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We prove a gluing theorem for linearised vacuum gravitational fields in Bondigauge on a class of characteristic hypersurfaces in static vacuum$(n+1)$-dimensional backgrounds with cosmological constant $ \Lambda \in\mathbb{R}$, $n\ge 4$. This generalises, in the linearised case, the pioneeringanalysis of Aretakis, Czimek and Rodnianski, carried-out on light cones infour-dimensional Minkowski spacetime.

Characteristic Gluing with $\Lambda$: II. Linearised equations in higher dimensions