The C³-Null Gluing Problem: Linear and Nonlinear Analysis

In this paper, we investigate the C³ 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 C³ 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 C² C 2 -null gluing problem analysed by Aretakis, Czimek and Rodnianski [2–4], and 10 are novel obstructions inherent to the C³ C 3 -null gluing problem. The 10 novel obstructions are linearly conserved charges calculated from third-order derivatives of the metric.