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196 JAIME RIPOLL [doCD] M.P. do Carmo and M. Dajczer, Rotational hypersurfaces in spaces of constant curvature, Transactions of the A.M.S., 2 (1983), 277. [CK] P. Collin and R. Krust, Le problème de Dirichlet pour l’équation des surfaces minimales sur de domaines non bornés, Bull. Soc. Math. de France, 119 (1991), 443-462. [ER] R.S. Earp and H. Rosenberg, The Dirichlet problem for the minimal surface equation on unbounded planar domains, Journal des Math. Pures et Apliquées, 68 (1989), 163-183. [ET] R.S. Earp and E. Toubiana, Some applications of maximum principle to hypersurface theory in euclidean and hyperbolic space, preprint. [EFR] N. do Espírito-Santo, K. Frensel and J. Ripoll, Characterization and existence results on H-domains, preprint. [F] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, 1983. [K] R.B. Kusner, Global geometry of extremal surfaces in three space, Dissertation at University of California, Berkeley. [KT] N. Kutev and F. Tomi, Existence and Nonexistence in the Exterior Dirichlet Problem for the Minimal Surface Equation in the Plane, to appear in the Journal of Differential and Integral Equations. [LM] R. López and S. Montiel, Constant mean curvature surfaces with planar boundary, Duke Math. Journal, 85(3) 1996, 583-605. [N] J.C.C. Nitsche, Lectures on Minimal Surfaces, Vol. I, Cambridge University Press, 1989. [PP] L.E. Payne and G.A. Philippin, Some maximum principles for nonlinear elliptic equations, Nonlinear Analysis, Theory, Methods and Applications, 3(2) (1979), 193-211. [R] J. Ripoll, Some existence and gradient estimates of solutions of the Dirichlet problem for the constant mean curvature equation in convex domains, to appear in the J. of Diff. Equations. [RT] J. Ripoll and F. Tomi, Some existence theorems for minimal graphs over nonconvex planar domains, preprint. [RR] A. Ros and H. Rosenberg, Constant mean curvature surfaces in a half-space of R 3 with boundary in the boundary of a half-space, Journal of Differential Geometry, 44(4) (1996), 807-817. [S] J. Serrin, The Dirichlet problem for surfaces of constant mean curvature, Proc. of London Math. Society, 21(3) (1970), 361-384. Received February 19, 1999. The author was partially supported by CNPq (Brazil). Universidade Federal do R. G. do Sul Instituto de Matemática Av. Bento Gonçalves 9500 90540-000 Porto Alegre RS Brazil E-mail address: ripoll@mat.ufrgs.br
PACIFIC JOURNAL OF MATHEMATICS Vol. 198, No. 1, 2001 SYMPLECTIC SUBMANIFOLDS FROM SURFACE FIBRATIONS Ivan Smith We give a simple construction yielding homology classes in (non-simply-connected) symplectic four-manifolds which admit infinitely many pairwise non-isotopic symplectic representatives. Examples are constructed in which the symplectic curves can have arbitrarily large genus. The examples are built from surface bundles over surfaces and involve only elementary techniques. As a corollary we see that a blow-up of any simply-connected complex projective surface contains a connected symplectic surface not isotopic to any complex curve. 1. Symplectic tori. The existence of symplectic submanifolds realising certain homology classes has had a significant impact on our understanding of symplectic topology, particularly in dimension four. This raises the natural question as to the uniqueness of symplectic representatives for homology classes in instances when they exist at all. In complex geometry the uniqueness (or finiteness) of complex representatives for (co)homology classes is well-known; for instance if X is a projective surface then an element α of H2(X) is dual to a cohomology class α∗ , and complex representatives of α correspond to transverse holomorphic sections of one of finitely many line bundles with first Chern class α∗ . (If the surface is simply connected the line bundle is unique.) These sections are points of certain projective spaces PH0 (X, Lα∗); since the locus of non-transverse sections in each such space is a complex subvariety, its complement is connected, and there are at most finitely many isotopy classes of submanifold. Our main result shows that this cannot be true in the symplectic category: Theorem 1.1. • <strong>For</strong> every odd genus g = 3 there is a symplectic four-manifold X and a homology class C ∈ H2(X, Z) such that C may be represented by infinitely many pairwise non-isotopic connected symplectic surfaces {Sa}a∈Z of genus g inside X. 197
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Pacific Journal of Mathematics Volu
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PACIFIC JOURNAL OF MATHEMATICS Vol.
