Gluing Initial Data Sets for General Relativity

TL;DR

This paper introduces an optimal local gluing method for general relativistic initial data, enabling the construction of complex space-times with specific properties while preserving data outside small neighborhoods.

Contribution

It presents a novel, fully local gluing technique applicable to generic initial data sets in general relativity, with natural hypotheses and broad applicability.

Findings

Constructed cosmological vacuum space-times without constant mean curvature surfaces.

Established a local gluing method applicable to generic initial data.

Demonstrated the existence of space-times with prescribed local modifications.

Abstract

We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.