Refined gluing for Vacuum Einstein constraint equations

TL;DR

This paper extends a gluing technique for vacuum Einstein initial data sets to asymptotically Euclidean and hyperbolic contexts, allowing localized modifications to generate new solutions.

Contribution

It adapts the connected sum gluing method to new geometric settings and demonstrates the possibility of localized solution modifications.

Findings

Gluing method extended to asymptotically Euclidean and hyperbolic cases.

Gluing procedure can be localized to produce new solutions.

New solutions match original data outside a neighborhood of the gluing locus.

Abstract

We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in any case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.