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Boundedness and Morawetz estimates on subextremal Kerr de Sitter

Georgios Mavrogiannis

TL;DR

This work analyzes the Klein–Gordon equation on subextremal Kerr–de Sitter spacetimes and proves boundedness and Morawetz (integrated decay) estimates under a mode-stability assumption for real frequencies of Carter’s radial ODE. The authors extend the Dafermos–Rodnianski framework by combining fixed-frequency analysis (Carter separation) with a continuity argument in the rotation parameter $a$, addressing superradiance and trapping in the de Sitter setting. A key innovation is the use of red-shift multipliers, twisted currents, and a carefully designed $Z^\star$-based multiplier to obtain non-degenerate estimates near the horizons and in the trapping region; axisymmetric data yield unconditional bounds, while the general case relies on (MS). These results pave the way for a companion paper proving a relatively non-degenerate integrated estimate and exponential decay, and a subsequent paper addressing nonlinear stability for quasilinear waves on Kerr–de Sitter backgrounds. The techniques contribute to the broader program of dynamical stability for waves on rotating cosmological black holes, with potential implications for nonlinear analysis and gravitational physics in de Sitter-like spacetimes.

Abstract

We study the Klein--Gordon equation $\Box_{g_{a,M,l}}ψ-μ^2_{\textit{KG}}ψ=0$ on subextremal Kerr--de Sitter black hole backgrounds with parameters $(a,M,l)$, where $l^2=\frac{3}Λ$. We prove boundedness and Morawetz estimates assuming an appropriate mode stability statement for real frequency solutions of Carter's radial ode. Our results in particular apply in the very slowly rotating case $|a|\ll M,l$, and in the case where the solution~$ψ$ is axisymmetric. This generalizes the work of Dafermos--Rodnianski \cite{DR3} on Schwarzschild--de~Sitter. The boundedness and Morawetz results of the present paper will be used in our companion \cite{mavrogiannis2} to prove a `relatively non-degenerate integrated estimate' for subextremal Kerr--de Sitter black holes~(and as a consequence exponential decay). In a forthcoming paper \cite{mavrogiannis3}, this will immediately yield nonlinear stability results for quasilinear wave equations on subextremal Kerr--de Sitter backgrounds.

Boundedness and Morawetz estimates on subextremal Kerr de Sitter

TL;DR

This work analyzes the Klein–Gordon equation on subextremal Kerr–de Sitter spacetimes and proves boundedness and Morawetz (integrated decay) estimates under a mode-stability assumption for real frequencies of Carter’s radial ODE. The authors extend the Dafermos–Rodnianski framework by combining fixed-frequency analysis (Carter separation) with a continuity argument in the rotation parameter , addressing superradiance and trapping in the de Sitter setting. A key innovation is the use of red-shift multipliers, twisted currents, and a carefully designed -based multiplier to obtain non-degenerate estimates near the horizons and in the trapping region; axisymmetric data yield unconditional bounds, while the general case relies on (MS). These results pave the way for a companion paper proving a relatively non-degenerate integrated estimate and exponential decay, and a subsequent paper addressing nonlinear stability for quasilinear waves on Kerr–de Sitter backgrounds. The techniques contribute to the broader program of dynamical stability for waves on rotating cosmological black holes, with potential implications for nonlinear analysis and gravitational physics in de Sitter-like spacetimes.

Abstract

We study the Klein--Gordon equation on subextremal Kerr--de Sitter black hole backgrounds with parameters , where . We prove boundedness and Morawetz estimates assuming an appropriate mode stability statement for real frequency solutions of Carter's radial ode. Our results in particular apply in the very slowly rotating case , and in the case where the solution~ is axisymmetric. This generalizes the work of Dafermos--Rodnianski \cite{DR3} on Schwarzschild--de~Sitter. The boundedness and Morawetz results of the present paper will be used in our companion \cite{mavrogiannis2} to prove a `relatively non-degenerate integrated estimate' for subextremal Kerr--de Sitter black holes~(and as a consequence exponential decay). In a forthcoming paper \cite{mavrogiannis3}, this will immediately yield nonlinear stability results for quasilinear wave equations on subextremal Kerr--de Sitter backgrounds.

Paper Structure

This paper contains 101 sections, 72 theorems, 758 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Previous results on Kerr(--de Sitter) backgrounds
  3. Review of DR3 and DR2
  4. The Morawetz (degenerate integrated decay) estimate of DR3 on Schwarzschild--de Sitter
  5. The Morawetz estimate of DR2 on g Kerr
  6. Preliminaries for the statement and proof of Theorem \ref{['main theorem 1']}
  7. Carter's radial ode in Kerr--de Sitter
  8. An energy identity for the solution u
  9. Superradiant frequencies and mode stability
  10. The rough version of Theorem \ref{['main theorem 1']}
  11. Discussion of the proof
  12. The de Sitter frequencies and trapping
  13. The continuity argument
  14. Constant solutions when PDFstring
  15. Outline
  16. ...and 86 more sections

Key Result

Theorem \ref{main theorem 1}

Let $l>0$, $\mu^2_{KG}\geq 0$ and let $(a,M,l)$ be subextremal Kerr--de Sitter black hole parameters. Moreover, assume that the following condition holds Let $\psi$ be a solution of the Klein--Gordon equation eq: kleingordon. Then, we have the following energy estimates for all $0\leq\tau_1\leq \tau_2$, where $Z^\star$ is a regular spacelike vector field on $\{t^\star\geq 0\}$ that extends smoot

Figures (5)

  • Figure 1: Carter--Penrose diagram of Kerr--de Sitter
  • Figure 2: The $\Delta$ polynomial with 4 real roots
  • Figure 3: The red graph is $h$, the green graph is $y$
  • Figure 4: The cases of \ref{['eq: sec: trapped frequencies, eq 1']} respectively.
  • Figure 5: The shaded region is $\tilde{\mathcal{B}}_l$

Theorems & Definitions (203)

  • Theorem \ref{main theorem 1}: rough version
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Definition 2.3
  • Remark 2.1
  • ...and 193 more