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

TL;DR

This paper establishes a nonlinear $C^k$-gluing theorem for vacuum gravitational fields in Bondi gauge, extending previous work to higher dimensions, arbitrary differentiability, and backgrounds with cosmological constant.

Contribution

It generalizes the $C^2$-gluing results to higher differentiability and dimensions, including backgrounds with cosmological constant, near Birmingham-Kottler solutions.

Findings

Proves a nonlinear characteristic $C^k$-gluing theorem for vacuum gravitational fields.

Extends previous $C^2$-gluing results to arbitrary $k$ and higher dimensions.

Applies to backgrounds with cosmological constant near Birmingham-Kottler solutions.

Abstract

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.