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

TL;DR

This paper introduces elementary estimates for the capacity of non-contractible annuli on cylinders with constant and variable negative curvature, providing sharp bounds and new comparison methods.

Contribution

It presents novel elementary bounds for annuli capacity, including a symmetrization process for constant curvature and a comparison approach for variable curvature.

Findings

Lower bounds depend only on the annulus area

Sharpness demonstrated through examples

Comparison annuli with minimal capacity constructed

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}.