Something like Picard for 1-forms

Recomm. for undergrads: no
Posted by: MathOMan
on: February 26th, 2010
Conjecture   Let $ D $ be the open unit disk in the complex plane and let $ U_1,\dots,U_n $ be open sets such that $ \bigcup_{j=1}^nU_j=D\setminus\{0\} $. Suppose there are injective holomorphic functions $ f_j : U_j \to \mathbb{C}, $ $ j=1,\ldots,n, $ such that for the differentials we have $ {\rm d}f_j={\rm d}f_k $ on any intersection $ U_j\cap U_k $. Then those differentials glue together to a meromorphic 1-form on $ D $.

It is an evidence that the 1-form is holomorphic on $ D\setminus\{0\} $. In the case that its residue at the origin vanishes we can use Picard's big theorem.


*B. Elsner: Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303–331

* indicates original appearance(s) of problem.