Spectral gaps for normally hyperbolic trapping

Dyatlov, Semyon

Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 55-82.

Cet article démontre l’existence d’une bande sans résonances pour des ensembles captés normalement hyperboliques de codimension 2, dont les variétés entrantes/sortantes sont lisses. Une application importante est la décroissance exponentielle des ondes pour les trous noirs de Kerr et Kerr–de Sitter. On retrouve la taille optimale de la bande et on y donne une borne $o\left(h^{-2}\right)$ de la résolvante. On démontre alors l’existence de bandes plus profondes sans résonances si l’ensemble capté est $r$-normalement hyperbolique et satisfait une condition de pincement. On donne aussi une borne inférieure sur la norme de la résolvante tronquée sur l’axe réel.

We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming/outgoing tails. An important application is wave decay on Kerr and Kerr–de Sitter black holes. We recover the optimal size of the strip and give an $o\left(h^{-2}\right)$ resolvent bound there. We next show existence of deeper resonance free strips under the $r$-normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the one-sided cutoff resolvent on the real line.

Classification : 35B34, 37D05
Keywords: spectral gaps, normally hyperbolic trapping, black holes
Mot clés : trous spectraux, ensembles captés normalement hyperboliques, trous noirs

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