Gluing and Wormholes for the Einstein Constraint Equations

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.

Generalized connected sum construction for scalar flat metrics

“…An important corollary of the second construction is that it provides a gluing construction for time symmetric initial data sets in the context of the Einstein Constraint equations. In this sense our result partially completes the work of Isenberg et al [8], which treats the point-wise connected sum of nontime symmetric Cauchy data.…”

“…They have been used to understand solutions to problems arising from the geometry (minimal and constant mean curvature surfaces [13,14], constant scalar curvature metrics [9,12,15], and recently even Einstein metrics [1]) and from the physic (Einstein constraint equations [7] and [8]). However most of the results are concerned with the connected sum at points (point-wise connected sum), whereas the case of connected sum along a submanifold (generalized connected sum or fiber sum) has received less attention.…”

“…Therefore Theorem 2 automatically provides a generalized gluing construction for non Ricci flat initial data sets, in the spirit of [8].…”

“…Before starting, let us remark that in the expression of F ε (v) (8) it is always possible to choose S = S(ε, v) in such a way that M F ε dvol g ε = 0, in fact it is sufficient to set…”

“…In fact we recall that our main purpose is to produce a solution to the Eq. (8). We also recall that F ε (v) represents the error term in this equation, and that we can think of λ F ε (v) as the rough projection of the error term along the first nonconstant eigenfunction β ε of the linearized operator g ε .…”