Something like Picard for 1-forms

Recomm. for undergrads: no
Posted by: MathOMan
on: February 26th, 2010

\begin{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 $D$. \end{conjecture}

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.