Abstract:
We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space carries a metric of positive sectional curvature, provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.

Abstract:
We establish an exponential decay estimate for linear waves on the Kerr-de Sitter slowly rotating black hole. Combining the cutoff resolvent estimate of arXiv:1003.6128 with the red-shift effect and a parametrix near the event horizons, we obtain exponential decay on the whole space.

Abstract:
We establish a Bohr-Sommerfeld type condition for quasi-normal modes of a slowly rotating Kerr-de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a=0 (Schwarzschild-de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.

Abstract:
We prove an asymptotic formula for the number of scattering resonances in a strip near the real axis when the trapped set is r-normally hyperbolic with r large and a pinching condition on the normal expansion rates holds. Our dynamical assumptions are stable under smooth perturbations and motivated by the setting of black holes. The key tool is a Fourier integral operator which microlocally projects onto the resonant states in the strip. In addition to Weyl law, this operator provides new information about microlocal concentration of resonant states.

Abstract:
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 $o(h^{-2})$ 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 ne-sided cutoff resolvent on the real line.

Abstract:
We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole rotates slowly enough, we show that these poles form a discrete subset of the complex plane. As an application we prove that the local energy of linear waves in that background decays exponentially once orthogonality to the zero resonance is imposed.

Abstract:
We apply the results of arXiv:1301.5633 to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a|> 1, then the energy norm of the solution is bounded by O(\lambda^{1/2} exp(-(\nu_min - \epsilon)t/2) + \lambda^(-\infty)), for t < C log\lambda, where \nu_min is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(\lambda exp(-(\nu_min -\epsilon)t) + \lambda^(-\infty)) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.

Abstract:
We consider a surface M with constant curvature cusp ends and its Eisenstein functions E_j(\lambda). These are the plane waves associated to the j-th cusp and the spectral parameter \lambda, (\Delta - 1/4 - \lambda^2)E_j = 0. We prove that as Re\lambda \to \infty and Im\lambda \to \nu > 0, E_j converges microlocally to a certain naturally defined measure decaying exponentially along the geodesic flow. In particular, for a sequence of \lambda's corresponding to scattering resonances, we find the microlocal limit of resonant states with energies away from the real line. This statement is similar to quantum unique ergodicity (QUE), which holds in certain other situations; however, the proof uses only the structure of the infinite ends, not the global properties of the geodesic flow. As an application, we also show that the scattering matrix tends to zero in strips separated from the real line.

Abstract:
We present dynamical properties of linear waves and null geodesics valid for Kerr and Kerr-de Sitter black holes and their stationary perturbations. The two are intimately linked by the geometric optics approximation. For the nullgeodesic flow the key property is the r-normal hyperbolicity of the trapped set and for linear waves it is the distribution of quasi-normal modes: the exact quantization conditions do not hold for perturbations but the bounds on decay rates and the statistics of frequencies are still valid.

Abstract:
In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework is relatively simple given modern microlocal analysis, and only takes a bit over a dozen pages after the statement of notation. It resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis, including semiclassical analysis. The rest of the paper is devoted to applications. Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose's b-structures. These include asymptotically Minkowski metrics, asymptotically de Sitter-type metrics on a blow-up of the natural compactification and Kerr-de Sitter-type metrics. The simplest application, however, is to provide a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces). The results include, in particular, a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. 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.