Using the results of the previous chapter, we first prove that a suitably modified Stokes operator on a manifold with straight cylindrical ends is invertible. As in the previous chapters, this invertibility result allows us to introduce the layer potentials of the modified Stokes operator. The properties of the layer potentials are established using the pseudodifferential operators and the results of the previous chapter. In particular, we provide complete proofs for the usual “jump” and “limit properties” for these layer potential operators (although, some of the more tedious, but elementary calculations are relegated to the Appendix). The invertibility of the relevant boundary integral operators then allows us to obtain well-posedness results for the modified Stokes system with Dirichlet boundary conditions on domains with straight cylindrical ends. We also obtain a description of the inverse operator using the pseudodifferential calculi developed in the previous chapter.