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

TL;DR

This paper proves energy boundedness for solutions to the wave equation on Kerr spacetimes and nearby metrics, avoiding traditional ILED methods and demonstrating robustness even with only $C^1$ regularity.

Contribution

It introduces a new proof technique for energy boundedness that does not rely on ILED, applicable to $C^1$ perturbations of Kerr.

Findings

Energy boundedness holds for high-frequency solutions on Kerr.

The new estimate applies to $C^1$ close metrics with trapped null geodesics.

Traditional ILED methods are circumvented in the proof.

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.