In this talk, I will describe a new approach to general relativistic initial data gluing based on explicit solution operators for the linearized constraint equation with prescribed support properties. In particular, we retrieve and optimize -- in terms of positivity, regularity, size and/or spatial decay requirements -- obstruction-free gluing originally put forth by Czimek-Rodnianski. Notably, our proof of the strengthened obstruction-free gluing theorem relies on purely spacelike techniques, rather than null gluing as in the original approach.
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