Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)
The reviewed article establishes meromorphic continuation of the scattering resolvent in a variety of situations which are united by a single set of microlocal assumptions. Meromorphic continuation of the scattering resolvent gives meromorphic continuation of the Fourier transform (in time) of solutions to the wave equation; together with additional high-energy estimates, it provides an expansion of waves into exponentially decaying and oscillating terms. The frequencies of these terms, called resonances, are the poles of the meromorphic continuation. The author uses microlocal estimates (ellipticity, propagation of singularities, and radial points estimates) to obtain the meromorphic continuation of the resolvent by expressing it as the solution to a Fredholm problem. The situations to which this approach applies include: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] Kerr-de Sitter black hole spacetimes in general relativity, where the reviewed work gives exponential decay of waves (under additional assumptions and relying on previous work by Wunsch-Zworski and Datchev-Vasy) and is the first step towards establishing stability of these spacetimes under Einstein equations; \item[{\(\bullet\)}] asymptotically hyperbolic spaces such as convex co-compact hyperbolic quotients, where the reviewed work has since been used to obtain fractal Weyl upper bounds and other fine results on resonances in the high frequency limit; \item[{\(\bullet\)}] asymptotically flat spaces, where the reviewed work has since been used to obtain a better understanding of wave decay and radiation patterns. \end{itemize}} The methods used are stable under perturbations and do not rely on the symmetries of the studied systems beyond their stationarity. What is more, the paper gives high frequency estimates which can be used to treat the infinite ends/event horizons of the system as a black box; this can be combined with detailed analysis of the trapping to obtain various high-frequency results on resonances.
meromorphic continuation of scattering resolvent
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