Biharmonic extensions in Riemannian manifolds (Q1572921)

Let \(R\) be a hyperbolic Riemannian manifold. It is known that given a harmonic function \(h\) outside a compact set, there always exists a harmonic function \(H\) on \(R\) such that \(H-h\) is bounded outside a compact set. The authors prove that a biharmonic extension in \(R\) is possible if and only if there exist potentials \(p>0\) and \(q>0\) in \(R\) (\(q\) is bounded outside a compact set) such that \(\Delta q=p\), where \(\Delta \) is the Laplace-Beltrami operator in \(R\).