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Boundedness for the wave equation on $C^1$ stationary axisymmetric perturbations of Kerr

Yakov Shlapentokh-Rothman, Mihai Tohaneanu

TL;DR

The paper develops a novel energy-boundedness framework for the wave equation on sub-extremal Kerr spacetimes by isolating high-frequency trapped modes with a pseudo-differential projection $ P_{HT}$. It avoids reliance on integrated local energy decay and extends boundedness results to stationary axisymmetric metrics that are merely $C^1$ close to Kerr, even when stable trapping is present. The approach combines $(t^*, extphi^*)$-Fourier analysis, twisted currents, and an interpolating operator $ox_ au$ to bridge perturbed and Kerr geometries, together with sophisticated physical-space currents and redshift/elliptic estimates. These linear estimates feed into potential quasilinear applications by providing top-order control and a robust mechanism to handle trapping without ILED, with implications for stability questions in Kerr and related spacetimes.

Abstract

On the full range of sub-extremal Kerr exterior spacetimes we give a new proof of energy boundedness for high-frequency projections of solutions to the wave equation onto trapped frequencies. A key feature of the new estimate is that it circumvents the use of an integrated local energy decay (ILED) statement. As an illustration of the robustness of the estimate, we use it to establish energy boundedness for solutions to the wave equation on stationary and axisymmetric metrics which are merely $C^1$ close to a sub-extremal Kerr spacetime. We show explicitly that such perturbed metrics may possess stably trapped null geodesics, and thus one does not expect ILED statements to hold.

Boundedness for the wave equation on $C^1$ stationary axisymmetric perturbations of Kerr

TL;DR

The paper develops a novel energy-boundedness framework for the wave equation on sub-extremal Kerr spacetimes by isolating high-frequency trapped modes with a pseudo-differential projection . It avoids reliance on integrated local energy decay and extends boundedness results to stationary axisymmetric metrics that are merely close to Kerr, even when stable trapping is present. The approach combines -Fourier analysis, twisted currents, and an interpolating operator to bridge perturbed and Kerr geometries, together with sophisticated physical-space currents and redshift/elliptic estimates. These linear estimates feed into potential quasilinear applications by providing top-order control and a robust mechanism to handle trapping without ILED, with implications for stability questions in Kerr and related spacetimes.

Abstract

On the full range of sub-extremal Kerr exterior spacetimes we give a new proof of energy boundedness for high-frequency projections of solutions to the wave equation onto trapped frequencies. A key feature of the new estimate is that it circumvents the use of an integrated local energy decay (ILED) statement. As an illustration of the robustness of the estimate, we use it to establish energy boundedness for solutions to the wave equation on stationary and axisymmetric metrics which are merely close to a sub-extremal Kerr spacetime. We show explicitly that such perturbed metrics may possess stably trapped null geodesics, and thus one does not expect ILED statements to hold.

Paper Structure

This paper contains 30 sections, 50 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Superradiance
  4. Stable/unstable trapping
  5. Quasilinear problems in the full sub-extremal range $|a| < M$
  6. Further directions
  7. Sketch of the proof
  8. Acknowledgments
  9. Preliminaries
  10. The Kerr spacetime
  11. Currents
  12. Standard Currents
  13. Twisted currents on Kerr spacetimes
  14. $(t^*,\phi^*)$-Fourier analysis
  15. The interpolating operator
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $M > 0$ and $a \in (-M,M)$. Then, for every $\epsilon > 0$, $k \geq 0$, and choice of small constants $\delta_1 > 0$ and $\delta_2 > 0$, we may choose the parameter $A_{\rm high}$ in Definition hightrappedpart sufficiently large and a time independent vector field $N$ which equals $T$ for $r$ la with $F$ smooth and compactly supported on $J^{+}(\Sigma_1)$ and $\left(n_{\Sigma_0}\psi,\psi\right

Theorems & Definitions (123)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.3
  • Theorem 1.2
  • ...and 113 more