Energy-Morawetz estimates for Teukolsky equations in perturbations of Kerr

TL;DR

This paper establishes energy and Morawetz estimates for solutions to Teukolsky equations in perturbed Kerr spacetimes, advancing the mathematical understanding necessary for proving the nonlinear stability of Kerr black holes.

Contribution

It introduces tensorial Teukolsky equations in perturbations of Kerr and develops new techniques for energy estimates, including a scalarization procedure, crucial for stability analysis.

Findings

Proved energy and Morawetz estimates for tensorial Teukolsky equations.

Extended microlocal multiplier techniques to tensorial waves.

Introduced a scalarization method for tensorial wave analysis.

Abstract

In this paper, we prove energy and Morawetz estimates for solutions to Teukolsky equations in spacetimes with metrics that are perturbations, compatible with nonlinear applications, of Kerr metrics in the full subextremal range. The Teukolsky equations are written in tensorial form using the non-integrable formalism in \cite{GKS22}, and we follow the approach in \cite{Ma} of relying on a Teukolsky wave/transport system. The estimates are proved by extending the ideas from our earlier result \cite{MaSz24} on the corresponding problem for the scalar wave, notably the use of $r$-foliation-adapted microlocal multipliers for the wave part, and by incorporating techniques from \cite{Ma} to control the linear coupling terms between the components of the Teukolsky wave/transport system. Additionally, in order to adapt the methodology of \cite{MaSz24} to tensorial waves, we introduce a well-suited regular scalarization procedure which is of independent interest. This result, alongside our companion paper \cite{MaSz24}, is an essential step towards extending the current proof of Kerr stability in \cite{GCM1} \cite{GCM2} \cite{KS:Kerr} \cite{GKS22} \cite{Shen}, valid in the slowly rotating case, to a complete resolution of the Kerr stability conjecture, i.e., the statement that the Kerr family of spacetimes is nonlinearly stable for all subextremal angular momenta.