Von Neumann spectra near the spectral gap
Identifieur interne : 002751 ( Istex/Corpus ); précédent : 002750; suivant : 002752Von Neumann spectra near the spectral gap
Auteurs : Alan L. Carey ; Thierry Coulhon ; Varghese Mathai ; John PhillipsSource :
- Bulletin des sciences mathematiques [ 0007-4497 ] ; 1998.
English descriptors
- KwdEn :
- 58G11, 58G18 and 58G25, Acyclic riemannian manifold, Analytic laplacian, Analytic torsion, Betti numbers, Carey, Casimir operator, Combinatorial, Combinatorial counterpart, Combinatorial laplacian, Compact subsets, Convergence, Converges, Determinant, Determinant class, Differential forms, Dimr, Fundamental group, Heat kernel, Heat kernels, Hyperbolic manifold, Hyperbolic manifolds, Inequality, Integral kernel, Laplacian, Large time asymptotics, Lemma, Mathai, Meromorphic continuation, Mesh, Metric, Neumann, Neumann algebra, Neumann determinant, Neumann determinants, Neumann spectra, Neumann trace, Novikov-Shubin invariants, Partial zeta functions, Plancherel measure, Positive decay, Previous theorem, Rham, Riemannian, Riemannian manifold, Right hand side, Sciences mathbmatiques, Spectral density function, Subset, Theta, Theta functions, Tome, Torsion, Triangulation, Zeta, Zeta function, Zeta functions, bottom of the spectrum, torsion, von Neumann determinants.
- Teeft :
- Acyclic riemannian manifold, Analytic laplacian, Analytic torsion, Betti numbers, Carey, Casimir operator, Combinatorial, Combinatorial counterpart, Combinatorial laplacian, Compact subsets, Convergence, Converges, Determinant, Determinant class, Differential forms, Dimr, Fundamental group, Heat kernel, Hyperbolic manifold, Hyperbolic manifolds, Inequality, Integral kernel, Laplacian, Large time asymptotics, Lemma, Mathai, Meromorphic continuation, Mesh, Metric, Neumann, Neumann algebra, Neumann determinant, Neumann determinants, Neumann spectra, Neumann trace, Partial zeta functions, Plancherel measure, Positive decay, Previous theorem, Rham, Riemannian, Riemannian manifold, Right hand side, Sciences mathbmatiques, Spectral density function, Subset, Theta, Theta functions, Tome, Torsion, Triangulation, Zeta, Zeta function, Zeta functions.
Abstract
Abstract: In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.
Url:
DOI: 10.1016/S0007-4497(98)80088-2
Links to Exploration step
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<front><div type="abstract" xml:lang="en">Abstract: In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</div>
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<head><ce:title>Von Neumann spectra near the spectral gap</ce:title>
<ce:author-group><ce:author><ce:given-name>Alan L.</ce:given-name>
<ce:surname>Carey</ce:surname>
<ce:cross-ref refid="AFF1"><ce:sup loc="post">1</ce:sup>
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<ce:author><ce:given-name>Thierry</ce:given-name>
<ce:surname>Coulhon</ce:surname>
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<ce:author><ce:given-name>Varghese</ce:given-name>
<ce:surname>Mathai</ce:surname>
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<ce:author><ce:given-name>John</ce:given-name>
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<ce:affiliation id="AFF1"><ce:label>a</ce:label>
<ce:textfn>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</ce:textfn>
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<ce:affiliation id="AFF2"><ce:label>b</ce:label>
<ce:textfn>Département de Mathématiques, Université de Cergy-Pontoise, 95033 Cergy-Pontoise cedex, France</ce:textfn>
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<ce:affiliation id="AFF3"><ce:label>c</ce:label>
<ce:textfn>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</ce:textfn>
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<ce:affiliation id="AFF4"><ce:label>d</ce:label>
<ce:textfn>Department of Mathematics, University of Victoria. Victoria, BC, Canada</ce:textfn>
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<ce:footnote id="FN1"><ce:label>(∗)</ce:label>
<ce:note-para>Manuscript presented by <ce:small-caps>G. Pisier</ce:small-caps>
, received in March 1996.</ce:note-para>
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<ce:abstract-sec><ce:simple-para view="all" id="simple-para.0010">In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on <ce:italic>L</ce:italic>
<ce:sup loc="post">2</ce:sup>
differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain <ce:italic>L</ce:italic>
<ce:sup loc="post">2</ce:sup>
theta and <ce:italic>L</ce:italic>
<ce:sup loc="post">2</ce:sup>
zera functions defined by metric dependent combinatorial Laplacians acting on <ce:italic>L</ce:italic>
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cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</ce:simple-para>
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<ce:keywords class="keyword"><ce:section-title>Keywords</ce:section-title>
<ce:keyword><ce:text>Heat kernels</ce:text>
</ce:keyword>
<ce:keyword><ce:text>bottom of the spectrum</ce:text>
</ce:keyword>
<ce:keyword><ce:text>Novikov-Shubin invariants</ce:text>
</ce:keyword>
<ce:keyword><ce:text>von Neumann determinants</ce:text>
</ce:keyword>
<ce:keyword><ce:text>torsion</ce:text>
</ce:keyword>
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<ce:keywords class="msc"><ce:section-title>MSC</ce:section-title>
<ce:keyword><ce:text>58G11</ce:text>
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<ce:keyword><ce:text>58G18 and 58G25</ce:text>
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<name type="personal"><namePart type="given">Alan L.</namePart>
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<abstract lang="en">Abstract: In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</abstract>
<subject lang="en"><genre>Keywords</genre>
<topic>Heat kernels</topic>
<topic>bottom of the spectrum</topic>
<topic>Novikov-Shubin invariants</topic>
<topic>von Neumann determinants</topic>
<topic>torsion</topic>
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<subject lang="en"><genre>MSC</genre>
<topic>58G11</topic>
<topic>58G18 and 58G25</topic>
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