## Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds (2005)

Venue: | Gui05c] [Gui06] [GZ95] [GZ97] [GZ99] [His94] [His00] [Jan79] Colin Guillarmou. Resonances and |

Citations: | 42 - 13 self |

### BibTeX

@INPROCEEDINGS{Guillarmou05meromorphicproperties,

author = {Colin Guillarmou},

title = {Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds},

booktitle = {Gui05c] [Gui06] [GZ95] [GZ97] [GZ99] [His94] [His00] [Jan79] Colin Guillarmou. Resonances and},

year = {2005}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N are at most some poles of finite multiplicity, and that the same 2 property holds for the points of n+1 − N if and only if the metric is ‘even’. On the other 2 hand, there exist some metrics for which R(λ) has an essential singularity on n+1 − N and 2 these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R(λ) approaching an essential singularity.

### Citations

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Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature
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- 1987
(Show Context)
Citation Context ... (∆g − λ(n − λ)) −1 on some non-compact spaces (Xn+1, g) called asymptotically hyperbolic manifolds. This meromorphic extension in C \ 1 2 (n − N) with finite rank poles, proved by Mazzeo and Melrose =-=[17]-=-, is a beautiful application of Melrose’s pseudodifferential calculus on manifolds with corners, which generalizes some well-known results on hyperbolic spaces. Meromorphic extensions of resolvents ha... |

89 |
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Citation Context ...ompact [10, 11, 12, 19], the De Sitter-Schwarzschild model [23], the almost-product type metrics [14]. The asymptotically Einstein manifolds of dimension n + 1 are only even modulo O(x n ) in general =-=[7, 8]-=-. Let us denote by Mah(X) the space of asymptotically hyperbolic metrics on X with the topology inherited from x−2C ∞ ( ¯ X, T ∗X¯ ∗ ⊗ T X). ¯ If the metric is not even, there is at least a point of Z... |

86 | Scattering matrix in conformal geometry
- Graham, Zworski
(Show Context)
Citation Context ...se we will explain later. In fact, our philosophy will be to use the properties of the scattering operator, whose poles are essentially the resonances (cf. [4]). The recent work of Graham and Zworski =-=[8]-=- gives indeed a simple and explicit presentation of the scattering operator S(λ) on asymptotically hyperbolic manifolds which allows us to study the nature of S(λ) near 1 2 (n + N). Thanks to their ca... |

77 |
The Hodge cohomology of a conformally compact metric
- Mazzeo
- 1988
(Show Context)
Citation Context ... to the scattering operator 3.1. Stretched products. To begin, let us introduce a few notations and recall some basic things on stretched products (the reader can refer to Mazzeo-Melrose [17], Mazzeo =-=[16]-=- or Melrose [18] for details). Let ¯ X a smooth compact manifold with boundary and x a boundary defining function. The manifold ¯ X × ¯ X is a smooth manifold with corners, whose boundary hypersurface... |

67 | The divisor of Selberg’s zeta function for Kleinian groups
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(Show Context)
Citation Context ...hism induced by the flow φt of the gradient gradx2g(x): { [0, ǫ) × ∂X¯ → φ([0, ǫ) × ∂X) ¯ ⊂ X¯ φ : (t, y) → φt(y) i=0 Using the relations between resolvent and scattering operator in a way similar to =-=[6, 9, 19]-=- and the calculus of the residues of S(λ) by Graham-Zworski [8] we find a necessary and sufficient condition on the metric to have a finite-meromorphic extension of the resolvent to C. Proposition 1.3... |

57 |
Analysis on manifolds with corners
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(Show Context)
Citation Context ...ng operator 3.1. Stretched products. To begin, let us introduce a few notations and recall some basic things on stretched products (the reader can refer to Mazzeo-Melrose [17], Mazzeo [16] or Melrose =-=[18]-=- for details). Let ¯ X a smooth compact manifold with boundary and x a boundary defining function. The manifold ¯ X × ¯ X is a smooth manifold with corners, whose boundary hypersurfaces are diffeomorp... |

43 |
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- Guillopé, Zworski
- 1997
(Show Context)
Citation Context ...orphic extension to C if and only if g is even modulo O(x ∞ ). Remark: as a matter of fact, the usual examples are some particular cases of even metrics: the hyperbolic metrics perturbed on a compact =-=[10, 11, 12, 19]-=-, the De Sitter-Schwarzschild model [23], the almost-product type metrics [14]. The asymptotically Einstein manifolds of dimension n + 1 are only even modulo O(x n ) in general [7, 8]. Let us denote b... |

42 |
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- 1995
(Show Context)
Citation Context ...orphic extension to C if and only if g is even modulo O(x ∞ ). Remark: as a matter of fact, the usual examples are some particular cases of even metrics: the hyperbolic metrics perturbed on a compact =-=[10, 11, 12, 19]-=-, the De Sitter-Schwarzschild model [23], the almost-product type metrics [14]. The asymptotically Einstein manifolds of dimension n + 1 are only even modulo O(x n ) in general [7, 8]. Let us denote b... |

