How do metrics behave under joining along a manifold embedded in the boundary?

This is, more-or-less, part of Problem 4.66 in Kirby's List:

**Problem 4.66** How do metrics (e.g. Riemannian, Lorentz, constant curvature) behave under standard topological constructions such as connected sum, plumbing, handle addition? Same question for $\eta$-invariants, moduli spaces, etc.

So, in theory it is an open problem. However, Kirby states of the problem:

*Much has probably been done on this open ended problem, and the editor has not attempted to update it.*

So, it would seem that this question may indeed have an answer in the literature. My question is: What is that answer and where does one find it?