Resonance expansions for tensor-valued waves on asymptotically Kerr-de Sitter spaces

Abstract

In recent joint work with Vasy, we analyze the low energy behavior of differential form-valued waves on black hole spacetimes. In order to deduce asymptotics and decay from this, one in addition needs high energy estimates for the wave operator acting on sections of the form bundle. The present paper provides these on perturbations of Schwarzschild-de Sitter spaces in all spacetime dimensions $n\ge 4$. In fact, we prove exponential decay, up to a finite-dimensional space of resonances, of waves valued in any finite rank subbundle of the tensor bundle, which in particular includes differential forms and symmetric tensors. As the main technical tool for working on vector bundles that do not have a natural positive definite inner product, we introduce pseudodifferential inner products, which are inner products depending on the position in phase space.

29 pages, 1 figure. v2 is the published version, with improved exposition

References in corpus (3)

• The wave equation on Schwarzschild-de Sitter spacetimes
• Asymptotics for the wave equation on differential forms on Kerr-de Sitter space
• Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry

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• The global non-linear stability of the Kerr-de Sitter family of black holes
• Strong cosmic censorship in charged black-hole spacetimes: still subtle
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• A sharp version of Price's law for wave decay on asymptotically flat spacetimes
• Asymptotics for the wave equation on differential forms on Kerr-de Sitter space
• Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
• Normally hyperbolic trapping on asymptotically stationary spacetimes
• An index theorem on asymptotically static spacetimes with compact Cauchy surface
• Uniqueness of Kerr-Newman-de Sitter black holes with small angular momenta
• Mathematical Study of Scattering Resonances
• Microlocal analysis near null infinity in asymptotically flat spacetimes