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15
A renormalized index theorem for some complete asymptotically regular metrics: the GaussBonnet theorem
, 2005
"... The GaussBonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the PoincaréEinstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L²cohomology spaces as well as to carry ..."
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Cited by 21 (2 self)
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The GaussBonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the PoincaréEinstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L²cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x m, the finite time supertrace of the heat kernel on conformally compact manifolds is shown to renormalize independently of the choice of special boundary defining function.
ANALYTIC CONTINUATION AND HIGH ENERGY ESTIMATES FOR THE RESOLVENT OF THE LAPLACIAN ON FORMS ON ASYMPTOTICALLY HYPERBOLIC SPACES
"... Abstract. We prove the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar ..."
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Cited by 8 (3 self)
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Abstract. We prove the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recently, roughly by extending the problem across the boundary of the compactification of the asymptotically hyperbolic space in a suitable manner. The main novelty is that the nonscalar nature of the operator is dealt with by relating it to a problem on an asymptotically Minkowski space to motivate the choice of the extension across the conformal boundary. 1.
ANALYTIC CONTINUATION OF RESOLVENT KERNELS ON NONCOMPACT SYMMETRIC SPACES
, 2003
"... Abstract. Let X = G/K be a symmetric space of noncompact type and let ∆ be the Laplacian associated with a Ginvariant metric on X. We show that the resolvent kernel of ∆ admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branc ..."
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Cited by 8 (4 self)
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Abstract. Let X = G/K be a symmetric space of noncompact type and let ∆ be the Laplacian associated with a Ginvariant metric on X. We show that the resolvent kernel of ∆ admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real axis removed. It has a branching point at the bottom of the spectrum of ∆. It is further shown that this branching point is quadratic if the rank of X is odd, and is logarithmic otherwise. In case G has only one conjugacy class of Cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of C with the only branching point being the bottom of the spectrum. Mathematics Subject Classification (2000): 58J50 (11F72) 1.
Diabatic limit, eta invariants and CauchyRiemann manifolds of dimension 3
, 2005
"... We relate a recently introduced nonlocal geometric invariant of compact strictly pseudoconvex CauchyRiemann (CR) manifolds of dimension 3 to various ηinvariants in CR geometry: on the one hand a renormalized ηinvariant appearing when considering a sequence of metrics converging to the CR struc ..."
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Cited by 5 (0 self)
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We relate a recently introduced nonlocal geometric invariant of compact strictly pseudoconvex CauchyRiemann (CR) manifolds of dimension 3 to various ηinvariants in CR geometry: on the one hand a renormalized ηinvariant 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 ηinvariant 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 KählerEinstein or an Einstein metric.
Rigidity of amalgamated product in negative curvature
 Journ. Diff. Geom
"... Abstract. 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 − ..."
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Cited by 2 (0 self)
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Abstract. 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. 1.
Rigidity and L2 cohomology of hyperbolic manifold, Annales de l’institut Fourier 60 (2011) no
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AMALGAMATED PRODUCTS, CRITICAL EXPONENTS AND UNIFORM GROWTH OF GROUPS: A UNIFIED APPROACH
"... Abstract. The aim of this note is to advertise a method which turns out to be powerful enough to be used successfully in problems which are apparently unrelated. It is based on a modification of a construction that we first introduced in [2]. 1. ..."
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Abstract. The aim of this note is to advertise a method which turns out to be powerful enough to be used successfully in problems which are apparently unrelated. It is based on a modification of a construction that we first introduced in [2]. 1.
AN INVARIANT OF CAUCHYRIEMANN SEIFERT 3MANIFOLDS AND APPLICATIONS
, 2004
"... Abstract. We compute a recently introduced geometric invariant of stricly pseudoconvex CR 3manifolds for certain circle invariant spherical CR structures on Seifert manifolds. We give applications to the problem of filling the CR manifold by a complex hyperbolic manifold, and more generally by a Kä ..."
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Abstract. We compute a recently introduced geometric invariant of stricly pseudoconvex CR 3manifolds for certain circle invariant spherical CR structures on Seifert manifolds. We give applications to the problem of filling the CR manifold by a complex hyperbolic manifold, and more generally by a KählerEinstein or an Einstein metric. 1. Introduction. In [7] we introduced a new invariant, called the νinvariant, of strictly pseudoconvex CauchyRiemann (CR) compact 3manifolds. This invariant is an analogue in CR geometry of the ηinvariant in conformal geometry. The definition of the νinvariant is rather abstract and