![](/files/happy5.png)
Something like Picard for 1-forms
Conjecture Let
be the open unit disk in the complex plane and let
be open sets such that
. Suppose there are injective holomorphic functions
such that for the differentials we have
on any intersection
. Then those differentials glue together to a meromorphic 1-form on
.
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ U_1,\dots,U_n $](/files/tex/1606100587fc5cb83e059d2991dd76eb805a6775.png)
![$ \bigcup_{j=1}^nU_j=D\setminus\{0\} $](/files/tex/8afae3051dcee56cf63468daeee944bb7f748d37.png)
![$ f_j : U_j \to \mathbb{C}, $](/files/tex/5ea158ab05a44d0e9936418b4a1d5d02f3dce48a.png)
![$ j=1,\ldots,n, $](/files/tex/b38f1c2a1f2d3d469c982a068d7c857016aacf7d.png)
![$ {\rm d}f_j={\rm d}f_k $](/files/tex/5d35704b0e0ac44eb5c78d5ba2aeb3412fe4ebf1.png)
![$ U_j\cap U_k $](/files/tex/1695a8119c26f14a983381dfe368313a8419a1f0.png)
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
It is an evidence that the 1-form is holomorphic on . In the case that its residue at the origin vanishes we can use Picard's big theorem.
Bibliography
*B. Elsner: Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303–331
* indicates original appearance(s) of problem.