Characteristic Gluing with $\mathrm{\Lambda }$ 1. Linearised Einstein equations on four-dimensional spacetimes

We establish a gluing theorem for linearised vacuum gravitational fields in Bondi gauge on a class of characteristic surfaces in static vacuum four-dimensional backgrounds with cosmological constant $\mathrm{\Lambda }\in \mathbb{R}$ and arbitrary topology of the compact cross-sections of the null hypersurface. This generalises and complements, in the linearised case, the pioneering analysis of Aretakis, Czimek and Rodnianski, carried-out on light-cones in Minkowski spacetime.