On the global stability of the wave-map equation in Kerr spaces with small angular momentum

Abstract

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $\Phi$ defined from a fixed Kerr solution $\KK(M,a)$, $0\le a<M$, with values in the two dimensional hyperbolic space $\HHH^2$. A particular such wave map is given by the complex Ernst potential associated to the axial Killing vectorfield $\mathbb{Z}$ of $\KK(M,a)$. We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of $\KK(M,a)$, for all $0\le a<M$ and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.

References in corpus (2)

• On the uniqueness of smooth, stationary black holes in vacuum
• Errata for ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'', ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'', and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation''

Cited by in corpus (1)

• Bounds for axially symmetric linear perturbations for the extreme Kerr black hole