Maskit combinations of Poincaré–Einstein metrics

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3-manifold which is a finite connected sum of quotients of S3 and S2×S1 bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.