Extremal problems and holomorphic mappings of Riemann surfaces

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Let R be a Riemann surface and let Γ be a group of conformal automorphisms of R acting properly discontinuously on R. Hyperbolically maximal domains on R for Γ are obtained by solving some extremal problems. They are closely related to Γ-invariant meromorphic quadratic differentials on R. This fact leads us to a new class of domains on R, that of Γ-circularizable domains. These domains have many interesting properties. If D is a Γ-cirucularizable domain, then it is simply connected. The boundary ∂D is a branched polygon unless it reduces to a point. In fact, ∂D consists of horizontal trajectories and critical points of some Γ-invariant meromorphic quadratic differential on R. If D is not compact, then for some point p in D the domain D is precisely invariant under the stabilizer of p in the whole group Γ.Now, every hyperbolically maximal domain for Γ is Γ-circularizable. Many interesting properties of hyperbolically maximal domains come from their circularizability. We give a necessary and sufficient condition for a Γ-circularizable domain to be hyperbolically maximal for Γ.Next, we generalize the above mentioned extremal problems and obtain generalized hyperbolically maximal domains. These generalized domains turns out to be more naturally related to circularizable domains than the original ones.