Regularity of the Future Event Horizon in Perturbations of Kerr

The goal of the paper is to show that the event horizons of the spacetimes constructed in \cite{KS}, see also \cite{KS-Schw}, in the proof of the nonlinear stability of slowly rotating Kerr spacetimes $\mathcal{K}(a_0,m_0)$, are necessarily smooth null hypersurfaces. Moreover we show that the result remains true for the entire range of $\mathrm{\mid }{a}_0\mathrm{\mid }\mathrm{/}{m}_0$ for which stability can be established.