The $C^3$-null gluing problem: linear and nonlinear analysis

In this paper, we investigate the $C^3$-null gluing problem for the Einstein vacuum equations, that is, we consider the null gluing of up to and including third-order derivatives of the metric. In the regime where the characteristic data is close to Minkowski data, we show that this $C^3$-null gluing problem is solvable up to a $20$-dimensional space of obstructions. The obstructions correspond to $20$ linearly conserved quantities: $10$ of which are already present in the $C^2$-null gluing problem analysed by Aretakis, Czimek and Rodnianski, and $10$ are novel obstructions inherent to the $C^3$-null gluing problem. The $10$ novel obstructions are linearly conserved charges calculated from third-order derivatives of the metric.