Abstract
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}.