Biharmonic classification of Riemannian spaces (Q1423452)

Let \(R\) be a Riemann surface or a Riemannian manifold of dimension \(\geq 2\). If \(R\) is hyperbolic, then it is said to be a biharmonic-extension space, if for a given biharmonic function \(b\) outside a compact set there exists a biharmonic function \(B\) on \(R\) such that \(b-B\) is bounded near infinity. An equivalent characterization of this property is given, and a lot of its consequences in connection with other properties (such as the space being tapered or of Almansi type) are studied. A short section is devoted to the wider class of hyperbolic spaces satisfying the so-called flux condition. The last section briefly deals with the case of a parabolic \(R\), for which in the definition of the biharmonic-extension property \(b\)-\(B\) has to be replaced by \(b\)-\(B\)-\(\alpha E\), where \(E\) is the Evans function with pole at some point and \(\alpha\) is a constant.