NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 119 [10] L. D. FADDEEV, Increasing solutions of the Schrödinger equation (in Russian), Dokl. Akad. Nauk SSSR 165 (1965), 514 – 517; English translation in Sov. Phys. Dokl. 10 (1966), 1033 – 1035. 85, 100 [11] H. M FARKAS and I. KRA, Riemann Surfaces, 2nd ed., Grad. Texts in Math. 71, Springer, New York, 1992. MR 1139765 86 [12] I. M. GELFAND, “Some aspects of functional analysis and algebra” in Proceedings of the International Congress of Mathematicians (Amsterdam, 1954), North-Holland, Amsterdam, 253 – 276. MR 0095423 83 [13] C. R. GRAHAM and M. ZWORSKI, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89 – 118. MR 1965361 111 [14] C. GUILLARMOU and L. GUILLOPÉ, The determinant of the Dirichlet-to-Neumann map for surfaces with boundary, Int. Math. Res. Not. IMRN 2007, no. 22, art. ID rnm099. MR 2376211 111 [15] C. GUILLARMOU and A. SÁ BARRETO, Inverse problems for Einstein manifolds, Inverse Probl. Imaging 3 (2009), 1 – 15. MR 2558301 84 [16] C. GUILLARMOU and L. TZOU,“Calderón inverse problem for the Schrödinger operator on Riemann surfaces” in The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis (Canberra, 2009), Proc. Centre Math. Appl. Austral. Nat. Univ. 44, Austral. Nat. Univ., Canberra, 2010, 129 – 141. MR 2655386 84 [17] G. HENKIN and V. MICHEL, Inverse conductivity problem on Riemann surfaces, J. Geom. Anal. 18 (2008), 1033 – 1052. MR 2438910 84, 85 [18] G. HENKIN and R. G. NOVIKOV, On the reconstruction of conductivity of bordered two-dimensional surface in R 3 from electrical currents measurements on its boundary, to appear in J. Geom. Anal., preprint, arXiv:1003.4897v1 [math.AP] 85 [19] O. Y. IMANUVILOV, G. UHLMANN, andM. YAMAMOTO, The Calderón problem with partial data in two dimensions, J.Amer.Math.Soc.23 (2010), 655 – 691. 84, 85, 86, 95, 96, 103, 105 [20] M. S. JOSHI and A. SÁ BARRETO, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41 – 86. MR 1756569 [21] C. E. KENIG, J. SJÖSTRAND,andG. UHLMANN, The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), 567 – 591. MR 2299741 111 84 [22] R. KOHN and M. VOGELIUS, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), 289 – 298. MR 0739921 114 [23] M. LASSAS, M. TAYLOR,andG. UHLMANN, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom. 11 (2003), 207 – 222. MR 2014876 84 [24] M. LASSAS and G. UHLMANN, On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), 771 – 787. MR 1862026 84 [25] R. MAZZEO and M. TAYLOR, Curvature and uniformization, Israel J. Math. 130 (2002), 323 – 346. MR 1919383 95 [26] D. MCDUFF and D. SALAMON, J -Holomorphic curves and symplectic topology, Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc., Providence, 2004. MR 2045629 87, 88
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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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42 CASIM ABBAS are contact forms on
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44 CASIM ABBAS the standard complex
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46 CASIM ABBAS and Jδε2 = xγ1(r)
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48 CASIM ABBAS (λ0,J0) with vanish
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50 CASIM ABBAS defined as H 1 j (S)
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52 CASIM ABBAS and denoting the cor
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54 CASIM ABBAS formula (2.2) for th
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56 CASIM ABBAS Restricting any of t
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58 CASIM ABBAS where fτ (∞) = li
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60 CASIM ABBAS for τ
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62 CASIM ABBAS Recalling our origin
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64 CASIM ABBAS THEOREM 3.7 Let µ,
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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 and 92: 92 GUILLARMOU and TZOU THEOREM 2.3.
Page 93 and 94: 94 GUILLARMOU and TZOU At a point (
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
Page 101 and 102: 102 GUILLARMOU and TZOU −1 where
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 =
Page 109 and 110: 110 GUILLARMOU and TZOU Proof of Le
Page 111 and 112: 112 GUILLARMOU and TZOU COROLLARY 6
Page 113 and 114: 114 GUILLARMOU and TZOU in x−k−
Page 115 and 116: 116 GUILLARMOU and TZOU Therefore,
Page 117: 118 GUILLARMOU and TZOU Proof of Pr
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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
Page 135 and 136: 136 MARGULIS and MOHAMMADI Indeed,
Page 137 and 138: 138 MARGULIS and MOHAMMADI For simp
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Page 141 and 142: 142 MARGULIS and MOHAMMADI (ii) The
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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