Точные формулы и оценки для экстремальных длин семейств канонических петель на компактных римановых поверхностях

The problem of the calculation of the extremal lengths of the homotopic classes of the closed curves on the compact Riemann surfaces is considered. The expressions and estimates for the extremal lengths on the torus and on the compact Riemann surfaces of the special kind are adduced. The connection of this extremal lengths with the problem of the modules of Riemann surfaces is analysed.