Gluing and wormholes for the Einstein constraint equations

TL;DR

This paper presents a new method for gluing solutions of the Einstein constraint equations, enabling the creation of complex initial data sets with handles or wormholes while maintaining proximity to original data.

Contribution

The authors develop a general gluing theorem for constant mean curvature solutions, allowing connected sums and wormhole attachments in various geometric settings.

Findings

Enables connected sum and wormhole constructions for Einstein initial data

Maintains closeness to original data away from the glued region

Applicable to compact, asymptotically Euclidean, and hyperbolic manifolds

Abstract

We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes.