Isolated singularities of quadratic differentials arising from a module problem

Abstract:

If $R \subset S$ are Riemann surfaces, we will say that ${z_0} \in S - R$ is an isolated point boundary component of $R$ if there exists a neighborhood $U$ of ${z_0}$ in $S$ such that $U - \{ {z_0}\} \subset R$. We prove that the quadratic differential $Q\left ( z \right )d{z^2}$ obtained by solving the module problem $P({a_1}, \ldots ,{a_k})$ applied to a free family of homotopy classes on $R$ can be extended to ${z_0} \in S$ so that either $Q\left ( z \right )$ is regular at ${z_0}$ or $Q(z)$ has a simple pole at ${z_0}$.