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This document proves global boundedness and decay for axisymmetricperturbations of a known solution to the wave map problem from a slowlyrotating $|a|\ll M$ Kerr spacetime to the hyperbolic plane. This problem ismotivated by the general axisymmetric stability of Kerr conjecture and wasfirst posed by Ionescu and Klainerman in [IK14]. Two particular developments inthis paper, the treatment of terms near the axis of symmetry and the use of adecay hierarchy for energy estimates on uniformly spacelike hypersurfaces, canbe used for a variety of similar problems.

Global Stability of the Nontrivial Solutions to the Wave Map Problem  from Kerr $|a|\ll M$ to the Hyperbolic Plane under Axisymmetric Perturbations  Preserving Angular Momentum