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

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