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
  This document proves global boundedness and decay for axisymmetric perturbations of a known solution to the wave map problem from a slowly rotating $|a|\ll M$ Kerr spacetime to the hyperbolic plane. This problem is motivated by the general axisymmetric stability of Kerr conjecture and was first posed by Ionescu and Klainerman in [IK14]. Two particular developments in this paper, the treatment of terms near the axis of symmetry and the use of a decay hierarchy for energy estimates on uniformly spacelike hypersurfaces, can be used for a variety of similar problems. 
John Stogin
math.AP