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174 W. MENASCO AND X. ZHANG [KT] J. Kalliongis and C. Tsau, Seifert fibered surgery manifolds of composite knots, Proc. Amer. Math. Soc., 108 (1990), 1047-1053. [Ma] D. Matignon, Dehn surgery on a knot with three bridges cannot yield P 3 , Osaka J. Math., 34 (1997), 133-134. [M] W. Menasco, Closed incompressible surfaces in alternating knot complements, Topology, 23 (1984), 37-44. [MT] W. Menasco and M. Thistlethwaite, Surfaces with boundary in alternating knot exteriors, J. Reine Angew. Math., 426 (1992), 47-65. [MM] K. Miyasaki and K. Motegi, Seifert fibered manifolds and Dehn surgery, Topology, 36 (1997), 579-603. [MS] K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann., 289 (1991), 143-167. [R] D. Rolfsen, Knots and Links, Publish or Perish Inc., 1976. [Sch] M. Scharlemann, Producing reducible 3-manifolds by surgery on a knot, Topology, 29 (1990), 481-500. [T] W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6 (1982), 357-381. [W] Y. Wu, Dehn surgery on arborescent knots, J. Diff. Geo., 43 (1996), 171-197. [W2] , Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Diff. Geo., 48 (1998), 407-437. [W3] , Incompressibility of surfaces in surgered 3-manifolds, Topology, 31 (1992), 271-279. Received May 28, 1999. The first author was partially supported by NSF grant DMS 9626884. The second author was partially supported by NSF grant DMS 9971561. Mathematics Department SUNY at Buffalo Buffalo, NY 14214-3093 E-mail address: menasco@math.buffalo.edu Mathematics Department SUNY at Buffalo Buffalo, NY 14214-3093 E-mail address: xinzhang@math.buffalo.edu
PACIFIC JOURNAL OF MATHEMATICS Vol. 198, No. 1, 2001 SOME CHARACTERIZATION, UNIQUENESS AND EXISTENCE RESULTS FOR EUCLIDEAN GRAPHS OF CONSTANT MEAN CURVATURE WITH PLANAR BOUNDARY Jaime Ripoll We establish the existence and uniqueness of solutions to the Dirichlet problem for the cmc surface equation, including the minimal one, for zero boundary data, in certain domains of the plane. We obtain results that characterize the sphere and cmc graphs among compact embedded cmc surfaces with planar boundary satisfying certain geometric conditions. We also find conditions that imply that a compact embedded cmc surface which is a graph near the boundary is indeed a global graph. 0. Introduction. In this paper we shall obtain some characterization, uniqueness and existence theorems to the Dirichlet’s problem for the constant mean curvature (cmc) equation with vanishing boundary data (1) QH(u) := div ∇u 1 + |∇u| 2 + 2H = 0, u|∂Ω = 0, u ∈ C 2 (Ω) ∩ C 0 (Ω) where Ω is a domain in the plane and H ≥ 0. Although being a very special case of boundary data, it is shown in [R] that the general case of arbitrary continuous boundary data can be reduced, in many situations, to the zero boundary data. In the minimal case, it is well known that if Ω is bounded and convex then there is a solution to Q0 = 0 in Ω which assumes any continuous prescribed data value at ∂Ω. <strong>For</strong> Ω convex but not bounded, and for some special cases of non-convex unbounded domains, R. Earp and H. Rosenberg ([ER]) proved the existence of solutions which take on any continuous bounded boundary value if Ω is not a half plane. This last case was treated by P. Collin and R. Krust ([CK]) who proved existence for a given continuous boundary data with growth at most linear. In [CK] a uniqueness theorem in arbitrary unbounded domains is also obtained. 175
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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
Page 128 and 129: 124 DANIEL J. MADDEN times. Further
Page 130 and 131: 126 DANIEL J. MADDEN Further, s t
Page 132 and 133: 128 DANIEL J. MADDEN Now the whole
Page 134 and 135: 130 DANIEL J. MADDEN and β = 1 −
Page 136 and 137: 132 DANIEL J. MADDEN We can simplif
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Page 140 and 141: 136 DANIEL J. MADDEN and d = d(u, v
Page 142 and 143: 138 DANIEL J. MADDEN (6l + 7) k +
Page 144 and 145: 140 DANIEL J. MADDEN Then the conti
Page 146 and 147: 142 DANIEL J. MADDEN Thus n + 1 1 =
Page 148 and 149: 144 DANIEL J. MADDEN If we take b =
Page 150 and 151: 146 DANIEL J. MADDEN b, 2b(2bn + 1)
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Page 212 and 213: 208 PAULO TIRAO Theorem. The affine
Page 214 and 215: 210 PAULO TIRAO It follows from the
Page 216 and 217: 212 PAULO TIRAO K ρ1 (0,−1,1) =
Page 218 and 219: 214 PAULO TIRAO Theorem 2.7. Let ρ
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224 PAULO TIRAO Lemma 4.2. Let Λ1
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226 PAULO TIRAO (c)1 B1 B2 B3 B1 0
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228 PAULO TIRAO ⎛ ⎜ B2 = ⎜
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230 PAULO TIRAO being those in whic
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232 PAULO TIRAO Hence, if ρ = m1χ
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(1.6) HÖLDER REGULARITY FOR ∂ 23
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HÖLDER REGULARITY FOR ∂ 239 can
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HÖLDER REGULARITY FOR ∂ 241 wher
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HÖLDER REGULARITY FOR ∂ 243 Theo
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Note (4.7) bD∩U ′ i dzA j J H
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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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