Peter Hintz

The global non-linear stability of the Kerr-de Sitter family of black holes

We establish the full global non-linear stability of the Kerr-de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extending the linear and non-linear analysis on black hole spacetimes described in a sequence of earlier papers by the authors: We develop a general framework which enables us to deal systematically with the diffeomorphism invariance of Einsteins equations. In particular, the iteration scheme used to solve Einsteins equations automatically finds the parameters of the Kerr-de Sitter black hole that the solution is asymptotic to, the exponentially decaying tail of the solution, and the gauge in which we are able to find the solution; the gauge here is a wave map/DeTurck type gauge, modified by source terms which are treated as unknowns, lying in a suitable finite-dimensional space.

Global Analysis of Quasilinear Wave Equations on Asymptotically Kerr-de Sitter Spaces

We consider quasilinear wave equations on manifolds for which infinity has a structure generalizing that of Kerr-de Sitter space; in particular the trapped geodesics form a normally hyperbolic invariant manifold. We prove the global existence and decay, to constants for the actual wave equation, of solutions. The key new ingredient compared to earlier work by the authors in the semilinear case [33] and by the first author in the non-trapping quasilinear case [30] is the use of the Nash-Moser iteration in our framework.

Analysis of linear waves near the Cauchy horizon of cosmological black holes

We show that linear scalar waves are bounded and continuous up to the Cauchy horizon of Reissner-Nordstrom-de Sitter and Kerr-de Sitter spacetimes, and in fact decay exponentially fast to a constant along the Cauchy horizon. We obtain our results by modifying the spacetime beyond the Cauchy horizon in a suitable manner, which puts the wave equation into a framework in which a number of standard as well as more recent microlocal regularity and scattering theory results apply. In particular, the conormal regularity of waves at the Cauchy horizon - which yields the boundedness statement - is a consequence of radial point estimates, which are microlocal manifestations of the blue-shift and red-shift effects.

Asymptotics for the wave equation on differential forms on Kerr–de Sitter space

We study asymptotics for solutions of Maxwells equations, in fact of the Hodge-de Rham equation

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

(d+\delta)u=0

Reconstruction of Lorentzian Manifolds from Boundary Light Observation Sets

without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which in particular include Schwarzschild-de Sitter spaces of all spacetime dimensions

Resonances for Obstacles in Hyperbolic Space

n\geq 4

Wave decay for star-shaped obstacles in ℝ 3 : papers of Morawetz and Ralston revisited

. We prove that solutions decay exponentially to

Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes

0

Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime

or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and in particular prove analogous results on Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwells equations, the Hodge-de Rham equation and the wave equation on differential forms on Kerr-de Sitter spacetimes with small angular momentum.