Completeness and existence of bounded biharmonic functions on a riemannian manifold

Abstract

A.S. Galbraith has communicated to us the following intriguing problem: does the completeness of a manifold imply, or is it implied by, the emptiness of the class H 2 B of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry H 2 B-functions, and one might suspect that this is always the case. We shall show, however, that there do exist complete manifolds of any dimension that carry H 2 B-functions. Moreover, there exist both complete and incomplete manifolds not permitting these functions, and, trivially, incomplete manifolds possessing them.