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
DOI: 10.2140/pmp.2021.2.71
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.

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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