Vol. 2, No. 1, 2021

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Normally hyperbolic trapping on asymptotically stationary spacetimes

Peter Hintz

Vol. 2 (2021), No. 1, 71–126
Abstract

We prove microlocal estimates at the trapped set of asymptotically Kerr spacetimes: these are spacetimes whose metrics decay inverse polynomially in time to a stationary subextremal Kerr metric. This combines two independent results. The first one is purely dynamical: we show that the stable and unstable manifolds of a decaying perturbation of a time-translation-invariant dynamical system with normally hyperbolic trapping are smooth and decay to their stationary counterparts. The second, independent, result provides microlocal estimates for operators whose null-bicharacteristic flow has a normally hyperbolic invariant manifold, under suitable nondegeneracy conditions on the stable and unstable manifolds; this includes operators on closed manifolds, as well as operators on spacetimes for which the invariant manifold lies at future infinity.

Keywords
normally hyperbolic trapping, propagation of singularities, wave equations, black holes
Mathematical Subject Classification 2010
Primary: 37D05, 58J47
Secondary: 35L05, 58J40, 83C57
Milestones
Received: 20 December 2019
Accepted: 25 November 2020
Published: 16 March 2021
Authors
Peter Hintz
Massachusetts Institute of Technology
Department of Mathematics
Cambridge, MA
United States