Extremal problems and holomorphic mappings of Riemann surfaces

设R为黎曼曲面，Γ为R的一组适当不连续作用于R的共形自同构，通过求解一些极值问题，得到了Γ在R上的双曲极大定义域。它们与R上的Γ-invariant亚纯二次微分密切相关，这一事实将我们引向R上的一类新的定义域，即Γ-circularizable定义域。这些域有许多有趣的性质。如果D是Γ-cirucularizable域，那么它是单连通的。边界∂D是一个分支多边形，除非它缩减为一个点。事实上，∂D由水平轨迹和r上某个Γ-invariant亚纯二次微分的临界点组成。如果D不紧，则对于D中的某个点p，在整个群的p稳定子Γ下，域D是精确不变的。现在，Γ的每个双曲极大定义域都是Γ-circularizable。双曲极大域的许多有趣性质来自于它们的可循环性。给出了Γ域双曲极大的一个充要条件。然后对上述极值问题进行推广，得到广义双曲极大域。结果表明，这些广义的定义域比原来的定义域更自然地与可循环的定义域联系在一起。

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.