In this paper, we extend the work in \cite{ D}\cite{ ChrusLiWe}\cite{ ChrusCo}\cite{ Co}. We weaken the asymptotic conditions on the second fundamental form, and we also give an $L^{ 6}-$norm bound for the difference between general data and Extreme Kerr data or Extreme Kerr-Newman data by proving convexity of the renormalized Dirichlet energy when the target has non-positive curvature. In particular, we give the first proof of the strict mass/angular momentum/charge inequality for axisymmetric Einstein/Maxwell data which is not identical with the extreme Kerr-Newman solution.
Categories: math.DG gr-qc