Circularizable domains on Riemann surfaces (Q2574392)

Let \(R\) be a Riemann surface and \(\Gamma\) a properly discontinuous group of conformal automorphisms of \(R\) such that every nontrivial element of \(\Gamma\) has no fixed points in \(R\). Suppose that \(R / \Gamma\) is conformally equivalent to neither the plane \(\mathbb C\) nor the sphere \(\widehat{\mathbb C}\). For \(p \in R\) let \(\mathcal D_p^{\Gamma} (R)\) denote the class of simply connected domains \(D\) on \(R\) containing \(p\) such that \(\gamma (D) \cap D = \emptyset\) for any \(\gamma \in \Gamma \setminus \{ \text{id}_R \} \) Then every \(D \in \mathcal D_p^{\Gamma} (R)\) carries a unique complete conformal metric \(ds_D\) with curvature \(-1\). In the previous paper [Q. J. Math. 53, 337--346 (2002; Zbl 1021.30045)] the authors considered the function \(D \to (ds_R / ds_D) (p)\) on \({\mathcal D}_p^{\Gamma}(R)\), where \(ds_R\) denotes a complete conformal metric on \(R\) with constant curvature, and proved that \({\mathcal D}_p^{\Gamma} (R)\) contains a unique element that maximize the function. Such an element is called a hyperbolically maximal domain for \(\Gamma\). A domain \(D\) on \(R\) is said to be a \(\Gamma\)-circularizable if there is a \(\Gamma\)-invariant meromorphic quadratic differential \(\varphi\) with poles in \(\bigcup_{\gamma \in \Gamma} \gamma(D)\) such that every noncritical point in \(D\) lies on a closed horizontal trajectory of \(\varphi\) that stays entirely in \(D\). The authors show that a circularizable domain \(D\) satisfies \(\gamma (D) \cap D = \emptyset\) for \(\gamma \in \Gamma \setminus \{ \text{id}_R \}\) provided \(D\) contains no fixed points of nontrivial elements of \(\Gamma\). Properties of boundaries of circularizable domains are investigated. Finally, the authors give a necessary and sufficient condition for a \(\Gamma\)-circularizable domain to be hyperbolically maximal for \(\Gamma\).