Inequalities for the capacity of non-contractible annuli on cylinders of constant and variable negative curvature

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. To obtain this result we use projection of gradients on curves to obtain a lower bound on the capacity, which we call directional capacity. In the case of constant curvature we then apply a symmetrization process that results in an annulus of minimal directional capacity. For this annulus the lower bound on the capacity is sharp. In the case of variable negative curvature we obtain the lower bound by constructing a comparison annulus with the same area but lower directional capacity on a cylinder of constant curvature. The methods developed here have been applied to estimate the energy of harmonic forms on Riemann surfaces in Muetzel (Math Zeitschrift, 2012, arXiv: 1202.0782).