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106 GILLES CARRON Dans le cas où les bouts de M étaient euclidiens, N. Borisov, W. Muller, R. Schrader avaient déjà obtenu ce résultat. Signalons aussi que dans ce cas, on peut calculer la dimension de l’espace des k-formes harmoniques L 2 ([C1]). Preuve. La proposition précédente et le Théorème 2.1 nous assurent que les opérateurs de Gauss-Bonnet et de Signature sont non-paraboliques à l’infini sur les variétés Σi × R ni , et puisqu’au voisinage de l’infini M est isométrique à une union de telles variétés, les opérateurs considérés sont bien non-paraboliques à l’infini. On applique alors le théorème de l’indice relatif entre l’opérateur de Gauss-Bonnet sur M et sur ∪i(Σi × R ni ), ceci donne inde DGB(M) − inde DGB(Σi × R ni ) = i On utilise alors le fait que sur R ni , il n’y a pas de formes harmoniques L 2 non-nulles, que si ni ≥ 3, il n’y en a pas dans W = H 1 0 et que pour ni = 1, 2 les formes harmoniques qui sont dans W sont les formes parallèles. L’analyse est la même pour l’opérateur de signature. References [An] A. Ancona, Théorie du potentiel sur des graphes et des variétés, Lectures Notes, 1427 (1990). [A1] N. Anghel, An abstract index theorem on non-compact Riemannian manifolds, Houston J. Math., 19(2) (1993), 223-237. [A2] , Index theory for short-ranged fields in higher dimensions, J. Funct. Anal., 119 (1994), 19-36. [A3] , On the index of Callias type operators, GAFA, 3(5) (1995), 432-438. [At] M.F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Soc. Math. France, Astérisque, 32, 33 (1976), 43-72. [A-P-S] M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. Soc., 77 (1975), 43-69. [B-M-S] N.V. Borisov, W. Müller and R. Schrader, Relative index theorems and supersymmetric scattering theory, Comm. Math. Phys., 114 (1988), 475-513. [Br] J. Brüning, L 2 -index theorems on a certain complete manifold, J. Differential Geometry, 32 (1990), 491-532. [B1] U. Bunke, Relative index theory, J. Funct. Anal., 105 (1992), 63-76. [B2] , A K-theoretic relative index theorem and Callias-type Dirac operator, Math. Ann., 303(2) (1995), 241-279. [Ca] C. Callias, Axial anomalies and index theorems on open spaces, Commun. Math. Phys., 62 (1978), 213-234. [C1] G. Carron, L 2 -cohomologie et inégalités de Sobolev, Math. Ann., 314 (1999), 613-639. M Ω.
UN THÉORÈME DE L’INDICE RELATIF 107 [C2] , Une suite exacte en L 2 -cohomologie, Duke Math. Journal, 95 (1998), 343-372. [C3] , Inégalités de Faber-Krahn et inclusion de Sobolev- Orlicz, Potential Analysis, 7 (1997), 555-575. [C4] , Inégalités de Sobolev-Orlicz non-uniformes, Colloq. Math., 77(2) (1998), 163-178. [C5] , Inégalité de Hardy sur les variétés riemanniennes, J. Math. Pures Appli., 76 (1997), 883-891. [D] H. Donnelly, Essential spectrum and heat kernel, J. Funct. Anal., 75 (1987), 362-381. [F] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math., 82 (1985), 121-132. [G-L] M. Gromov and H.B. Lawson, Jr, Positive scalar curvature and the Dirac operator on a complete Riemannian manifold, Publ. Math. I.H.E.S., 58 (1983), 83-196. [H-S] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 7156-727. [L] J. Lott, L 2 -cohomology of geometrically infinite hyperbolic 3-manifold, Geom. Funct. Anal., 7 (1997), 81-119. Received May 26, 1999 and revised September 10, 1999. Universite de Nantes 44322 Nantes cedex 02 France E-mail address: Gilles.Carron@math.univ-nantes.fr
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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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Page 100 and 101: 98 GILLES CARRON Proposition 2.7. N
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Page 128 and 129: 124 DANIEL J. MADDEN times. Further
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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
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Page 144 and 145: 140 DANIEL J. MADDEN Then the conti
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TANGLES, 2-HANDLE ADDITIONS AND DEH
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m n n n n TANGLES, 2-HANDLE ADDITIO
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SOME CHARACTERIZATION, UNIQUENESS A
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and SOME CHARACTERIZATION, UNIQUENE
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SYMPLECTIC SUBMANIFOLDS FROM SURFAC
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208 PAULO TIRAO Theorem. The affine
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210 PAULO TIRAO It follows from the
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212 PAULO TIRAO K ρ1 (0,−1,1) =
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214 PAULO TIRAO Theorem 2.7. Let ρ
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216 PAULO TIRAO Notation: By A = [
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218 PAULO TIRAO Now is not difficul
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220 PAULO TIRAO Regard H n (Z2 ⊕
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222 PAULO TIRAO is an isomorphism.
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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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