On the differential form spectrum of hyperbolic manifolds

Carron, Gilles; Pedon, Emmanuel

doi:10.2422/2036-2145.2004.4.03

Cited by 15 publications

(13 citation statements)

References 54 publications

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“…The following lemma is due to G.Carron and E.Pedon, [8]. For a complete riemannian manifold Y , we denote H 1 c (Y, R) the first cohomology group generated by diferential forms with compact support.…”

Section: Proof Of the Equality Casementioning

confidence: 99%

Rigidity of amalgamated products in negative curvature

Besson

Courtois

Gallot

2008

J. Differential Geom.

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Let Γ be the fundamental group of a compact riemannian manifold X of sectional curvature K ≤ −1 and dimension n ≥ 3. We suppose that Γ = A * C B is the free product of its subgroups A and B over the amalgamated subgroup C. We prove that the critical exponent δ(C) of C satisfies δ(C) ≥ n − 2. The equality happens if and only if there exist an embedded compact hypersurface Y ⊂ X, totally geodesic, of constant sectional curvature −1, whose fundamental group is C and which separates X in two connected components whose fundamental groups are A and B. Similar results hold if Γ is an HNN extension, or more generally if Γ acts on a simplicial tree without fixed point.Date: November 16, 2018.

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Section: Proof Of the Equality Casementioning

confidence: 99%

Rigidity of amalgamated products in negative curvature

Besson

Courtois

Gallot

2008

J. Differential Geom.

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“…In the case where the torsion of (M, θ) vanishes, the terms η i (M, θ) in ( 12) for i > 0 vanish, so that, when ε goes to infinity instead of 0, one has (17) η…”

Section: The Renormalized Eta Invariantmentioning

confidence: 99%

“…From [17,Theorem 5.12], one knows that pseudoconvex complex hyperbolic surfaces N have vanishing third homology group H 3 (N, Z). Hence no multiple ends can occur, but one expects orbifold singularities or cusps to appear in the interior of a complex hyperbolic filling.…”

Section: Proof Of the Corollariesmentioning

confidence: 99%

“…Theorem 1.1 gives a simple formula relating the ν-invariant and the renormalized η-invariant η 0 of the contact-rescaling. According to (17), η 0 coincides with the adiabatic limit of η in the case the CR manifold has vanishing torsion, and this enables computations, for explicit expressions of the adiabatic limit are known in a number of cases. But a deeper question is to relate directly the ν-invariant to the geometry and spectral theory of the CR or pseudohermitian manifold.…”

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confidence: 94%

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Diabatic limit, eta invariants and Cauchy–Riemann manifolds of dimension 3☆

Biquard¹,

Herzlich²,

Rumin³

2007

Annales Scientifiques de l’École Normale Supérieure

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We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various η-invar-iants in CR geometry: on the one hand a renormalized η-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the η-invariant of the middle degree operator of the contact complex. We then provide explicit computations for a class of examples: transverse circle invariant CR structures on Seifert manifolds. Applications are given to the problem of filling a CR manifold by a complex hyperbolic manifold, and more generally by a Kähler-Einstein or an Einstein metric.1991 Mathematics Subject Classification. 32V05, 32V20, 53C20, 58J28. Key words and phrases. CR manifolds of dimension 3, pseudohermitian structures, eta invariants, contact complex.The second author is supported in part by a Young Researchers aci program of the French Ministry of Research.

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“…In fact, R. Mazzeo has shown that the cohomology with compact support of a convex cocompact hyperbolic n−manifold is isomorphic to the space of harmonic L 2 forms in degree p < n/2 [17]: If X = Γ\H n is convex cocompact and if p < n/2 then H p c (X) ≃ H p (X) := {α ∈ L 2 (Λ p T * X), dα = d * α = 0}. In [12], with E. Pedon, we obtained the following result : Theorem 1.2. Let X = Γ\H n be a hyperbolic manifold, assume that for p < n/2 :…”

Section: Introductionmentioning

confidence: 99%

Rigidity and L^2 cohomology of hyperbolic manifolds

Carron¹

2010

Ann. inst. Fourier

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When X = Γ\H n is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L 2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results. R ÉSUM É : La petitesse de l'exposant critique du groupe fondamental d'une variété hyperbolique implique des résultats d'annulation pour certains espaces de cohomologie et de formes harmoniques L 2 . Nous obtenons ici des résultats de rigidité reliés à ces résultats d'annulations. Ceci est une généralisation de résultats déjà connus dans le cas convexe co-compact.

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On the differential form spectrum of hyperbolic manifolds

Cited by 15 publications

References 54 publications

Rigidity of amalgamated products in negative curvature

Rigidity of amalgamated products in negative curvature

Diabatic limit, eta invariants and Cauchy–Riemann manifolds of dimension 3☆

Rigidity and L^2 cohomology of hyperbolic manifolds

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