  Using a new method we give elementary estimates for the capacity of
non-contractible annuli on cylinders and provide examples, where these
inequalities are sharp. Here the lower bound depends only on the area of the
annulus. In the case of constant curvature this lower bound is obtained with
the help of a symmetrization process that results in an annulus of minimal
capacity. In the case of variable negative curvature we obtain the lower bound
by constructing a comparison annulus with the same area but lower capacity on a
cylinder of constant curvature. The methods developed here have been applied to
estimated the energy of harmonic forms on Riemann surfaces in \cite{mu}.
