In this paper we consider the equivariant 2+1 dimensional Einsteinwave map system and show that if the target satisfies the so called Grillakis condition, then global existence holds. In view of the fact that the 3+1 vacuum Einstein equations with a spacelike translational Killing field reduce to a 2+1 dimensional Einstein-wave map system with target the hyperbolic plane, which in particular satisfies the Grillakis condition, this work proves global existence for the equivariant class of such spacetimes.

We show that there exists a 1-parameter family of positive-definite and conserved energy functionals for axially symmetric Newman-Penrose-Maxwell scalars on the maximal spacelike hypersurfaces in the exterior of Kerr black holes. It is also shown that the Poisson bracket within this 1-parameter family of energies vanishes on the maximal hypersurfaces.

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We consider the Cauchy problem of 2+1 equivariant wave maps coupled to Einstein's equations of general relativity and prove that two separate (nonlinear) subclasses of the system disperse to their corresponding linearized equations in the large. Global asymptotic behaviour of 2+1 Einstein-wave map system is relevant because the system occurs naturally in 3+1 vacuum Einstein's equations.

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