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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Isolated singularities of quadratic differentials arising from a module problem


Author: Jeffrey Clayton Wiener
Journal: Proc. Amer. Math. Soc. 55 (1976), 47-51
DOI: https://doi.org/10.1090/S0002-9939-1976-0393465-1
MathSciNet review: 0393465
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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}$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0393465-1
Keywords: Homotopy class, point cycle, module, module problem, quadratic differential, extremal metric, canonical exhaustion, Huber module
Article copyright: © Copyright 1976 American Mathematical Society