36 |
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- 1991
(Show Context)
Citation Context ...t Patterson-Perry arguments [16, Lem. 4.9] prove that R(λ) does not have poles on the line {ℜ(λ) = n n 2 }, except maybe λ = 2 , it is a consequence of the absence of embedded eigenvalues (see Mazzeo =-=[13]-=-). The only poles of R(λ) in the half plane {ℜ(λ) > n 2 } is the finite set of λe such that λe(n − λe) ∈ σpp(∆g), they are some first order poles and their residue is (3.4) ResλeR(λ) = (2λe − n) −1 p∑... |

34 |
Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity. Asymptotic Anal
- Guillopé, Zworski
- 1995
(Show Context)
Citation Context ...ese points are poles with infinite rank residues, or even essential singularities of R(λ). However, if the manifold has constant negative sectional curvature away from a compact, Guillopé and Zworski =-=[11]-=- did show the meromorphic continuation of the resolvent to C with finite rank poles. The key to analyze the points of 1 2 (n − N) is the special structure of the metric near infinity, in the sense tha... |

32 |
The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix
- Perry
- 1989
(Show Context)
Citation Context ... we have the following holomorphic identity on L(xNL2 (X), x−NL2 (X)): (3.15) R(λ) − R(n − λ) = (2λ − n) t E(n − λ)S(λ)E(n − λ). Observe that the proof of Green’s formula obtained by Agmon [2], Perry =-=[20]-=- or Guillopé [9] for hyperbolic quotients remains true in our framework (see also Borthwick [3, Prop. 4.5] in our setting): for λ, n − λ /∈ (R ∪ Z1 − ∪ Z2 − ), m, m′ ∈ X and m ̸= m ′ (3.16) r(λ; m, m ... |

29 |
Sá Barreto. Inverse scattering on asymptotically hyperbolic manifolds
- Joshi, A
(Show Context)
Citation Context ..., the usual examples are some particular cases of even metrics: the hyperbolic metrics perturbed on a compact [10, 11, 12, 19], the De Sitter-Schwarzschild model [23], the almost-product type metrics =-=[14]-=-. The asymptotically Einstein manifolds of dimension n + 1 are only even modulo O(x n ) in general [7, 8]. Let us denote by Mah(X) the space of asymptotically hyperbolic metrics on X with the topology... |

18 |
Fonctions zeta de Selberg et surfaces de géométrie finie
- Guillopé
- 1990
(Show Context)
Citation Context ...owing holomorphic identity on L(xNL2 (X), x−NL2 (X)): (3.15) R(λ) − R(n − λ) = (2λ − n) t E(n − λ)S(λ)E(n − λ). Observe that the proof of Green’s formula obtained by Agmon [2], Perry [20] or Guillopé =-=[9]-=- for hyperbolic quotients remains true in our framework (see also Borthwick [3, Prop. 4.5] in our setting): for λ, n − λ /∈ (R ∪ Z1 − ∪ Z2 − ), m, m′ ∈ X and m ̸= m ′ (3.16) r(λ; m, m ′ ) − r(n − λ; m... |

17 | Scattering poles for asymptotically hyperbolic manifolds
- Borthwick, Perry
(Show Context)
Citation Context ...aracter to eigenvalues of a compact manifold. As far as we are concerned, the construction of [17] does not treat the special points ( ) n−k 2 and, as Borthwick k∈N and Perry noticed in their article =-=[4]-=-, it seems possible that these points are poles with infinite rank residues, or even essential singularities of R(λ). However, if the manifold has constant negative sectional curvature away from a com... |

16 | Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group
- Bunke, Olbrich
- 1999
(Show Context)
Citation Context ...phic in U \ {λ0} but not in U, we will say that λ0 is an essential singularity of M(λ). At last, note that all these definitions extend to locally convex vector spaces (see for instance Bunke-Olbrich =-=[5]-=-). Here is an interpretation of the result of Mazzeo and Melrose [17, Th. 7.1] : Theorem 1.1. Let (X, g) be an asymptotically hyperbolic manifold, ∆g its Laplacian acting on functions and x a boundary... |

11 |
The Selberg zeta function and a local trace formula for Kleinian groups
- Perry
- 1990
(Show Context)
Citation Context ...¯ the principal symbol of S(λ) is given by Joshi and Sá Barreto [14]: ( 2λ−n (3.10) σ0 (S(λ)) = c(λ)σ0 Λ ) Λ := (1 + ∆h0) 1 − λ) 2 , c(λ) := 2 Γ(λ − n 2 ) which leads us to set the factorization (see =-=[21, 12, 19]-=- for a similar approach) n n−2λΓ( 2 n n ˜S(λ) −λ+ := c(n − λ)Λ 2 −λ+ S(λ)Λ 2 . So ˜ S(λ) can be expressed by ˜S(λ) = 1 + K(λ) where K(λ) is a compact operator for λ ∈ C \ (R ∪ 1 2Z). Notice that the p... |

