIM ABBAS for τ
Page 61 and 62: 62 CASIM ABBAS Recalling our origin
Page 63 and 64: 64 CASIM ABBAS THEOREM 3.7 Let µ,
Page 65 and 66: 66 CASIM ABBAS Since w2 − w1C 1,
Page 67 and 68: 68 CASIM ABBAS be the conformal tra
Page 69 and 70: 70 CASIM ABBAS so that, for z ∈ B
Page 71 and 72: 72 CASIM ABBAS and, for every R>0,
Page 73 and 74: 74 CASIM ABBAS W 1,p loc (C) since
Page 75 and 76: 76 CASIM ABBAS (follows from 0 =
Page 77 and 78: 78 CASIM ABBAS is finite, the image
Page 79 and 80: 80 CASIM ABBAS Converting from coor
Page 81 and 82: 82 CASIM ABBAS [27] D. MCDUFF and D
Page 83 and 84: 84 GUILLARMOU and TZOU where ∂ν
Page 85 and 86: 86 GUILLARMOU and TZOU Carleman est
Page 87 and 88: 88 GUILLARMOU and TZOU where F ⊂
Page 89 and 90: 90 GUILLARMOU and TZOU LEMMA 2.2.4
Page 91: 92 GUILLARMOU and TZOU THEOREM 2.3.
Page 95 and 96: 96 GUILLARMOU and TZOU enough, we h
Page 97 and 98: 98 GUILLARMOU and TZOU Using the fa
Page 99 and 100: 100 GUILLARMOU and TZOU certain exp
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Page 103 and 104: 104 GUILLARMOU and TZOU factor, or
Page 105 and 106: 106 GUILLARMOU and TZOU Proof of Pr
Page 107 and 108: 108 GUILLARMOU and TZOU where I1 =
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Page 111 and 112: 112 GUILLARMOU and TZOU COROLLARY 6
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Page 115 and 116: 116 GUILLARMOU and TZOU Therefore,
Page 117 and 118: 118 GUILLARMOU and TZOU Proof of Pr
Page 119 and 120: 120 GUILLARMOU and TZOU [27] R. B.
Page 121 and 122: 122 MARGULIS and MOHAMMADI multiple
Page 123 and 124: 124 MARGULIS and MOHAMMADI As in [E
Page 125 and 126: 126 MARGULIS and MOHAMMADI planes
Page 127 and 128: 128 MARGULIS and MOHAMMADI α ∈ R
Page 129 and 130: 130 MARGULIS and MOHAMMADI We fix s
Page 131 and 132: 132 MARGULIS and MOHAMMADI Hence, i
Page 133 and 134: 134 MARGULIS and MOHAMMADI Hence th
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144 MARGULIS and MOHAMMADI needs to
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146 MARGULIS and MOHAMMADI Proof Le
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148 MARGULIS and MOHAMMADI all j>s,
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152 MARGULIS and MOHAMMADI Some imp
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158 MARGULIS and MOHAMMADI PROPOSIT
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160 MARGULIS and MOHAMMADI [EMM2]
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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