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EIGENVALUES ASYMPTOTICS 3 (i) If pm
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EIGENVALUES ASYMPTOTICS 5 Corollary
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Thus, by the Itô formula, we have
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Therefore, K1(t) ≡ t 2−d/2 ≤
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EIGENVALUES ASYMPTOTICS 11 The chan
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EIGENVALUES ASYMPTOTICS 13 Here Res
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IMAGINARY QUADRATIC FIELDS 17 Table
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IMAGINARY QUADRATIC FIELDS 19 and t
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IMAGINARY QUADRATIC FIELDS 21 Propo
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〈σ 〈σ〉 〈σ, τ〉 2 〈1
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IMAGINARY QUADRATIC FIELDS 25 Then
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IMAGINARY QUADRATIC FIELDS 27 were
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IMAGINARY QUADRATIC FIELDS 29 Thus
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IMAGINARY QUADRATIC FIELDS 31 [3] ,
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34 CARINA BOYALLIAN spherical serie
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36 CARINA BOYALLIAN Here I G MAN (
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38 CARINA BOYALLIAN where k ≤ [ n
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40 CARINA BOYALLIAN The case G = Sp
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42 CARINA BOYALLIAN we can, since i
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44 CARINA BOYALLIAN and this formul
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46 CARINA BOYALLIAN Here, J(ν) int
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48 CARINA BOYALLIAN [V] D. Vogan, R
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50 ROBERT F. BROWN AND HELGA SCHIRM
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82 GILLES CARRON à l’infini s’
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84 GILLES CARRON isométriques au d
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86 GILLES CARRON Comme D est ellipt
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88 GILLES CARRON Réciproquement si
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90 GILLES CARRON où H est la proje
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92 GILLES CARRON 2.a. Exemples repo
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94 GILLES CARRON alors si σ ∈ C
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96 GILLES CARRON Ainsi s’il exist
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98 GILLES CARRON Proposition 2.7. N
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100 GILLES CARRON l’espace des 1
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102 GILLES CARRON compact. De même
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104 GILLES CARRON 4. Application du
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106 GILLES CARRON Dans le cas où l
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BRAIDED OPERATOR COMMUTATORS 111 Re
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BRAIDED OPERATOR COMMUTATORS 113 Re
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BRAIDED OPERATOR COMMUTATORS 115 3.
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Then, for 1 < k ≤ m + 1, we have
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BRAIDED OPERATOR COMMUTATORS 119 Mo
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BRAIDED OPERATOR COMMUTATORS 121 It
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E-mail address: prosk@imath.kiev.ua
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124 DANIEL J. MADDEN times. Further
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126 DANIEL J. MADDEN Further, s t
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128 DANIEL J. MADDEN Now the whole
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130 DANIEL J. MADDEN and β = 1 −
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132 DANIEL J. MADDEN We can simplif
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134 DANIEL J. MADDEN surd, but we n
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136 DANIEL J. MADDEN and d = d(u, v
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138 DANIEL J. MADDEN (6l + 7) k +
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140 DANIEL J. MADDEN Then the conti
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142 DANIEL J. MADDEN Thus n + 1 1 =
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144 DANIEL J. MADDEN If we take b =
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Page 155 and 156: TANGLES, 2-HANDLE ADDITIONS AND DEH
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Page 212 and 213: 208 PAULO TIRAO Theorem. The affine
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Page 226 and 227: 222 PAULO TIRAO is an isomorphism.
Page 228 and 229: 224 PAULO TIRAO Lemma 4.2. Let Λ1
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Page 232 and 233: 228 PAULO TIRAO ⎛ ⎜ B2 = ⎜
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HÖLDER REGULARITY FOR ∂ 247 p. 1
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similarly, we can prove HÖLDER REG
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HÖLDER REGULARITY FOR ∂ 251 For
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Now if n1 ≥ 2, l ∈ S1, then (4.
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HÖLDER REGULARITY FOR ∂ 255 Refe
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