Biharmonic Green functions in a Riemannian manifold (Q1303601)

The main result of this paper asserts that there exists a biharmonic Green function on a Riemannian manifold \(R\) if and only if there exists a non-harmonic positive superharmonic function \(s\) on \(R\) such that \(\Delta s\) is also superharmonic on \(R\). Here \(\Delta\) denotes the Laplace-Beltrami operator on \(R\). A further equivalent condition is that there is a potential \(g>0\) on \(R\) such that \(\Delta g\) is also a potential. The author then relates this result to work of \textit{L. Sario} [J. Aust. Math. Soc., Ser. A 21, 155-165 (1976; Zbl 0316.31008)].