  We prove a nonlinear characteristic $C^k$-gluing theorem for vacuum
gravitational fields in Bondi gauge for a class of characteristic hypersurfaces
near static vacuum $n$-dimensional backgrounds, $n\ge 3$, with any finite $k$,
with cosmological constant $ \Lambda \in \mathbb{R}$, near Birmingham-Kottler
backgrounds. This generalises the $C^2$-gluing of Aretakis, Czimek and
Rodnianski, carried-out near light cones in four-dimensional Minkowski
spacetime.
