Jenkins - Strebel differentials
In this largely expository paper, the authors revisit a fundamental result of Strebel. Theorem (Strebel). Let \(S=\overline S\setminus\{y_1,\dots,y_m\}\) be a Riemann surface of finite type and of genus \(g\). Let \(x_1,\dots,x_n\in S\). Assume that \(\chi(S\setminus\{x_1,\dots,x_n\})<0\). Let \(a_1,\dots,a_n\) be positive real numbers. Then there exists a unique meromorphic Jenkins-Strebel quadratic differential \(\omega\) on \(S\) having the following properties: i) \(\omega\) is holomorphic on \(S\setminus\{x_1,\dots,x_n\}\) and has poles of order two at the points \(x_i\), with the residues \(\left(a_i/2\pi i\right)^2\), \(i=1,\dots,n\). ii) If \(\Gamma\) is the critical graph of \(\omega\), then \(\overline S \setminus\Gamma\) is the union of disjoint disks \(\Delta_1,\dots,\Delta_n\) with \(x_i\in\Delta_i\), \(i=1,\dots,n\). Given a meromorphic Jenkins-Strebel differential \(\omega\), the vertices of its critical graph \(\,\Gamma\) are its critical points. These are the zeros and the simple poles of \(\omega\) in \(\overline S\). Moreover, a vertex \(p\) of \(\Gamma\) has the valency \(\nu\) (\(\geq1\)) if and only if \(\nu=\text{ord}_p\,\omega-2\). Finally, \(\text{ord}_p\,\omega\geq 1\) for \(i=1,\dots,m\), and the only vertices of valency \(\leq 2\) are among the points \(y_1,\dots,y_m\). The authors give a self-contained and somewhat shortened proof of Strebel's result by using ideas already partly contained in the paper [\textit{F. Natoli,} Il grafo di Strebel, Tesi di Laurea, Università di Pisa, Pisa 2000].
scientific article; zbMATH DE number 5714228