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 \(\Z\) 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.