Characteristic Gluing with $\mathrm{\Lambda }$: III. High-differentiability nonlinear gluing

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 $\mathrm{\Lambda }\in \mathbb{R}$, 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.