Gluing constructions of initial data sets play an important role in general relativity. Earlier in 1979, Schoen-Yau used gluing constructions with conformal deformations as a crucial step in their proof of the famous positive mass theorem. Corvino later refined this approach by introducing localized deformations that preserve the manifold’s asymptotic structure.

In this talk, I will survey recent theorems on localized deformation and their applications regarding rigidity and non-rigidity type results. I then outline extensions of these results to manifolds with boundary, including asymptotically flat regions outside black-hole horizons, and conclude with a brief discussion of the analytic challenges that arise in this boundary setting.