Crucial counterexamples in
the biharmonic classification theory of Riemannian 2-manifolds have been
deduced from certain general principles. The present note is methodological in
nature: the aim is to supplement the theory by showing that very simple
counterexamples can be directly constructed. Whereas earlier work has been
devoted to the class H2 of nonharmonic biharmonic functions, here the class
W of all biharmonic functions is discussed. This is of interest, since the
classes OWB and OWD of Riemannian manifolds without (nonconstant)
bounded or Dirichlet finite biharmonic functions are strictly contained in the
corresponding classes OH2B and OH2D, as is seen by endowing the unit disk
with a suitable conformal metric. Moreover, for W-functions the biharmonic
equation need not be reduced to the Poisson equation but can be dealt with
directly.
