Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

By C. Guillarmou. 2003

171 citations recorded by the source corpus.

We show that the resolvent of the Laplacian on asymptotically hyperbolic spaces extends meromorphically with finite rank poles to the complex plane if and only if the metric is `even' (in a sense). If it is not even, there exist some cases where the resolvent has an essential singularity in the non-physical sheet.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/acf5c4212392a0e8fb47e6f37d04fa1b6ebb7a54

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Resonance projectors and asymptotics for r -normally hyperbolic trapped sets (2013, methodology): “poles of the meromorphic continuation of the Schwartz kernel of the resolvent R g(!) := (h2( g 1(n 1)2=4) !2) : L2(M) !L2(M); Im!>0; to the entire C, rst constructed in [MaMe] with improvements by [Gu] { see [Va12, Theorem 5.1]. We can now proceed similarly to x4.3, using that the regions fx>&”
Fractional Laplacian in conformal geometry (2010, background): “MazzeoMelrose [23] for most of the values of the parameter s, and Guillarmou [18] for a closer look at the remaining poles).”
Gluing Semiclassical Resolvent Estimates via Propagation of Singularities (2010, methodology): “is now (H z2) 1 for Imz>0 or its meromorphic continuation for Imz<0. This follows from the substitution h 2z = 1 + ; Rez= h 1: On the other hand, work of Mazzeo-Melrose [MaMe87] and Guillarmou [Gui05] (see also [Vas10, Vas11]) shows that the weighted resolvent x 5 =2+ 2R(z)x continues meromorp”
Fractal Weyl laws for asymptotically hyperbolic manifolds (2012, unknown): “AppendixBfor more details. Res= n 1 2 . As an operator from L 2 comp to L loc it extends meromorphically to C, with nite rank poles called resonances, as shown by Mazzeo{Melrose [MaMe] and Guillarmou [Gu] for asymptotically hyperbolic manifolds, Guillope{Zworski [GuZw95a] when curvature near innity ”
Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov) (2010, methodology): “The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.”
Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole (2010, methodology): “It should be noted that, unlike [11] or [31], the construction of Rg(ω) in the present paper does not use the theorem of Mazzeo–Melrose [28] on the meromorphic continuation of the resolvent on spaces with asymptotically constant negative curvature (see also [21]).”