Holomorphic mappings and moduli of Riemann surfaces

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Let R be a closed Riemann surface of positive genus g. In the previous research we have shown that the extremal lengths of homology classes satisfy an algebraic equation of degree 2g depending on the genus. In the present research we prove a more primitive algebraic equation of degree 2, which easily leads us to the previously shown equation. Moreover, applying the equation of degree 2, we find a new symmetry of closed Riemann surfaces of genus 2. Also, we give a criterion for an open Riemann surface of positive finite genus to belong the class O_<AD> in terms of the conjugate operator of the space of harmonic differentials.Next, let Γ be a properly discontinuous group of conformal automorphisms of a Riemann surface R. We introduce a quasi order in a class of simply connected subdomains of R ; the quasi order is defined in terms of hyperbolic metrics on the simply connected subdomains. We show that the class has a unique maximal element. The maximum D is a locally finite fundamental domain for Γ. If R is simply connected, then any connected component of the boundary of D is piecewise analytic simple arc or curve. Furthermore, our fundamental domains are different from Dirichlet or Ford fundamental domains. We thus obtain a new natural method to construct a fundamental domain for Γ.