Mode stability for the wave and conformally coupled wave on subextremal Kerr–de Sitter

Prove mode stability on the real axis for Carter’s radial ordinary differential equation associated with the wave equation (μ^2_KG = 0) and with the conformally coupled wave equation (μ^2_KG = 2Λ/3) on all subextremal Kerr–de Sitter black hole backgrounds (a, M, l). Equivalently, establish the absence of nontrivial real-frequency mode solutions satisfying the ingoing/outgoing boundary conditions at the event and cosmological horizons for every real triplet (ω, m, ℓ), i.e., show that the corresponding Wronskian is nonzero for all such frequencies.

Background

The paper’s main results (boundedness and Morawetz estimates) are proved under an assumed mode stability condition (MS) for real-frequency solutions of Carter’s radial ODE. While mode stability is known in the asymptotically flat Kerr case through Whiting and quantitative refinements by Shlapentokh-Rothman, the corresponding results on Kerr–de Sitter are not established for the physically important massless (μ^2_KG = 0) and conformally coupled (μ^2_KG = 2Λ/3) wave equations.

The authors highlight that, unlike the Λ = 0 case, it is unknown whether mode stability holds across the full subextremal parameter range in Kerr–de Sitter. They note partial progress and related results but emphasize that a general proof in the Kerr–de Sitter setting remains open and would remove the need for assuming (MS) in their framework.