Abstract:  We present results concerning the norm convergence of
resolvents for wild perturbations of the Laplace-Beltrami operator.  We
study here manifolds with an increasing number of small (i.e., short and
thin) handles added. We consider two situations: if the small handles
are distributed too sparse the limit operator is the unperturbed one on
the initial manifold, the handles are fading. On the other hand, if the
small handles are dense in certain regions the limit operator is the
Laplace-Beltrami operator acting on functions which are identical on the
two parts joined by the handles, the handles hence produce adhesion. Our
results also apply to non-compact manifolds. Our work is based on a norm
convergence result for operators acting in varying Hilbert spaces.