Let $R$ be a Riemannian manifold without a biharmonic Green function defined on it and $\Omega$ a domain in $R$. A necessary and sufficient condition is given for the existence of a biharmonic Green function on $\Omega$.

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis and Kral-Wedland . We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the...