  We study the surface diffusion flow acting on a class of general
(non--axisymmetric) perturbations of cylinders $\mathcal{C}_r$ in ${\rm I \!
R}^3$. Using tools from parabolic theory on uniformly regular manifolds, and
maximal regularity, we establish existence and uniqueness of solutions to
surface diffusion flow starting from (spatially--unbounded) surfaces defined
over $\mathcal{C}_r$ via scalar height functions which are uniformly bounded
away from the central cylindrical axis. Additionally, we show that
$\mathcal{C}_r$ is normally stable with respect to $2 \pi$--axially--periodic
perturbations if the radius $r > 1$,and unstable if $0 < r < 1$. Stability is
also shown to hold in settings with axial Neumann boundary conditions.