10 | Scattering theory and deformations of asymptotically hyperbolic metrics.” dg-ga/9711016
- Borthwick
- 1997
(Show Context)
Citation Context ... the collar U := (0, 1) × Sn which carries the metric g := x −2 (dx 2 + d(x)h0) ( d(x) := 1 − x2 )2 + χ(x)x 4 2k+1 where h0 := gSn is the canonical metric on the n-dimensional sphere Sn , and χ ∈ C∞ (=-=[0, 1]-=-) a non negative function such that χ(x) = 1 for x ∈ [0, 1 2 ] and χ(x) = 0 for x ∈ [1 2 , 1]. We easily check that for Bn+1 := {m ∈ Rn+1 ; |m| < 1}, the diffeomorphism S n { × (0, 1) → Bn+1 \ |m| ≤ ψ... |

10 | The Analysis of Partial Differential Operators III - Hörmander - 1986 |

9 |
A Perturbation Theory of Resonances
- Agmon
- 1998
(Show Context)
Citation Context ...ases are the first examples (as far as we know) of essential singularities coming from the meromorphic extension of the resolvent. If the resonances are interpreted as eigenvalues of an operator (see =-=[1]-=- when the extension is finite-meromorphic), we could think that these essential singularities are some isolated points in the essential spectrum of this operator. Z 1 − = n+1 2 − N ℜ(λ) = n 2 Z 1 + = ... |

9 |
Zworski Distribution of resonances for spherical black
- Barreto, M
(Show Context)
Citation Context ...rk: as a matter of fact, the usual examples are some particular cases of even metrics: the hyperbolic metrics perturbed on a compact (cf. [8], [9], [10], [16]), the De Sitter-Schwarzschild model (cf. =-=[18]-=-), the asymptotically Einstein manifolds of even dimension (cf. [6], [5]), the almost-product type metrics (cf. [12]). Let us denote by Mah(X) the space of asymptotically hyperbolic metrics on X with ... |

8 |
On the spectral theory of the Laplacian on non-compact hyperbolic manifolds, Journes “Equations aux derives partielles” (Saint Jean de
- Agmon
- 1987
(Show Context)
Citation Context ... Z2 − ∪ R)} we have the following holomorphic identity on L(xNL2 (X), x−NL2 (X)): (3.15) R(λ) − R(n − λ) = (2λ − n) t E(n − λ)S(λ)E(n − λ). Observe that the proof of Green’s formula obtained by Agmon =-=[2]-=-, Perry [20] or Guillopé [9] for hyperbolic quotients remains true in our framework (see also Borthwick [3, Prop. 4.5] in our setting): for λ, n − λ /∈ (R ∪ Z1 − ∪ Z2 − ), m, m′ ∈ X and m ̸= m ′ (3.16... |

8 | Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds.” math.AP/0312108
- Barreto
- 2003
(Show Context)
Citation Context ...inition with respect to the choice of the boundary defining function x. That is dropped for notations, because it does not play an important role for what we study. We can also notice that Sá Barreto =-=[22]-=- has recently adapted the radiation fields theory to this setting, which provides a new definition for the operators E(λ), S(λ) in terms of these radiation fields. 3.6. Meromorphic equivalences. Let U... |

5 |
Sharp bounds on the number of resonances for conformally compact manifolds with constant negative curvature near infinity
- Cuevas, Vodev
(Show Context)
Citation Context ...hism induced by the flow φt of the gradient gradx2g(x): { [0, ǫ) × ∂X¯ → φ([0, ǫ) × ∂X) ¯ ⊂ X¯ φ : (t, y) → φt(y) i=0 Using the relations between resolvent and scattering operator in a way similar to =-=[6, 9, 19]-=- and the calculus of the residues of S(λ) by Graham-Zworski [8] we find a necessary and sufficient condition on the metric to have a finite-meromorphic extension of the resolvent to C. Proposition 1.3... |

5 |
Distribution of resonances for spherical black holes
- Barreto, Zworski
- 1997
(Show Context)
Citation Context ...o O(x ∞ ). Remark: as a matter of fact, the usual examples are some particular cases of even metrics: the hyperbolic metrics perturbed on a compact [10, 11, 12, 19], the De Sitter-Schwarzschild model =-=[23]-=-, the almost-product type metrics [14]. The asymptotically Einstein manifolds of dimension n + 1 are only even modulo O(x n ) in general [7, 8]. Let us denote by Mah(X) the space of asymptotically hyp... |