In this talk I will present a characterization of Kerr-de Sitter among spacetimes satisfying the Einstein vacuum field equations with positive cosmological constant, admitting a smooth conformal compactification and a non-trivial Killing vector field. The characterization will be given in terms of the so-called asymptotic Killing data at null infinity.

We will discuss new developments in the analysis of asymptotic foliations by prescribed curvature in relativistic initial data sets with prescribed asymptotic decay, generalizing results by Huisken and Yau. We will relate these foliations to the definition of the center of mass of the initial data sets under consideration. The results presented are joint work with Cortier--Sakovich and with Nerz.